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Examples from Timofeev (1948) Ch. 1, divided into sections kindly provided by clicliclic vel Martin on sci.math.symbolic

fricas

(1) -> INT ==> integrate

Type: Void

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pi ==> %pi

Type: Void

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SIN ==> sin

Type: Void

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COS ==> cos

Type: Void

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TAN ==> tan

Type: Void

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COT ==> cot

Type: Void

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SEC ==> sec

Type: Void

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CSC ==> csc

Type: Void

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LN ==> log

Type: Void

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SQRT ==> sqrt

Type: Void

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ATAN ==> atan

Type: Void

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ACOS ==> acos

Type: Void

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ASIN ==> asin

Type: Void

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-- Timofeev (1948) Ch. 1, examples 1 - 21
INT(1/(a^2-b^2*x^2),x)

$\label{eq1}\frac{{\log \left({{b \ x}+ a}\right)}-{\log \left({{b \ x}- a}\right)}}{2 \ a \ b}$

(1)

Type: Union(Expression(Integer),...)

fricas

INT(1/(a^2+b^2*x^2),x)

$\label{eq2}\frac{\arctan \left({\frac{b \ x}{a}}\right)}{a \ b}$

(2)

Type: Union(Expression(Integer),...)

fricas

INT(SEC(2*a*x),x)

$\label{eq3}\frac{{\log \left({{\sin \left({2 \ a \ x}\right)}+ 1}\right)}-{\log \left({-{\sin \left({2 \ a \ x}\right)}+ 1}\right)}}{4 \ a}$

(3)

Type: Union(Expression(Integer),...)

fricas

INT(1/(4*SIN(x/3)),x)

$\label{eq4}\frac{-{3 \ {\log \left({\frac{{\cos \left({\frac{x}{3}}\right)}+ 1}{2}}\right)}}+{3 \ {\log \left({\frac{-{\cos \left({\frac{x}{3}}\right)}+ 1}{2}}\right)}}}{8}$

(4)

Type: Union(Expression(Integer),...)

fricas

INT(1/COS(3/4*pi-2*x),x)

$\label{eq5}\frac{{\log \left({{\sin \left({\frac{{8 \ x}-{3 \ \pi}}{4}}\right)}+ 1}\right)}-{\log \left({-{\sin \left({\frac{{8 \ x}-{3 \ \pi}}{4}}\right)}+ 1}\right)}}{4}$

(5)

Type: Union(Expression(Integer),...)

fricas

INT(SEC(x)*TAN(x),x)

$\label{eq6}\frac{1}{\cos \left({x}\right)}$

(6)

Type: Union(Expression(Integer),...)

fricas

INT(CSC(x)*COT(x),x)

$\label{eq7}-{\frac{1}{\sin \left({x}\right)}}$

(7)

Type: Union(Expression(Integer),...)

fricas

INT(TAN(x)/SIN(2*x),x)

$\label{eq8}\frac{\tan \left({x}\right)}{2}$

(8)

Type: Union(Expression(Integer),...)

fricas

INT(1/(1+COS(x)),x)

$\label{eq9}\frac{\sin \left({x}\right)}{{\cos \left({x}\right)}+ 1}$

(9)

Type: Union(Expression(Integer),...)

fricas

INT(1/(1-COS(x)),x)

$\label{eq10}\frac{-{\cos \left({x}\right)}- 1}{\sin \left({x}\right)}$

(10)

Type: Union(Expression(Integer),...)

fricas

INT(SIN(x)/(a-b*COS(x)),x)

$\label{eq11}\frac{\log \left({-{b \ {\cos \left({x}\right)}}+ a}\right)}{b}$

(11)

Type: Union(Expression(Integer),...)

fricas

INT(COS(x)/(a^2+b^2*SIN(x)^2),x)

$\label{eq12}\frac{\arctan \left({\frac{b \ {\sin \left({x}\right)}}{a}}\right)}{a \ b}$

(12)

Type: Union(Expression(Integer),...)

fricas

INT(COS(x)/(a^2-b^2*SIN(x)^2),x)

