MathAction SandBox Integration

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Integration

Let's do some integration examples:

fricas

(1) -> integrate(%e^x, x)

$\label{eq1}{e}^{x}$

(1)

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

int(cos(x),x,0,pi);

reduce

$\displaylines{\qdd 0 \cr}$

fricas

integrate(x^2/sqrt(4-x^2),x)

$\label{eq2}\frac{{{\left(-{{32}\ {\sqrt{-{{x}^{2}}+ 4}}}-{8 \ {{x}^{2}}}+{64}\right)}\ {\arctan \left({\frac{{\sqrt{-{{x}^{2}}+ 4}}- 2}{x}}\right)}}+{{\left(-{{x}^{3}}+{8 \ x}\right)}\ {\sqrt{-{{x}^{2}}+ 4}}}+{4 \ {{x}^{3}}}-{{16}\ x}}{{8 \ {\sqrt{-{{x}^{2}}+ 4}}}+{2 \ {{x}^{2}}}-{16}}$

(2)

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

Below FriCAS gives up because sign of a is unknown:

fricas

integrate(exp(-a*x^2),x=0..%plusInfinity)

$\label{eq3}\verb#"failed"#$

(3)

Type: Union(fail: failed,...)

The following won't "work", see CommonMistakes:

fricas

integrate(exp(-a::PositiveInteger*x^2),x=0..%plusInfinity)

   Cannot convert the value from type Variable(a) to PositiveInteger .

fricas

integrate((x^3+x^2+2)/(x*(x^2-1)^2), x)

$\label{eq4}\frac{{{\left(-{5 \ {{x}^{2}}}+ 5 \right)}\ {\log \left({x + 1}\right)}}+{{\left({8 \ {{x}^{2}}}- 8 \right)}\ {\log \left({x}\right)}}+{{\left(-{3 \ {{x}^{2}}}+ 3 \right)}\ {\log \left({x - 1}\right)}}-{2 \ x}- 6}{{4 \ {{x}^{2}}}- 4}$

(4)

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

fricas

integrate(2*x/sin(x)^2,x)

$\label{eq5}\frac{{2 \ {\sin \left({x}\right)}\ {\log \left({\frac{\sin \left({x}\right)}{2}}\right)}}-{2 \ x \ {\cos \left({x}\right)}}}{\sin \left({x}\right)}$

(5)

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

Comparing FriCAS and Reduce:

fricas

integrate(sin(1/x),x)

$\label{eq6}{x \ {\sin \left({\frac{1}{x}}\right)}}-{Ci \left({\frac{1}{x}}\right)}$

(6)

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

int(sin(1/x),x);

reduce

$\displaylines{\qdd \int {\sin \(\frac{1}{ x}$

load_package algint;

int(sin(1/x),x);

reduce

$\displaylines{\qdd \int {\sin \(\frac{1}{ x}$

A different problem, where FriCAS has to give up:

fricas

integrate(sqrt(sin(1/x)),x)

   >> Error detected within library code:
   integrate: implementation incomplete (has polynomial part)

In Reduce:

load_package algint;

int(sqrt(sin(1/x)),x);

reduce

$\displaylines{\qdd \frac{2\cdot \sqrt{\sin \(\frac{1}{ x}$

fricas

integrate(exp(-x^2),x)

$\label{eq7}\frac{{\erf \left({x}\right)}\ {\sqrt{\pi}}}{2}$

(7)

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

fricas

integrate(sin(x)/x,x)

$\label{eq8}Si \left({x}\right)$

(8)

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

fricas

differentiate(%,x)

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

(9)

Type: Expression(Integer)

fricas

integrate(sin(1/x),x=%minusInfinity..%plusInfinity,"noPole")

   >> Error detected within library code:
   integrate: pole in path of integration

fricas

integrate(2*x/sin(x)^2,x=1/2..1)

$\label{eq10}\verb#"potentialPole"#$

(10)

Type: Union(pole: potentialPole,...)

fricas

integrate(sin(x),x=0..%pi/2)

$\label{eq11}1$

(11)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

fricas

integrate(atan(x/a)/x,x)

$\label{eq12}\int^{ \displaystyle x}{{\frac{\arctan \left({\frac{\%A}{a}}\right)}{\%A}}\ {d \%A}}$

(12)

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

fricas

integrate(1/(a+z^3), z=0..1,"noPole")

