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Timofeev Chapter 9 Integration Problems

fricas
-- Problems 1 - 6 (p. 391-392)
integrate(acos(x)^2/x^5, x)

\label{eq1}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({4 \ {{x}^{3}}}+{2 \  x}\right)}\ {\arccos \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{4 \ {{x}^{4}}\ {\log \left({x}\right)}}- 
\
\
\displaystyle
{3 \ {{\arccos \left({x}\right)}^{2}}}-{{x}^{2}}
(1)
Type: Union(Expression(Integer),...)
fricas
integrate(x^2*asin(x)^2, x)

\label{eq2}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({6 \ {{x}^{2}}}+{12}\right)}\ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{9 \ {{x}^{3}}\ {{\arcsin \left({x}\right)}^{2}}}- 
\
\
\displaystyle
{2 \ {{x}^{3}}}-{{12}\  x}
(2)
Type: Union(Expression(Integer),...)
fricas
integrate(x^3*atan(x)^2, x)

\label{eq3}{\left(
\begin{array}{@{}l}
\displaystyle
-{4 \ {\log \left({{{x}^{2}}+ 1}\right)}}+{{\left({3 \ {{x}^{4}}}- 3 \right)}\ {{\arctan \left({x}\right)}^{2}}}+ 
\
\
\displaystyle
{{\left(-{2 \ {{x}^{3}}}+{6 \  x}\right)}\ {\arctan \left({x}\right)}}+{{x}^{2}}
(3)
Type: Union(Expression(Integer),...)
fricas
integrate(atan(x)^2/x^5, x)

\label{eq4}{\left(
\begin{array}{@{}l}
\displaystyle
{4 \ {{x}^{4}}\ {\log \left({{{x}^{2}}+ 1}\right)}}-{8 \ {{x}^{4}}\ {\log \left({x}\right)}}+ 
\
\
\displaystyle
{{\left({3 \ {{x}^{4}}}- 3 \right)}\ {{\arctan \left({x}\right)}^{2}}}+{{\left({6 \ {{x}^{3}}}-{2 \  x}\right)}\ {\arctan \left({x}\right)}}-{{x}^{2}}
(4)
Type: Union(Expression(Integer),...)
fricas
integrate(x^3*acsc(x)^2, x)

\label{eq5}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({2 \ {{x}^{2}}}+ 4 \right)}\ {acsc \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+{4 \ {\log \left({x}\right)}}+ 
\
\
\displaystyle
{3 \ {{x}^{4}}\ {{acsc \left({x}\right)}^{2}}}+{{x}^{2}}
(5)
Type: Union(Expression(Integer),...)
fricas
integrate(asec(x)^4/x^5, x)

\label{eq6}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{\left({{48}\ {{x}^{2}}}+{32}\right)}\ {{asec \left({x}\right)}^{3}}}+ 
\
\
\displaystyle
{{\left(-{{90}\ {{x}^{2}}}-{12}\right)}\ {asec \left({x}\right)}}
(6)
Type: Union(Expression(Integer),...)
fricas
-- Problems 7 - 18 (p. 397-398)
integrate(asin(x)*sqrt(1-x^2), x)

\label{eq7}{{2 \  x \ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{{\arcsin \left({x}\right)}^{2}}-{{x}^{2}}}\over 4(7)
Type: Union(Expression(Integer),...)
fricas
integrate(acos(x)*sqrt(1-x^2), x)

\label{eq8}{{2 \  x \ {\arccos \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}-{{\arccos \left({x}\right)}^{2}}+{{x}^{2}}}\over 4(8)
Type: Union(Expression(Integer),...)
fricas
integrate(x*acos(x)*sqrt(1-x^2), x)

\label{eq9}{{{\left({3 \ {{x}^{2}}}- 3 \right)}\ {\arccos \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{{x}^{3}}-{3 \  x}}\over 9(9)
Type: Union(Expression(Integer),...)
fricas
integrate((1-x^2)^(3/2)*asin(x), x)

\label{eq10}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left(-{4 \ {{x}^{3}}}+{{10}\  x}\right)}\ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+ 
\
\
\displaystyle
{3 \ {{\arcsin \left({x}\right)}^{2}}}+{{x}^{4}}-{5 \ {{x}^{2}}}
(10)
Type: Union(Expression(Integer),...)
fricas
integrate(x*(1-x^2)^(3/2)*asin(x), x)