$\label{eq13}\frac{{\log \left({{b \ {\sin \left({x}\right)}}+ a}\right)}-{\log \left({-{b \ {\sin \left({x}\right)}}+ a}\right)}}{2 \ a \ b}$

(13)

Type: Union(Expression(Integer),...)

fricas

INT(SIN(2*x)/(b^2*SIN(x)^2+a^2),x)

$\label{eq14}\frac{\log \left({-{{{b}^{2}}\ {{\cos \left({x}\right)}^{2}}}+{{b}^{2}}+{{a}^{2}}}\right)}{{b}^{2}}$

(14)

Type: Union(Expression(Integer),...)

fricas

INT(SIN(2*x)/(b^2*SIN(x)^2-a^2),x)

$\label{eq15}\frac{\log \left({{{{b}^{2}}\ {{\cos \left({x}\right)}^{2}}}-{{b}^{2}}+{{a}^{2}}}\right)}{{b}^{2}}$

(15)

Type: Union(Expression(Integer),...)

fricas

INT(SIN(2*x)/(b^2*COS(x)^2+a^2),x)

$\label{eq16}-{\frac{\log \left({{{{b}^{2}}\ {{\cos \left({x}\right)}^{2}}}+{{a}^{2}}}\right)}{{b}^{2}}}$

(16)

Type: Union(Expression(Integer),...)

fricas

INT(SIN(2*x)/(b^2*COS(x)^2-a^2),x)

$\label{eq17}-{\frac{\log \left({{{{b}^{2}}\ {{\cos \left({x}\right)}^{2}}}-{{a}^{2}}}\right)}{{b}^{2}}}$

(17)

Type: Union(Expression(Integer),...)

fricas

INT(1/(4-COS(x)^2),x)

$\label{eq18}-{\frac{\arctan \left({\frac{{7 \ {\sqrt{3}}\ {{\cos \left({x}\right)}^{2}}}-{4 \ {\sqrt{3}}}}{{12}\ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}\right)}{4 \ {\sqrt{3}}}}$

(18)

Type: Union(Expression(Integer),...)

fricas

INT(%e^x/(%e^(2*x)-1),x)

$\label{eq19}\frac{-{\log \left({{{e}^{x}}+ 1}\right)}+{\log \left({{{e}^{x}}- 1}\right)}}{2}$

(19)

Type: Union(Expression(Integer),...)

fricas

INT(1/(x*LN(x)),x)

$\label{eq20}\log \left({\log \left({x}\right)}\right)$

(20)

Type: Union(Expression(Integer),...)

fricas

INT(1/(x*(1+LN(x)^2)),x)

$\label{eq21}\arctan \left({\log \left({x}\right)}\right)$

(21)

Type: Union(Expression(Integer),...)

fricas

INT(1/(x*(1-LN(x))),x)

$\label{eq22}-{\log \left({{\log \left({x}\right)}- 1}\right)}$

(22)

Type: Union(Expression(Integer),...)

fricas

INT(1/(x*(1+LN(x/a))),x)

$\label{eq23}\log \left({{\log \left({\frac{x}{a}}\right)}+ 1}\right)$

(23)

Type: Union(Expression(Integer),...)

fricas

-- Timofeev (1948) Ch. 1, examples 22 - 40 (p. 27-28)
INT(((1-SQRT(x)+x)/x)^2,x)

$\label{eq24}\frac{{6 \ x \ {\log \left({\sqrt{x}}\right)}}+{{\left(-{4 \ x}+ 4 \right)}\ {\sqrt{x}}}+{{x}^{2}}- 1}{x}$

(24)

Type: Union(Expression(Integer),...)

fricas

INT((2-x^(2/3))*(x+SQRT(x))/x^(3/2),x)

$\label{eq25}\frac{{{168}\ {\log \left({\root{6}\of{x}}\right)}}-{{21}\ {{\root{6}\of{x}}^{4}}}+{{56}\ {{\root{6}\of{x}}^{3}}}-{{12}\ x \ {\root{6}\of{x}}}}{1 4}$

(25)

Type: Union(Expression(Integer),...)