$\label{eq13}\frac{-{{\sqrt{3}}\ {\log \left({{3 \ {{a}^{2}}\ {{\root{3}\of{{a}^{2}}}^{2}}}+{{\left(-{2 \ {{a}^{3}}}+{{a}^{2}}\right)}\ {\root{3}\of{{a}^{2}}}}+{{a}^{4}}-{2 \ {{a}^{3}}}}\right)}}+{2 \ {\sqrt{3}}\ {\log \left({{{\root{3}\of{{a}^{2}}}^{2}}+{2 \ a \ {\root{3}\of{{a}^{2}}}}+{{a}^{2}}}\right)}}+{{12}\ {\arctan \left({\frac{{2 \ {\sqrt{3}}\ {\root{3}\of{{a}^{2}}}}-{a \ {\sqrt{3}}}}{3 \ a}}\right)}}+{{\sqrt{3}}\ {\log \left({{a}^{4}}\right)}}-{2 \ {\sqrt{3}}\ {\log \left({{a}^{2}}\right)}}+{2 \ \pi}}{{12}\ {\sqrt{3}}\ {\root{3}\of{{a}^{2}}}}$

(13)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

fricas

integrate(x^3+x^2/4+x,x)

$\label{eq14}{{\frac{1}{4}}\ {{x}^{4}}}+{{\frac{1}{12}}\ {{x}^{3}}}+{{\frac{1}{2}}\ {{x}^{2}}}$

(14)

Type: Polynomial(Fraction(Integer))

You cannot integrate Expression Float

fricas

integrate(50*%e^(-0.02*t),t)

   There are 9 exposed and 11 unexposed library operations named 
      integrate having 2 argument(s) but none was determined to be 
      applicable. Use HyperDoc Browse, or issue
                            )display op integrate
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named 
      integrate with argument type(s) 
                              Expression(Float)
                                 Variable(t)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

But symbolic integration works with integer expressions

fricas

integrate(50*%e^(-0.02*t)::Expression Fraction Integer,t)

$\label{eq15}-{{2500}\ {{e}^{-{{\frac{1}{50}}\ t}}}}$

(15)

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

fricas

integrate(exp(cos(x)),x)

$\label{eq16}\int^{ \displaystyle x}{{{e}^{\cos \left({\%A}\right)}}\ {d \%A}}$

(16)

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

fricas

integrate(sin(x),x)

$\label{eq17}-{\cos \left({x}\right)}$

(17)

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

fricas

integrate(%,x)

$\label{eq18}-{\sin \left({x}\right)}$

(18)

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

fricas

integrate(a/h - c*h/12 + (b/h)*r + (c/h)*r^2,r)

$\label{eq19}\frac{{4 \ c \ {{r}^{3}}}+{6 \ b \ {{r}^{2}}}+{{\left(-{c \ {{h}^{2}}}+{{12}\ a}\right)}\ r}}{{12}\ h}$

(19)

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

fricas

integrate(exp(-(a+b*t)^2/2),t)

$\label{eq20}\frac{{\sqrt{\pi}}\ {\erf \left({\frac{{\left({b \ t}+ a \right)}\ {\sqrt{\frac{{b}^{2}}{2}}}}{b}}\right)}}{2 \ {\sqrt{\frac{{b}^{2}}{2}}}}$

(20)

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

fricas

integrate(exp(-(a+b*t)^2/t),t)

$\label{eq21}\int^{ \displaystyle t}{{{e}^{\frac{-{{{\%A}^{2}}\ {{b}^{2}}}-{2 \ \%A \ a \ b}-{{a}^{2}}}{\%A}}}\ {d \%A}}$

(21)

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

fricas

integrate(exp(-1/t),t)

$\label{eq22}{t \ {{e}^{-{\frac{1}{t}}}}}+{Ei \left({-{\frac{1}{t}}}\right)}$

(22)

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

fricas

integrate(exp(-1/t),t=1..x)

$\label{eq23}\verb#"potentialPole"#$

(23)

Type: Union(pole: potentialPole,...)