\label{eq11}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left(-{{15}\ {{x}^{4}}}+{{30}\ {{x}^{2}}}-{15}\right)}\ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{3 \ {{x}^{5}}}- 
\
\
\displaystyle
{{10}\ {{x}^{3}}}+{{15}\  x}
(11)
Type: Union(Expression(Integer),...)
fricas
integrate(x^3*(1-x^2)^(3/2)*acos(x), x)

\label{eq12}{\left(
\begin{array}{@{}l}
\displaystyle
{
\begin{array}{@{}l}
\displaystyle
{\left(-{{525}\ {{x}^{6}}}+{{840}\ {{x}^{4}}}-{{105}\ {{x}^{2}}}-{210}\right)}\ {\arccos \left({x}\right)}\  \cdot 
\
\
\displaystyle
{\sqrt{-{{x}^{2}}+ 1}}
(12)
Type: Union(Expression(Integer),...)
fricas
integrate((1-x^2)^(3/2)*acos(x)/x, x)

\label{eq13}\int^{
\displaystyle
x}{{{{\left(-{{\%A}^{2}}+ 1 \right)}\ {\arccos \left({\%A}\right)}\ {\sqrt{-{{\%A}^{2}}+ 1}}}\over \%A}\ {d \%A}}(13)
Type: Union(Expression(Integer),...)
fricas
integrate((1-x^2)^(3/2)*asin(x)/x^6, x)

\label{eq14}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left(-{4 \ {{x}^{4}}}+{8 \ {{x}^{2}}}- 4 \right)}\ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+ 
\
\
\displaystyle
{4 \ {{x}^{5}}\ {\log \left({x}\right)}}+{4 \ {{x}^{3}}}- x 
(14)
Type: Union(Expression(Integer),...)
fricas
integrate(x^2*asin(x)/sqrt(1-x^2), x)

\label{eq15}{-{2 \  x \ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{{\arcsin \left({x}\right)}^{2}}+{{x}^{2}}}\over 4(15)
Type: Union(Expression(Integer),...)
fricas
integrate(x^4*asin(x)/sqrt(1-x^2), x)

\label{eq16}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left(-{4 \ {{x}^{3}}}-{6 \  x}\right)}\ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{3 \ {{\arcsin \left({x}\right)}^{2}}}+ 
\
\
\displaystyle
{{x}^{4}}+{3 \ {{x}^{2}}}
(16)
Type: Union(Expression(Integer),...)
fricas
integrate(x*asin(x)/(1-x^2)^(3/2), x)

\label{eq17}{\left(
\begin{array}{@{}l}
\displaystyle
-{2 \ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{{\left(-{{x}^{2}}+ 1 \right)}\ {\log \left({x + 1}\right)}}+ 
\
\
\displaystyle
{{\left({{x}^{2}}- 1 \right)}\ {\log \left({x - 1}\right)}}
(17)
Type: Union(Expression(Integer),...)
fricas
integrate(x*acos(x)/(1-x^2)^(3/2), x)

\label{eq18}{\left(
\begin{array}{@{}l}
\displaystyle
-{2 \ {\arccos \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{{\left({{x}^{2}}- 1 \right)}\ {\log \left({x + 1}\right)}}+ 
\
\
\displaystyle
{{\left(-{{x}^{2}}+ 1 \right)}\ {\log \left({x - 1}\right)}}
(18)
Type: Union(Expression(Integer),...)
fricas
integrate(asin(x)/(1-x^2)^(5/2), x)

\label{eq19}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left(-{4 \ {{x}^{3}}}+{6 \  x}\right)}\ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+ 
\
\
\displaystyle
{{\left({2 \ {{x}^{4}}}-{4 \ {{x}^{2}}}+ 2 \right)}\ {\log \left({{{x}^{2}}- 1}\right)}}+{{x}^{2}}- 1 
(19)
Type: Union(Expression(Integer),...)
fricas
integrate(x^3*asin(x)/(1-x^2)^(3/2), x)