fricas

INT((2*x-1)/(2*x+3),x)

$\label{eq26}-{2 \ {\log \left({{2 \ x}+ 3}\right)}}+ x$

(26)

Type: Union(Expression(Integer),...)

fricas

INT((2*x-5)/(3*x^2-2),x)

$\label{eq27}\frac{{{15}\ {\log \left({\frac{{{\left({3 \ {{x}^{2}}}+ 2 \right)}\ {\sqrt{6}}}+{{12}\ x}}{{3 \ {{x}^{2}}}- 2}}\right)}}+{2 \ {\sqrt{6}}\ {\log \left({{3 \ {{x}^{2}}}- 2}\right)}}}{6 \ {\sqrt{6}}}$

(27)

Type: Union(Expression(Integer),...)

fricas

INT((2*x-5)/(3*x^2+2),x)

$\label{eq28}\frac{{{\sqrt{6}}\ {\log \left({{3 \ {{x}^{2}}}+ 2}\right)}}-{{15}\ {\arctan \left({\frac{x \ {\sqrt{6}}}{2}}\right)}}}{3 \ {\sqrt{6}}}$

(28)

Type: Union(Expression(Integer),...)

fricas

INT(SIN(x)*SIN(x/4),x)

$\label{eq29}\frac{{\left(-{{96}\ {{\cos \left({\frac{x}{4}}\right)}^{4}}}+{{112}\ {{\cos \left({\frac{x}{4}}\right)}^{2}}}-{16}\right)}\ {\sin \left({\frac{x}{4}}\right)}}{15}$

(29)

Type: Union(Expression(Integer),...)

fricas

INT(COS(3*x)*COS(4*x),x)

$\label{eq30}\frac{{\left({{32}\ {{\cos \left({x}\right)}^{6}}}-{{40}\ {{\cos \left({x}\right)}^{4}}}+{{12}\ {{\cos \left({x}\right)}^{2}}}+ 3 \right)}\ {\sin \left({x}\right)}}{7}$

(30)

Type: Union(Expression(Integer),...)

fricas

INT(TAN(x)*TAN(x-a),x)

$\label{eq31}\frac{{{\left(-{\cos \left({2 \ a}\right)}- 1 \right)}\ {\log \left({\frac{1}{{{\tan \left({x}\right)}^{2}}+ 1}}\right)}}+{{\left({\cos \left({2 \ a}\right)}+ 1 \right)}\ {\log \left({\frac{{{\left(-{\cos \left({2 \ a}\right)}+ 1 \right)}\ {{\tan \left({x}\right)}^{2}}}+{2 \ {\sin \left({2 \ a}\right)}\ {\tan \left({x}\right)}}+{\cos \left({2 \ a}\right)}+ 1}{{{\left({\cos \left({2 \ a}\right)}+ 1 \right)}\ {{\tan \left({x}\right)}^{2}}}+{\cos \left({2 \ a}\right)}+ 1}}\right)}}-{2 \ x \ {\sin \left({2 \ a}\right)}}}{2 \ {\sin \left({2 \ a}\right)}}$

(31)

Type: Union(Expression(Integer),...)

fricas

INT(SIN(x)^2,x)

$\label{eq32}\frac{-{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ x}{2}$

(32)

Type: Union(Expression(Integer),...)

fricas

INT(COS(x)^2,x)

$\label{eq33}\frac{{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ x}{2}$

(33)

Type: Union(Expression(Integer),...)

fricas

INT(SIN(x)*COS(x)^3,x)

$\label{eq34}-{\frac{{\cos \left({x}\right)}^{4}}{4}}$

(34)

Type: Union(Expression(Integer),...)

fricas

INT(COS(x)^3/SIN(x)^4,x)

$\label{eq35}\frac{{3 \ {{\cos \left({x}\right)}^{2}}}- 2}{{\left({3 \ {{\cos \left({x}\right)}^{2}}}- 3 \right)}\ {\sin \left({x}\right)}}$

(35)

Type: Union(Expression(Integer),...)

fricas

INT(1/(SIN(x)^2*COS(x)^2),x)