Unfortunately, there is currently no easy way to make "assumptions" about variables. Thus, The following won't work:

  \begin{axiom}
   assume(x, real)
   integrate(exp(-1/t),t=1..x)
   \end{axiom}

fricas

integrate(t*exp(-(a+b*t)^2/2),t)

$\label{eq24}\frac{-{a \ b \ {\sqrt{\pi}}\ {\erf \left({\frac{{\left({b \ t}+ a \right)}\ {\sqrt{\frac{{b}^{2}}{2}}}}{b}}\right)}}-{2 \ {{e}^{\frac{-{{{b}^{2}}\ {{t}^{2}}}-{2 \ a \ b \ t}-{{a}^{2}}}{2}}}\ {\sqrt{\frac{{b}^{2}}{2}}}}}{2 \ {{b}^{2}}\ {\sqrt{\frac{{b}^{2}}{2}}}}$

(24)

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

fricas

integrate(1/(a+z^3), z=0..1,"noPole")

$\label{eq25}\frac{-{{\sqrt{3}}\ {\log \left({{3 \ {{a}^{2}}\ {{\root{3}\of{{a}^{2}}}^{2}}}+{{\left(-{2 \ {{a}^{3}}}+{{a}^{2}}\right)}\ {\root{3}\of{{a}^{2}}}}+{{a}^{4}}-{2 \ {{a}^{3}}}}\right)}}+{2 \ {\sqrt{3}}\ {\log \left({{{\root{3}\of{{a}^{2}}}^{2}}+{2 \ a \ {\root{3}\of{{a}^{2}}}}+{{a}^{2}}}\right)}}+{{12}\ {\arctan \left({\frac{{2 \ {\sqrt{3}}\ {\root{3}\of{{a}^{2}}}}-{a \ {\sqrt{3}}}}{3 \ a}}\right)}}+{{\sqrt{3}}\ {\log \left({{a}^{4}}\right)}}-{2 \ {\sqrt{3}}\ {\log \left({{a}^{2}}\right)}}+{2 \ \pi}}{{12}\ {\sqrt{3}}\ {\root{3}\of{{a}^{2}}}}$

(25)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

From the ReduceProblem?:

fricas

integrate(1/sqrt(2*%pi)*exp(-1/2*log(x)^2),x=0..%plusInfinity)

$\label{eq26}\frac{{{e}^{\frac{1}{2}}}\ {\sqrt{2}}\ {\sqrt{\pi}}}{\sqrt{2 \ \pi}}$

(26)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

fricas

integrate(1/sqrt(2*%pi)*exp(-1/2*log(x)^2),x)

$\label{eq27}\frac{{{e}^{\frac{1}{2}}}\ {\sqrt{2}}\ {\sqrt{\pi}}\ {\erf \left({\frac{{\log \left({x}\right)}- 1}{\sqrt{2}}}\right)}}{2 \ {\sqrt{2 \ \pi}}}$

(27)

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

If you would get a result, you could use limit afterwards, of course.

Mathematical Paradox?: Thu, 10 Feb 2005 17:45:57 -0600 reply

Area under the curve:

fricas

integrate(1/x,x=1..%plusInfinity)

$\label{eq28}+ \infty$

(28)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

Paradox Part2:

Thu, 10 Feb 2005 17:47:59 -0600 replyVolume under that curve:

fricas

integrate(%pi*((1/x)^2), x=1..%plusInfinity)

$\label{eq29}\pi$

(29)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

Curve has an infinite area...but a finite volume (I think I did this correctly)!

... --unknown, Tue, 17 May 2005 18:52:42 -0500 reply

fricas

integrate(1/x,x)

$\label{eq30}\log \left({x}\right)$

(30)

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

fricas

integrate(sqrt(x),x)

$\label{eq31}\frac{2 \ x \ {\sqrt{x}}}{3}$

(31)

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

fricas

integrate(sqrt(x^3+x),x)

$\label{eq32}\frac{-{4 \ {weierstrassZeta \left({- 4, \: 0, \:{weierstrassPInverse \left({- 4, \: 0, \: x}\right)}}\right)}}+{2 \ x \ {\sqrt{{{x}^{3}}+ x}}}}{5}$

(32)

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

a turning moving body --unknown, Sun, 19 Jun 2005 20:16:03 -0500 reply

fricas

)set output tex off

fricas

)set output algebra on

integrate(( a*sin( m + n*t + o*t*t/2 ) )/( n + ot ) + ( b*cos( m + n*t + o*t*t/2 ) )/( n + ot ), t)