\label{eq20}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({2 \ {{x}^{2}}}- 4 \right)}\ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+ 
\
\
\displaystyle
{{\left(-{{x}^{2}}+ 1 \right)}\ {\log \left({x + 1}\right)}}+{{\left({{x}^{2}}- 1 \right)}\ {\log \left({x - 1}\right)}}-{2 \ {{x}^{3}}}+ 
\
\
\displaystyle
{2 \  x}
(20)
Type: Union(Expression(Integer),...)
fricas
-- Problems 19 - 22 (p. 401)
integrate(asin(x)/(x*(1-x^2)^(3/2)), x)

\label{eq21}\int^{
\displaystyle
x}-{{{\arcsin \left({\%A}\right)}\over{{\left({{\%A}^{3}}- \%A \right)}\ {\sqrt{-{{\%A}^{2}}+ 1}}}}\ {d \%A}}(21)
Type: Union(Expression(Integer),...)
fricas
integrate(acos(x)/(x^4*sqrt(1-x^2)), x)

\label{eq22}{{{\left(-{4 \ {{x}^{2}}}- 2 \right)}\ {\arccos \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}-{4 \ {{x}^{3}}\ {\log \left({x}\right)}}+ x}\over{6 \ {{x}^{3}}}(22)
Type: Union(Expression(Integer),...)
fricas
integrate(x*acos(x)^2*sqrt(1-x^2), x)

\label{eq23}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({{\left({9 \ {{x}^{2}}}- 9 \right)}\ {{\arccos \left({x}\right)}^{2}}}-{2 \ {{x}^{2}}}+{14}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+ \
\
\displaystyle
{{\left({6 \ {{x}^{3}}}-{{18}\  x}\right)}\ {\arccos \left({x}\right)}}
(23)
Type: Union(Expression(Integer),...)
fricas
integrate(x^2*asin(x)^3/sqrt(1-x^2), x)

\label{eq24}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left(-{4 \  x \ {{\arcsin \left({x}\right)}^{3}}}+{6 \  x \ {\arcsin \left({x}\right)}}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+ 
\
\
\displaystyle
{{\arcsin \left({x}\right)}^{4}}+{{\left({6 \ {{x}^{2}}}- 3 \right)}\ {{\arcsin \left({x}\right)}^{2}}}-{3 \ {{x}^{2}}}
(24)
Type: Union(Expression(Integer),...)
fricas
-- Problems 23 - 26 (p. 404-405)
integrate(x*atan(x)/(1+x^2)^2, x)

\label{eq25}{{{\left({{x}^{2}}- 1 \right)}\ {\arctan \left({x}\right)}}+ x}\over{{4 \ {{x}^{2}}}+ 4}(25)
Type: Union(Expression(Integer),...)
fricas
integrate(x*atan(x)/(1+x^2)^3, x)

\label{eq26}{{{\left({3 \ {{x}^{4}}}+{6 \ {{x}^{2}}}- 5 \right)}\ {\arctan \left({x}\right)}}+{3 \ {{x}^{3}}}+{5 \  x}}\over{{{32}\ {{x}^{4}}}+{{64}\ {{x}^{2}}}+{32}}(26)
Type: Union(Expression(Integer),...)
fricas
integrate(x^2*atan(x)/(1+x^2), x)

\label{eq27}{-{\log \left({{{x}^{2}}+ 1}\right)}-{{\arctan \left({x}\right)}^{2}}+{2 \  x \ {\arctan \left({x}\right)}}}\over 2(27)
Type: Union(Expression(Integer),...)
fricas
integrate(x^3*atan(x)/(1+x^2), x)

\label{eq28}\int^{
\displaystyle
x}{{{{{\%A}^{3}}\ {\arctan \left({\%A}\right)}}\over{{{\%A}^{2}}+ 1}}\ {d \%A}}(28)
Type: Union(Expression(Integer),...)
fricas
-- Problems 27 - 32 (p. 407-408)
integrate(x^2*atan(x)/(1+x^2)^2, x)

\label{eq29}{{{\left({{x}^{2}}+ 1 \right)}\ {{\arctan \left({x}\right)}^{2}}}-{2 \  x \ {\arctan \left({x}\right)}}- 1}\over{{4 \ {{x}^{2}}}+ 4}(29)
Type: Union(Expression(Integer),...)
fricas
integrate(x^3*atan(x)/(1+x^2)^2, x)