$\label{eq36}\frac{-{2 \ {{\cos \left({x}\right)}^{2}}}+ 1}{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}$

(36)

Type: Union(Expression(Integer),...)

fricas

INT(COT(3/4*x)^2,x)

$\label{eq37}\frac{-{3 \ x \ {\sin \left({\frac{3 \ x}{2}}\right)}}-{4 \ {\cos \left({\frac{3 \ x}{2}}\right)}}- 4}{3 \ {\sin \left({\frac{3 \ x}{2}}\right)}}$

(37)

Type: Union(Expression(Integer),...)

fricas

INT((1+TAN(2*x))^2,x)

$\label{eq38}\frac{-{\log \left({\frac{1}{{{\tan \left({2 \ x}\right)}^{2}}+ 1}}\right)}+{\tan \left({2 \ x}\right)}}{2}$

(38)

Type: Union(Expression(Integer),...)

fricas

INT((TAN(x)-COT(x))^2,x)

$\label{eq39}\frac{{{\tan \left({x}\right)}^{2}}-{4 \ x \ {\tan \left({x}\right)}}- 1}{\tan \left({x}\right)}$

(39)

Type: Union(Expression(Integer),...)

fricas

INT((TAN(x)-SEC(x))^2,x)

$\label{eq40}\frac{{{\left(- x + 2 \right)}\ {\sin \left({x}\right)}}+{{\left(- x - 2 \right)}\ {\cos \left({x}\right)}}- x - 2}{{\sin \left({x}\right)}+{\cos \left({x}\right)}+ 1}$

(40)

Type: Union(Expression(Integer),...)

fricas

INT(SIN(x)/(1+SIN(x)),x)

$\label{eq41}\frac{{{\left(x - 1 \right)}\ {\sin \left({x}\right)}}+{{\left(x + 1 \right)}\ {\cos \left({x}\right)}}+ x + 1}{{\sin \left({x}\right)}+{\cos \left({x}\right)}+ 1}$

(41)

Type: Union(Expression(Integer),...)

fricas

INT(COS(x)/(1-COS(x)),x)

$\label{eq42}\frac{-{x \ {\sin \left({x}\right)}}-{\cos \left({x}\right)}- 1}{\sin \left({x}\right)}$

(42)

Type: Union(Expression(Integer),...)

fricas

INT((%e^(x/2)-1)^3*%e^(-x/2),x)

$\label{eq43}\frac{{{{e}^{\frac{x}{2}}}^{3}}-{6 \ {{{e}^{\frac{x}{2}}}^{2}}}+{3 \ x \ {{e}^{\frac{x}{2}}}}+ 2}{{e}^{\frac{x}{2}}}$

(43)

Type: Union(Expression(Integer),...)

fricas

-- Timofeev (1948) Ch. 1, examples 41 - 65 (p. 35-37)

INT(1/(x^2-6*x+5),x)

$\label{eq44}\frac{-{\log \left({x - 1}\right)}+{\log \left({x - 5}\right)}}{4}$

(44)

Type: Union(Expression(Integer),...)

fricas

INT(x^2/(13-6*x^3+x^6),x)

$\label{eq45}\frac{\arctan \left({\frac{{{x}^{3}}- 3}{2}}\right)}{6}$

(45)

Type: Union(Expression(Integer),...)

fricas

INT((x+2)/(x^2-4*x-1),x)

$\label{eq46}\frac{{4 \ {\log \left({\frac{{{\left({{x}^{2}}-{4 \ x}+ 9 \right)}\ {\sqrt{5}}}-{{10}\ x}+{20}}{{{x}^{2}}-{4 \ x}- 1}}\right)}}+{{\sqrt{5}}\ {\log \left({{{x}^{2}}-{4 \ x}- 1}\right)}}}{2 \ {\sqrt{5}}}$

(46)

Type: Union(Expression(Integer),...)

fricas

INT(1/(1+(x+1)^(1/3)),x)

$\label{eq47}\frac{{6 \ {\log \left({{\root{3}\of{x + 1}}+ 1}\right)}}+{3 \ {{\root{3}\of{x + 1}}^{2}}}-{6 \ {\root{3}\of{x + 1}}}}{2}$