   (33)
                                                                   +---+
                                                                   | o
                         2                   2           (o t + n) |---
                2 m o - n           2 m o - n                     \|%pi
       (- b sin(----------) + a cos(----------))fresnelS(---------------)
                    2 o                 2 o                     o
     + 
                                                                 +---+
                                                                 | o
                       2                   2           (o t + n) |---
              2 m o - n           2 m o - n                     \|%pi
       (a sin(----------) + b cos(----------))fresnelC(---------------)
                  2 o                 2 o                     o
  /
              +---+
              | o
     (ot + n) |---
             \|%pi

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

a turning accelerating body --unknown, Sun, 19 Jun 2005 23:16:48 -0500 reply

fricas

integrate(( a*cos( m + n*t + o*t*t/2 ) )- ( b*sin( m + n*t + o*t*t/2 ) ), t)

   (34)
                                                                   +---+
                                                                   | o
                         2                   2           (o t + n) |---
                2 m o - n           2 m o - n                     \|%pi
       (- a sin(----------) - b cos(----------))fresnelS(---------------)
                    2 o                 2 o                     o
     + 
                                                                   +---+
                                                                   | o
                         2                   2           (o t + n) |---
                2 m o - n           2 m o - n                     \|%pi
       (- b sin(----------) + a cos(----------))fresnelC(---------------)
                    2 o                 2 o                     o
  /
      +---+
      | o
      |---
     \|%pi

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

fricas

)set output algebra off

fricas

)set output tex on

... --unknown, Sun, 26 Jun 2005 16:02:47 -0500 reply

fricas

integrate(-2*(3-3*t)^2*(3*t),t)

$\label{eq33}-{{\frac{27}{2}}\ {{t}^{4}}}+{{36}\ {{t}^{3}}}-{{27}\ {{t}^{2}}}$

(33)

Type: Polynomial(Fraction(Integer))

... --unknown, Sun, 30 Oct 2005 09:29:01 -0600 reply

fricas

integrate(1/(1+x^2),x=-u..u)

$\label{eq34}\verb#"potentialPole"#$

(34)

Type: Union(pole: potentialPole,...)

fricas

integrate(1/(1+x^2),x=-u..u, "noPole")

$\label{eq35}2 \ {\arctan \left({u}\right)}$

(35)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

... --unknown, Tue, 01 Nov 2005 10:35:26 -0600 reply

fricas

integrate(x^6*exp(-x^2), x=0..%plusInfinity)

$\label{eq36}\frac{{15}\ {\sqrt{\pi}}}{16}$

(36)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

... --unknown, Wed, 02 Nov 2005 17:00:21 -0600 reply

fricas

integrate(1/sqrt(1/x+1),x)

$\label{eq37}\frac{-{\log \left({{\sqrt{\frac{x + 1}{x}}}+ 1}\right)}+{\log \left({{\sqrt{\frac{x + 1}{x}}}- 1}\right)}+{2 \ x \ {\sqrt{\frac{x + 1}{x}}}}}{2}$

(37)

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

... --unknown, Wed, 23 Nov 2005 08:30:18 -0600 reply

fricas

integrate(sin(sin x), x)

$\label{eq38}\int^{ \displaystyle x}{{\sin \left({\sin \left({\%A}\right)}\right)}\ {d \%A}}$

(38)

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

... --unknown, Fri, 25 Nov 2005 16:28:37 -0600 reply

fricas

integrate(a/2*(1-cos(b*t)),t)

$\label{eq39}\frac{-{a \ {\sin \left({b \ t}\right)}}+{a \ b \ t}}{2 \ b}$

(39)

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

yet another test that shall work but not in maple ? --unknown, Thu, 09 Mar 2006 09:21:47 -0600 reply

fricas

integrate(tan(atan(x)/3),x)

$\label{eq40}\frac{{8 \ {\log \left({{3 \ {{\tan \left({\frac{\arctan \left({x}\right)}{3}}\right)}^{2}}}- 1}\right)}}-{3 \ {{\tan \left({\frac{\arctan \left({x}\right)}{3}}\right)}^{2}}}+{{18}\ x \ {\tan \left({\frac{\arctan \left({x}\right)}{3}}\right)}}}{1 8}$

(40)

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

... --unknown, Sat, 11 Mar 2006 12:41:33 -0600 reply

fricas

integrate(x, x)

$\label{eq41}{\frac{1}{2}}\ {{x}^{2}}$

(41)