\label{eq30}\int^{
\displaystyle
x}{{{{{\%A}^{3}}\ {\arctan \left({\%A}\right)}}\over{{{\%A}^{4}}+{2 \ {{\%A}^{2}}}+ 1}}\ {d \%A}}(30)
Type: Union(Expression(Integer),...)
fricas
integrate(x^5*atan(x)/(1+x^2)^2, x)

\label{eq31}\int^{
\displaystyle
x}{{{{{\%A}^{5}}\ {\arctan \left({\%A}\right)}}\over{{{\%A}^{4}}+{2 \ {{\%A}^{2}}}+ 1}}\ {d \%A}}(31)
Type: Union(Expression(Integer),...)
fricas
integrate((1+x^2)*atan(x)/x^2, x)

\label{eq32}{-{x \ {\log \left({{{x}^{2}}+ 1}\right)}}+{x \ {\log \left({x}\right)}}+{{\left({{x}^{2}}- 1 \right)}\ {\arctan \left({x}\right)}}}\over x(32)
Type: Union(Expression(Integer),...)
fricas
integrate((1+x^2)*atan(x)/x^5, x)

\label{eq33}{{{\left(-{3 \ {{x}^{4}}}-{6 \ {{x}^{2}}}- 3 \right)}\ {\arctan \left({x}\right)}}-{3 \ {{x}^{3}}}- x}\over{{12}\ {{x}^{4}}}(33)
Type: Union(Expression(Integer),...)
fricas
integrate((1+x^2)^2*atan(x)/x^5, x)

\label{eq34}\int^{
\displaystyle
x}{{{{\left({{\%A}^{4}}+{2 \ {{\%A}^{2}}}+ 1 \right)}\ {\arctan \left({\%A}\right)}}\over{{\%A}^{5}}}\ {d \%A}}(34)
Type: Union(Expression(Integer),...)
fricas
-- Problems 33 - 36 (p. 409)
integrate(atan(x)/(x^2*(1+x^2)), x)

\label{eq35}{-{x \ {\log \left({{{x}^{2}}+ 1}\right)}}+{2 \  x \ {\log \left({x}\right)}}-{x \ {{\arctan \left({x}\right)}^{2}}}-{2 \ {\arctan \left({x}\right)}}}\over{2 \  x}(35)
Type: Union(Expression(Integer),...)
fricas
integrate(atan(x)^2/x^3, x)

\label{eq36}{\left(
\begin{array}{@{}l}
\displaystyle
-{{{x}^{2}}\ {\log \left({{{x}^{2}}+ 1}\right)}}+{2 \ {{x}^{2}}\ {\log \left({x}\right)}}+ 
\
\
\displaystyle
{{\left(-{{x}^{2}}- 1 \right)}\ {{\arctan \left({x}\right)}^{2}}}-{2 \  x \ {\arctan \left({x}\right)}}
(36)
Type: Union(Expression(Integer),...)
fricas
integrate((1+x^2)*atan(x)^2/x^5, x)

\label{eq37}{\left(
\begin{array}{@{}l}
\displaystyle
-{2 \ {{x}^{4}}\ {\log \left({{{x}^{2}}+ 1}\right)}}+{4 \ {{x}^{4}}\ {\log \left({x}\right)}}+ 
\
\
\displaystyle
{{\left(-{3 \ {{x}^{4}}}-{6 \ {{x}^{2}}}- 3 \right)}\ {{\arctan \left({x}\right)}^{2}}}+ 
\
\
\displaystyle
{{\left(-{6 \ {{x}^{3}}}-{2 \  x}\right)}\ {\arctan \left({x}\right)}}-{{x}^{2}}
(37)
Type: Union(Expression(Integer),...)
fricas
integrate(x^3*atan(x)^2/(1+x^2)^3, x)

\label{eq38}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({5 \ {{x}^{4}}}-{6 \ {{x}^{2}}}- 3 \right)}\ {{\arctan \left({x}\right)}^{2}}}+ 
\
\
\displaystyle
{{\left({{10}\ {{x}^{3}}}+{6 \  x}\right)}\ {\arctan \left({x}\right)}}+{5 \ {{x}^{2}}}+ 4 
(38)
Type: Union(Expression(Integer),...)
fricas
-- Problems 37 - 43 (p. 412-414)
integrate(asec(x)*sqrt(-1+x^2)/x^2, x)