(47)

Type: Union(Expression(Integer),...)

fricas

INT(1/((a*x+b)*SQRT(x)),x)

$\label{eq48}\begin{array}{@{}l} \displaystyle \left[{\frac{\log \left({\frac{{2 \ a \ b \ {\sqrt{x}}}+{{\left({a \ x}- b \right)}\ {\sqrt{-{a \ b}}}}}{{a \ x}+ b}}\right)}{\sqrt{-{a \ b}}}}, \: \right. \ \ \displaystyle \left.-{\frac{2 \ {\arctan \left({\frac{b}{{\sqrt{a \ b}}\ {\sqrt{x}}}}\right)}}{\sqrt{a \ b}}}\right]$

(48)

Type: Union(List(Expression(Integer)),...)

fricas

INT(x^3*SQRT(1+x^2),x)

$\label{eq49}\frac{{{\left(-{{48}\ {{x}^{9}}}-{{76}\ {{x}^{7}}}-{3 \ {{x}^{5}}}+{{35}\ {{x}^{3}}}+{{10}\ x}\right)}\ {\sqrt{{{x}^{2}}+ 1}}}+{{48}\ {{x}^{10}}}+{{100}\ {{x}^{8}}}+{{35}\ {{x}^{6}}}-{{40}\ {{x}^{4}}}-{{25}\ {{x}^{2}}}- 2}{{{\left({{240}\ {{x}^{4}}}+{{180}\ {{x}^{2}}}+{15}\right)}\ {\sqrt{{{x}^{2}}+ 1}}}-{{240}\ {{x}^{5}}}-{{300}\ {{x}^{3}}}-{{75}\ x}}$

(49)

Type: Union(Expression(Integer),...)

fricas

INT(x/SQRT(a^4-x^4),x)

$\label{eq50}-{\arctan \left({\frac{{\sqrt{-{{x}^{4}}+{{a}^{4}}}}-{{a}^{2}}}{{x}^{2}}}\right)}$

(50)

Type: Union(Expression(Integer),...)

fricas

INT(1/(x*SQRT(x^2-a^2)),x)

$\label{eq51}\frac{2 \ {\arctan \left({\frac{{\sqrt{{{x}^{2}}-{{a}^{2}}}}- x}{a}}\right)}}{a}$

(51)

Type: Union(Expression(Integer),...)

fricas

INT(1/(x*SQRT(a^2-x^2)),x)

$\label{eq52}\frac{\log \left({\frac{{\sqrt{-{{x}^{2}}+{{a}^{2}}}}- a}{x}}\right)}{a}$

(52)

Type: Union(Expression(Integer),...)

fricas

INT(1/(x*SQRT(x^2+a^2)),x)

$\label{eq53}\frac{-{\log \left({{\sqrt{{{x}^{2}}+{{a}^{2}}}}- x + a}\right)}+{\log \left({{\sqrt{{{x}^{2}}+{{a}^{2}}}}- x - a}\right)}}{a}$

(53)

Type: Union(Expression(Integer),...)

fricas

INT(1/SQRT(2+x-x^2),x)

$\label{eq54}\arctan \left({\frac{{2 \ x}- 1}{2 \ {\sqrt{-{{x}^{2}}+ x + 2}}}}\right)$

(54)

Type: Union(Expression(Integer),...)

fricas

INT(1/SQRT(3*x^2-4*x+5),x)

$\label{eq55}\frac{\log \left({{{\left(-{{18}\ x}+{12}\right)}\ {\sqrt{{3 \ {{x}^{2}}}-{4 \ x}+ 5}}}+{{\left(-{{18}\ {{x}^{2}}}+{{24}\ x}-{19}\right)}\ {\sqrt{3}}}}\right)}{2 \ {\sqrt{3}}}$

(55)

Type: Union(Expression(Integer),...)

fricas

INT(1/SQRT(x-x^2),x)

$\label{eq56}-{2 \ {\arctan \left({\frac{\sqrt{-{{x}^{2}}+ x}}{x}}\right)}}$

(56)

Type: Union(Expression(Integer),...)

fricas

INT((2*x+1)/SQRT(2+x-x^2),x)

$\label{eq57}{2 \ {\arctan \left({\frac{{2 \ x}- 1}{2 \ {\sqrt{-{{x}^{2}}+ x + 2}}}}\right)}}-{2 \ {\sqrt{-{{x}^{2}}+ x + 2}}}$

(57)

Type: Union(Expression(Integer),...)

fricas

INT(1/(x*SQRT(2+x-x^2)),x)

$\label{eq58}\frac{{\sqrt{2}}\ {\log \left({\frac{{{\left(-{4 \ x}-{16}\right)}\ {\sqrt{2}}\ {\sqrt{-{{x}^{2}}+ x + 2}}}-{7 \ {{x}^{2}}}+{{16}\ x}+{32}}{{x}^{2}}}\right)}}{4}$

(58)

Type: Union(Expression(Integer),...)

fricas

INT(1/((x-2)*SQRT(2+x-x^2)),x)

$\label{eq59}\frac{2 \ {\sqrt{-{{x}^{2}}+ x + 2}}}{{3 \ x}- 6}$

(59)

Type: Union(Expression(Integer),...)

fricas

INT((2+3*SIN(x))/(SIN(x)*(1-COS(x))),x)

$\label{eq60}\frac{{{\left(-{\cos \left({x}\right)}+ 1 \right)}\ {\log \left({\frac{{\cos \left({x}\right)}+ 1}{2}}\right)}}+{{\left({\cos \left({x}\right)}- 1 \right)}\ {\log \left({\frac{-{\cos \left({x}\right)}+ 1}{2}}\right)}}+{6 \ {\sin \left({x}\right)}}+ 2}{{2 \ {\cos \left({x}\right)}}- 2}$

(60)

Type: Union(Expression(Integer),...)

fricas

INT(1/(2+3*COS(x)^2),x)

$\label{eq61}\begin{array}{@{}l} \displaystyle - \ \ \displaystyle {\frac{\arctan \left({\frac{{7 \ {\sqrt{10}}\ {{\cos \left({x}\right)}^{2}}}-{2 \ {\sqrt{10}}}}{{20}\ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}\right)}{2 \ {\sqrt{10}}}}$

(61)

Type: Union(Expression(Integer),...)

fricas

INT((1-TAN(x))/SIN(2*x),x)

$\label{eq62}\frac{{\log \left({\frac{{\tan \left({x}\right)}^{2}}{{{\tan \left({x}\right)}^{2}}+ 1}}\right)}-{\log \left({\frac{1}{{{\tan \left({x}\right)}^{2}}+ 1}}\right)}-{2 \ {\tan \left({x}\right)}}}{4}$

(62)

Type: Union(Expression(Integer),...)

fricas

INT((1+TAN(x)^2)/(1-TAN(x)^2),x)

$\label{eq63}\frac{{\log \left({\frac{{{\tan \left({x}\right)}^{2}}+{2 \ {\tan \left({x}\right)}}+ 1}{{{\tan \left({x}\right)}^{2}}+ 1}}\right)}-{\log \left({\frac{{{\tan \left({x}\right)}^{2}}-{2 \ {\tan \left({x}\right)}}+ 1}{{{\tan \left({x}\right)}^{2}}+ 1}}\right)}}{4}$

(63)

Type: Union(Expression(Integer),...)

fricas

INT((a^2-4*COS(x)^2)^(3/4)*SIN(2*x),x)

$\label{eq64}\frac{{\left(-{4 \ {{\cos \left({x}\right)}^{2}}}+{{a}^{2}}\right)}\ {{\root{4}\of{-{4 \ {{\cos \left({x}\right)}^{2}}}+{{a}^{2}}}}^{3}}}{7}$

(64)

Type: Union(Expression(Integer),...)

fricas

INT(SIN(2*x)/(a^2-4*SIN(x)^2)^(1/3),x)