\label{eq39}\int^{
\displaystyle
x}{{{{asec \left({\%A}\right)}\ {\sqrt{{{\%A}^{2}}- 1}}}\over{{\%A}^{2}}}\ {d \%A}}(39)
Type: Union(Expression(Integer),...)
fricas
integrate((-1+x^2)^(5/2)*acsc(x)/x^3, x)

\label{eq40}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({4 \ {{x}^{4}}}-{{28}\ {{x}^{2}}}- 6 \right)}\ {acsc \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}-{{28}\ {{x}^{2}}\ {\log \left({x}\right)}}- 
\
\
\displaystyle
{{15}\ {{x}^{2}}\ {{acsc \left({x}\right)}^{2}}}+{2 \ {{x}^{4}}}+ 3 
(40)
Type: Union(Expression(Integer),...)
fricas
integrate(asec(x)*sqrt(-1+x^2)/x^4, x)

\label{eq41}{{{\left({3 \ {{x}^{2}}}- 3 \right)}\ {asec \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+{3 \ {{x}^{2}}}- 1}\over{9 \ {{x}^{3}}}(41)
Type: Union(Expression(Integer),...)
fricas
integrate(asec(x)/(-1+x^2)^(5/2), x)

\label{eq42}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({8 \ {{x}^{3}}}-{{12}\  x}\right)}\ {asec \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+ 
\
\
\displaystyle
{{\left({5 \ {{x}^{4}}}-{{10}\ {{x}^{2}}}+ 5 \right)}\ {\log \left({x + 1}\right)}}+ 
\
\
\displaystyle
{{\left(-{5 \ {{x}^{4}}}+{{10}\ {{x}^{2}}}- 5 \right)}\ {\log \left({x - 1}\right)}}-{2 \ {{x}^{3}}}+{2 \  x}
(42)
Type: Union(Expression(Integer),...)
fricas
integrate(x^2*asec(x)/(-1+x^2)^(5/2), x)

\label{eq43}{\left(
\begin{array}{@{}l}
\displaystyle
-{4 \ {{x}^{3}}\ {asec \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+ 
\
\
\displaystyle
{{\left(-{{x}^{4}}+{2 \ {{x}^{2}}}- 1 \right)}\ {\log \left({x + 1}\right)}}+ 
\
\
\displaystyle
{{\left({{x}^{4}}-{2 \ {{x}^{2}}}+ 1 \right)}\ {\log \left({x - 1}\right)}}-{2 \ {{x}^{3}}}+{2 \  x}
(43)
Type: Union(Expression(Integer),...)
fricas
integrate(x^3*asec(x)/(-1+x^2)^(5/2), x)

\label{eq44}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left(-{6 \ {{x}^{2}}}+ 4 \right)}\ {asec \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+ 
\
\
\displaystyle
{{\left({2 \ {{x}^{4}}}-{4 \ {{x}^{2}}}+ 2 \right)}\ {\log \left({{{x}^{2}}- 1}\right)}}+ 
\
\
\displaystyle
{{\left(-{4 \ {{x}^{4}}}+{8 \ {{x}^{2}}}- 4 \right)}\ {\log \left({x}\right)}}-{{x}^{2}}+ 1 
(44)
Type: Union(Expression(Integer),...)
fricas
integrate(x^6*asec(x)/(-1+x^2)^(5/2), x)

\label{eq45}\int^{
\displaystyle
x}{{{{{\%A}^{6}}\ {asec \left({\%A}\right)}}\over{{\left({{\%A}^{4}}-{2 \ {{\%A}^{2}}}+ 1 \right)}\ {\sqrt{{{\%A}^{2}}- 1}}}}\ {d \%A}}(45)
Type: Union(Expression(Integer),...)
fricas
-- Problems 44 - 48 (p. 416-417)
integrate(asec(x)/(x^2*sqrt(-1+x^2)), x)

\label{eq46}{{{asec \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+ 1}\over x(46)
Type: Union(Expression(Integer),...)
fricas
integrate(acsc(x)/(x^2*(-1+x^2)^(5/2)), x)