$\label{eq65}-{\frac{3 \ {{\root{3}\of{{4 \ {{\cos \left({x}\right)}^{2}}}+{{a}^{2}}- 4}}^{2}}}{8}}$

(65)

Type: Union(Expression(Integer),...)

fricas

INT(1/SQRT(a^(2*x)-1),x)

$\label{eq66}\frac{\arctan \left({\sqrt{{{e}^{2 \ x \ {\log \left({a}\right)}}}- 1}}\right)}{\log \left({a}\right)}$

(66)

Type: Union(Expression(Integer),...)

fricas

INT(%e^(x/2)/SQRT(%e^x-1),x)

$\label{eq67}-{2 \ {\log \left({{\sqrt{{{{e}^{\frac{x}{2}}}^{2}}- 1}}-{{e}^{\frac{x}{2}}}}\right)}}$

(67)

Type: Union(Expression(Integer),...)

fricas

INT(ATAN(x)^n/(1+x^2),x)

$\label{eq68}\frac{{\arctan \left({x}\right)}\ {{e}^{n \ {\log \left({\arctan \left({x}\right)}\right)}}}}{n + 1}$

(68)

Type: Union(Expression(Integer),...)

fricas

INT(ASIN(x/a)^(3/2)/SQRT(a^2-x^2),x)

$\label{eq69}\frac{2 \ {{\arctan \left({\frac{x}{\sqrt{-{{x}^{2}}+{{a}^{2}}}}}\right)}^{2}}\ {\sqrt{2 \ {\arctan \left({\frac{x}{\sqrt{-{{x}^{2}}+{{a}^{2}}}}}\right)}}}}{5 \ {\sqrt{2}}}$

(69)

Type: Union(Expression(Integer),...)

fricas

INT(1/(ACOS(x)^3*SQRT(1-x^2)),x)

$\label{eq70}\frac{1}{2 \ {{\arccos \left({x}\right)}^{2}}}$

(70)

Type: Union(Expression(Integer),...)

fricas

-- Timofeev (1948) Ch. 1, examples 66 - 81 (p. 41-42)

INT(LN(x)^2*x,x)

$\label{eq71}\frac{{2 \ {{x}^{2}}\ {{\log \left({x}\right)}^{2}}}-{2 \ {{x}^{2}}\ {\log \left({x}\right)}}+{{x}^{2}}}{4}$

(71)

Type: Union(Expression(Integer),...)

fricas

INT(LN(x)/x^5,x)

$\label{eq72}\frac{-{4 \ {\log \left({x}\right)}}- 1}{{16}\ {{x}^{4}}}$

(72)

Type: Union(Expression(Integer),...)

fricas

INT(x^2*LN((x-1)/x),x)

$\label{eq73}\frac{-{2 \ {\log \left({x - 1}\right)}}+{2 \ {{x}^{3}}\ {\log \left({\frac{x - 1}{x}}\right)}}-{{x}^{2}}-{2 \ x}}{6}$

(73)

Type: Union(Expression(Integer),...)

fricas

INT(COS(x)^5,x)

$\label{eq74}\frac{{\left({3 \ {{\cos \left({x}\right)}^{4}}}+{4 \ {{\cos \left({x}\right)}^{2}}}+ 8 \right)}\ {\sin \left({x}\right)}}{1 5}$

(74)

Type: Union(Expression(Integer),...)

fricas

INT(SIN(x)^2*COS(x)^4,x)

$\label{eq75}\frac{{{\left(-{8 \ {{\cos \left({x}\right)}^{5}}}+{2 \ {{\cos \left({x}\right)}^{3}}}+{3 \ {\cos \left({x}\right)}}\right)}\ {\sin \left({x}\right)}}+{3 \ x}}{48}$

(75)

Type: Union(Expression(Integer),...)

fricas

INT(1/SIN(x)^5,x)

$\label{eq76}\frac{{{\left(-{3 \ {{\cos \left({x}\right)}^{4}}}+{6 \ {{\cos \left({x}\right)}^{2}}}- 3 \right)}\ {\log \left({\frac{{\cos \left({x}\right)}+ 1}{2}}\right)}}+{{\left({3 \ {{\cos \left({x}\right)}^{4}}}-{6 \ {{\cos \left({x}\right)}^{2}}}+ 3 \right)}\ {\log \left({\frac{-{\cos \left({x}\right)}+ 1}{2}}\right)}}+{6 \ {{\cos \left({x}\right)}^{3}}}-{{10}\ {\cos \left({x}\right)}}}{{{16}\ {{\cos \left({x}\right)}^{4}}}-{{32}\ {{\cos \left({x}\right)}^{2}}}+{16}}$

(76)

Type: Union(Expression(Integer),...)

fricas

INT(SIN(x)/%e^x,x)

$\label{eq77}\frac{-{\sin \left({x}\right)}-{\cos \left({x}\right)}}{2 \ {{e}^{x}}}$

(77)

Type: Union(Expression(Integer),...)

fricas

INT(%e^(2*x)*SIN(3*x),x)

$\label{eq78}\frac{{2 \ {{e}^{2 \ x}}\ {\sin \left({3 \ x}\right)}}-{3 \ {\cos \left({3 \ x}\right)}\ {{e}^{2 \ x}}}}{13}$

(78)

Type: Union(Expression(Integer),...)

fricas

INT(a^x*COS(x),x)

$\label{eq79}\frac{{\left({\sin \left({x}\right)}+{{\cos \left({x}\right)}\ {\log \left({a}\right)}}\right)}\ {{e}^{x \ {\log \left({a}\right)}}}}{{{\log \left({a}\right)}^{2}}+ 1}$

(79)

Type: Union(Expression(Integer),...)

fricas

INT(COS(LN(x)),x)

$\label{eq80}\frac{{x \ {\sin \left({\log \left({x}\right)}\right)}}+{x \ {\cos \left({\log \left({x}\right)}\right)}}}{2}$

(80)

Type: Union(Expression(Integer),...)

fricas

INT(SEC(x)^2*LN(COS(x)),x)

$\label{eq81}\frac{{{\sin \left({x}\right)}\ {\log \left({\cos \left({x}\right)}\right)}}+{\sin \left({x}\right)}-{x \ {\cos \left({x}\right)}}}{\cos \left({x}\right)}$

(81)

Type: Union(Expression(Integer),...)

fricas

INT(x*TAN(x)^2,x)

$\label{eq82}\frac{{\log \left({\frac{1}{{{\tan \left({x}\right)}^{2}}+ 1}}\right)}+{2 \ x \ {\tan \left({x}\right)}}-{{x}^{2}}}{2}$

(82)

Type: Union(Expression(Integer),...)

fricas

INT(ASIN(x)/x^2,x)

$\label{eq83}\frac{-{x \ {\log \left({{\sqrt{-{{x}^{2}}+ 1}}+ 1}\right)}}+{x \ {\log \left({{\sqrt{-{{x}^{2}}+ 1}}- 1}\right)}}-{2 \ {\arcsin \left({x}\right)}}}{2 \ x}$

(83)

Type: Union(Expression(Integer),...)

fricas

INT(ASIN(x)^2,x)

$\label{eq84}{2 \ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{x \ {{\arcsin \left({x}\right)}^{2}}}-{2 \ x}$

(84)

Type: Union(Expression(Integer),...)

fricas

INT(x^2*ATAN(x)/(1+x^2),x)

$\label{eq85}\frac{-{\log \left({{{x}^{2}}+ 1}\right)}-{{\arctan \left({x}\right)}^{2}}+{2 \ x \ {\arctan \left({x}\right)}}}{2}$

(85)

Type: Union(Expression(Integer),...)

fricas

INT(ACOS(SQRT(x/(x+1))),x)

$\label{eq86}\frac{{{\left(x + 1 \right)}\ {\sqrt{\frac{1}{x + 1}}}\ {\arccos \left({\sqrt{\frac{x}{x + 1}}}\right)}}+{\sqrt{\frac{x}{x + 1}}}}{\sqrt{\frac{1}{x + 1}}}$

(86)

Type: Union(Expression(Integer),...)