\label{eq47}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({{32}\ {{x}^{4}}}-{{48}\ {{x}^{2}}}+{12}\right)}\ {acsc \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+ 
\
\
\displaystyle
{{\left(-{{11}\ {{x}^{5}}}+{{22}\ {{x}^{3}}}-{{11}\  x}\right)}\ {\log \left({x + 1}\right)}}+ 
\
\
\displaystyle
{{\left({{11}\ {{x}^{5}}}-{{22}\ {{x}^{3}}}+{{11}\  x}\right)}\ {\log \left({x - 1}\right)}}-{{10}\ {{x}^{4}}}+{{22}\ {{x}^{2}}}- 
\
\
\displaystyle
{12}
(47)
Type: Union(Expression(Integer),...)
fricas
integrate(acsc(x)^4/(x^2*sqrt(-1+x^2)), x)

\label{eq48}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({{acsc \left({x}\right)}^{4}}-{{12}\ {{acsc \left({x}\right)}^{2}}}+{24}\right)}\ {\sqrt{{{x}^{2}}- 1}}}-{4 \ {{acsc \left({x}\right)}^{3}}}+ 
\
\
\displaystyle
{{24}\ {acsc \left({x}\right)}}
(48)
Type: Union(Expression(Integer),...)
fricas
integrate((-1+x^2)^(3/2)*asec(x)^2/x^5, x)

\label{eq49}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({{\left(-{{40}\ {{x}^{2}}}+{16}\right)}\ {{asec \left({x}\right)}^{2}}}+{{17}\ {{x}^{2}}}- 2 \right)}\ {\sqrt{{{x}^{2}}- 1}}}+ \
\
\displaystyle
{8 \ {{x}^{4}}\ {{asec \left({x}\right)}^{3}}}+{{\left({{17}\ {{x}^{4}}}-{{40}\ {{x}^{2}}}+ 8 \right)}\ {asec \left({x}\right)}}
(49)
Type: Union(Expression(Integer),...)
fricas
integrate(asec(x)^3*sqrt(-1+x^2)/x^4, x)

\label{eq50}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{\left({9 \ {{x}^{2}}}- 9 \right)}\ {{asec \left({x}\right)}^{3}}}+ 
\
\
\displaystyle
{{\left(-{{42}\ {{x}^{2}}}+ 6 \right)}\ {asec \left({x}\right)}}
(50)
Type: Union(Expression(Integer),...)
fricas
-- Problems 49 - 56 (p. 422)
integrate(asin(sqrt((-a+x)/(a+x))), x)

\label{eq51}{{\left(x + a \right)}\ {\arcsin \left({\sqrt{{x - a}\over{x + a}}}\right)}}+{{\left(- x - a \right)}\ {\sqrt{{2 \  a}\over{x + a}}}\ {\sqrt{{x - a}\over{x + a}}}}(51)
Type: Union(Expression(Integer),...)
fricas
integrate(atan(sqrt((-a+x)/(a+x))), x)

\label{eq52}{\left(
\begin{array}{@{}l}
\displaystyle
-{a \ {\log \left({{\sqrt{{x - a}\over{x + a}}}+ 1}\right)}}+{a \ {\log \left({{\sqrt{{x - a}\over{x + a}}}- 1}\right)}}+ \
\
\displaystyle
{2 \  x \ {\arctan \left({\sqrt{{x - a}\over{x + a}}}\right)}}
(52)
Type: Union(Expression(Integer),...)
fricas
integrate(atan(x)/(1+x)^3, x)

\label{eq53}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left(-{{x}^{2}}-{2 \  x}- 1 \right)}\ {\log \left({{{x}^{2}}+ 1}\right)}}+ 
\
\
\displaystyle
{{\left({2 \ {{x}^{2}}}+{4 \  x}+ 2 \right)}\ {\log \left({x + 1}\right)}}-{4 \ {\arctan \left({x}\right)}}-{2 \  x}- 2 
(53)
Type: Union(Expression(Integer),...)
fricas
integrate(atan(-a+x)/(a+x), x)

\label{eq54}\int^{
\displaystyle
x}-{{{\arctan \left({a - \%A}\right)}\over{a + \%A}}\ {d \%A}}(54)
Type: Union(Expression(Integer),...)
fricas
integrate(asin(sqrt(1-x^2))/sqrt(1-x^2), x)

\label{eq55}-{{{\arcsin \left({\sqrt{-{{x}^{2}}+ 1}}\right)}^{2}}\over 2}(55)
Type: Union(Expression(Integer),...)
fricas
integrate(x*atan(sqrt(1+x^2))/sqrt(1+x^2), x)

\label{eq56}{{2 \ {\sqrt{{{x}^{2}}+ 1}}\ {\arctan \left({\sqrt{{{x}^{2}}+ 1}}\right)}}-{\log \left({{{x}^{2}}+ 2}\right)}}\over 2(56)
Type: Union(Expression(Integer),...)
fricas
integrate(asin(x)/(1-x)^(5/2), x)

\label{eq57}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({{x}^{2}}-{2 \  x}+ 1 \right)}\ {\log{\left({{-{4 \ {\sqrt{-{{x}^{2}}+ 1}}\ {\sqrt{- x + 1}}}+{{\left(-{{x}^{2}}-{2 \  x}+ 3 \right)}\ {\sqrt{2}}}}\over{{{x}^{2}}-{2 \  x}+ 1}}\right)}}}+ 
\
\
\displaystyle
{{\left(-{2 \ {\sqrt{2}}\ {\sqrt{-{{x}^{2}}+ 1}}}+{4 \ {\sqrt{2}}\ {\arcsin \left({x}\right)}}\right)}\ {\sqrt{- x + 1}}}
(57)
Type: Union(Expression(Integer),...)
fricas
integrate((-1+x)^(5/2)*acsc(x), x)

\label{eq58}{\left(
\begin{array}{@{}l}
\displaystyle
{{30}\ {\sqrt{x - 1}}\ {\sqrt{{{x}^{2}}- 1}}\ {\log \left({{{{\sqrt{x - 1}}\ {\sqrt{{{x}^{2}}- 1}}}+{{x}^{2}}- 1}\over{{{x}^{2}}- 1}}\right)}}- 
\
\
\displaystyle
{{30}\ {\sqrt{x - 1}}\ {\sqrt{{{x}^{2}}- 1}}\ {\log \left({{{{\sqrt{x - 1}}\ {\sqrt{{{x}^{2}}- 1}}}-{{x}^{2}}+ 1}\over{{{x}^{2}}- 1}}\right)}}+ 
\
\
\displaystyle
{{\left({{30}\ {{x}^{4}}}-{{120}\ {{x}^{3}}}+{{180}\ {{x}^{2}}}-{{120}\  x}+{30}\right)}\ {acsc \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+ 
\
\
\displaystyle
{{12}\ {{x}^{4}}}-{{76}\ {{x}^{3}}}+{{320}\ {{x}^{2}}}+{{76}\  x}-{332}
(58)
Type: Union(Expression(Integer),...)
fricas
-- Problems 57 - 59 (p. 427)
fricas
)set output tex off
 
fricas
)set output algebra on
integrate(asin(sinh(x))*sech(x)^4, x)
(59) 6 5 2 4 - 4sinh(x) - 24cosh(x)sinh(x) + (- 60cosh(x) - 12)sinh(x) + 3 3 (- 80cosh(x) - 48cosh(x))sinh(x) + 4 2 2 (- 60cosh(x) - 72cosh(x) - 12)sinh(x) + 5 3 6 (- 24cosh(x) - 48cosh(x) - 24cosh(x))sinh(x) - 4cosh(x) + 4 2 - 12cosh(x) - 12cosh(x) - 4 * +-------------------------------------+ | 2 2 | - 2sinh(x) - 2cosh(x) + 6 |------------------------------------- | 2 2 \|sinh(x) - 2cosh(x)sinh(x) + cosh(x) atan(-------------------------------------------) 2 2 3sinh(x) + 6cosh(x)sinh(x) + 3cosh(x) - 1 + 2 2 (- 24sinh(x) - 48cosh(x)sinh(x) - 24cosh(x) - 8) * 2 2 sinh(x) + 2cosh(x)sinh(x) + cosh(x) - 1 atan(-----------------------------------------) +-------------------------------------+ | 2 2 | - 2sinh(x) - 2cosh(x) + 6 |------------------------------------- | 2 2 \|sinh(x) - 2cosh(x)sinh(x) + cosh(x) + 4 3 2 2 sinh(x) + 4cosh(x)sinh(x) + (6cosh(x) + 2)sinh(x) + 3 4 2 (4cosh(x) + 4cosh(x))sinh(x) + cosh(x) + 2cosh(x) + 1 * +-------------------------------------+ | 2 2 | - 2sinh(x) - 2cosh(x) + 6 |------------------------------------- | 2 2 \|sinh(x) - 2cosh(x)sinh(x) + cosh(x) / 6 5 2 4 6sinh(x) + 36cosh(x)sinh(x) + (90cosh(x) + 18)sinh(x) + 3 3 (120cosh(x) + 72cosh(x))sinh(x) + 4 2 2 (90cosh(x) + 108cosh(x) + 18)sinh(x) + 5 3 6 4 (36cosh(x) + 72cosh(x) + 36cosh(x))sinh(x) + 6cosh(x) + 18cosh(x) + 2 18cosh(x) + 6
Type: Union(Expression(Integer),...)
fricas
integrate(acot(cosh(x))*cosh(x)/sinh(x)^4, x)
(60) 6 5 2 4 sinh(x) + 6cosh(x)sinh(x) + (15cosh(x) - 3)sinh(x) + 3 3 (20cosh(x) - 12cosh(x))sinh(x) + 4 2 2 (15cosh(x) - 18cosh(x) + 3)sinh(x) + 5 3 6 4 (6cosh(x) - 12cosh(x) + 6cosh(x))sinh(x) + cosh(x) - 3cosh(x) + 2 3cosh(x) - 1 * log +-+ 2 +-+ (9\|2 - 12)sinh(x) + (- 16\|2 + 24)cosh(x)sinh(x) + +-+ 2 +-+ (9\|2 - 12)cosh(x) + 3\|2 - 4 / 2 2 sinh(x) + cosh(x) + 3 + +-+ 3 +-+ 2 +-+ 2 - 32\|2 sinh(x) - 96\|2 cosh(x)sinh(x) - 96\|2 cosh(x) sinh(x) + +-+ 3 - 32\|2 cosh(x) * 2sinh(x) + 2cosh(x) atan(-----------------------------------------) 2 2 sinh(x) + 2cosh(x)sinh(x) + cosh(x) + 1 + +-+ 4 +-+ 3 +-+ 2 +-+ 2 4\|2 sinh(x) + 16\|2 cosh(x)sinh(x) + (24\|2 cosh(x) - 8\|2 )sinh(x) + +-+ 3 +-+ +-+ 4 +-+ 2 (16\|2 cosh(x) - 16\|2 cosh(x))sinh(x) + 4\|2 cosh(x) - 8\|2 cosh(x) + +-+ 4\|2 / +-+ 6 +-+ 5 12\|2 sinh(x) + 72\|2 cosh(x)sinh(x) + +-+ 2 +-+ 4 (180\|2 cosh(x) - 36\|2 )sinh(x) + +-+ 3 +-+ 3 (240\|2 cosh(x) - 144\|2 cosh(x))sinh(x) + +-+ 4 +-+ 2 +-+ 2 (180\|2 cosh(x) - 216\|2 cosh(x) + 36\|2 )sinh(x) + +-+ 5 +-+ 3 +-+ (72\|2 cosh(x) - 144\|2 cosh(x) + 72\|2 cosh(x))sinh(x) + +-+ 6 +-+ 4 +-+ 2 +-+ 12\|2 cosh(x) - 36\|2 cosh(x) + 36\|2 cosh(x) - 12\|2
Type: Union(Expression(Integer),...)
fricas
)set output algebra off
 
fricas
)set output tex on
integrate(%e^x*asin(tanh(x)), x)

\label{eq59}\begin{array}{@{}l}
\displaystyle
-{\log \left({-{{2 \ {\cosh \left({x}\right)}}\over{{\sinh \left({x}\right)}-{\cosh \left({x}\right)}}}}\right)}+ 
\
\
\displaystyle
{{\left({\sinh \left({x}\right)}+{\cosh \left({x}\right)}\right)}\ {\arctan \left({\sinh \left({x}\right)}\right)}}
(59)
Type: Union(Expression(Integer),...)




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