Type: Polynomial(Fraction(Integer))

fricas

simplify((1/(2*z))*z^2)

$\label{eq42}\frac{z}{2}$

(42)

Type: Expression(Integer)

fricas

integrate((1/(2*z))*z^2, z)

$\label{eq43}\frac{{z}^{2}}{4}$

(43)

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

fricas

integrate(log(x),x)

$\label{eq44}{x \ {\log \left({x}\right)}}- x$

(44)

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

fricas

integrate(1/x,x)

$\label{eq45}\log \left({x}\right)$

(45)

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

fricas

integrate(0^0,x)

$\label{eq46}x$

(46)

Type: Polynomial(Fraction(Integer))

from fr.sci.maths --unknown, Tue, 09 May 2006 09:58:11 -0500 reply

fricas

integrate( log(y)^3/(y*(y-1)),y)

$\label{eq47}\int^{ \displaystyle y}{{\frac{{\log \left({\%A}\right)}^{3}}{{{\%A}^{2}}- \%A}}\ {d \%A}}$

(47)

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

... --unknown, Mon, 29 May 2006 14:20:28 -0500 reply

fricas

integrate(exp(-x^2),x=0..%plusInfinity)

$\label{eq48}\frac{\sqrt{\pi}}{2}$

(48)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

... --unknown, Mon, 29 May 2006 14:21:39 -0500 reply

fricas

integrate(x^2*exp(-x^2),x=0..%plusInfinity)

$\label{eq49}\frac{\sqrt{\pi}}{4}$

(49)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

test --unknown, Wed, 19 Jul 2006 11:59:36 -0500 reply

No ; after command or else output is supressed.

fricas

integrate(exp(%i*2*%pi*f*t), t=0..T)

$\label{eq50}\frac{-{i \ {{e}^{2 \ i \ T \ f \ \pi}}}+ i}{2 \ f \ \pi}$

(50)

Type: Union(f1: OrderedCompletion?(Expression(Complex(Integer))),...)

FriCAS and Maxima not capable of this integrand --WinnieThePooh?, Tue, 29 May 2007 17:24:44 -0500 reply

int(exp(sin(x)),x);

reduce

$\displaylines{\qdd \int {e^{\sin \(x$

fricas

integrate(exp(sin(x)),x)

$\label{eq51}\int^{ \displaystyle x}{{{e}^{\sin \left({\%A}\right)}}\ {d \%A}}$

(51)

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

An integral which Maxima can't do, but FriCAS can --amca01, Tue, 05 Jan 2010 15:08:53 -0800 reply

fricas

integrate(sqrt(x+sqrt(1+x^2))/x,x)

$\label{eq52}\begin{array}{@{}l} \displaystyle -{\log \left({{\sqrt{{\sqrt{{{x}^{2}}+ 1}}+ x}}+ 1}\right)}+{\log \left({{\sqrt{{\sqrt{{{x}^{2}}+ 1}}+ x}}- 1}\right)}- \ \ \displaystyle {2 \ {\arctan \left({\sqrt{{\sqrt{{{x}^{2}}+ 1}}+ x}}\right)}}+{2 \ {\sqrt{{\sqrt{{{x}^{2}}+ 1}}+ x}}}$

(52)

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

Exp Integral --pabjr, Tue, 14 Apr 2020 22:01:58 +0000 reply

fricas

)set output algebra on

r2:=integrate((1 - x)*%e^((-b*(x - 1))/(x + d)),x)

   (55)
                                                             - b x + b
                                                             ---------
           2                                2                  x + d
       (- x  + (- b d - b + 2)x + (- b + 1)d  + (- b + 2)d)%e
     + 
          2        2       2            2          b d + b   - b
       ((b  - 2 b)d  + (2 b  - 4 b)d + b  - 2 b)Ei(-------)%e
                                                    x + d
  /
     2

$\label{eq53}\frac{{{\left(-{{x}^{2}}+{{\left(-{b \ d}- b + 2 \right)}\ x}+{{\left(- b + 1 \right)}\ {{d}^{2}}}+{{\left(- b + 2 \right)}\ d}\right)}\ {{e}^{\frac{-{b \ x}+ b}{x + d}}}}+{{\left({{\left({{b}^{2}}-{2 \ b}\right)}\ {{d}^{2}}}+{{\left({2 \ {{b}^{2}}}-{4 \ b}\right)}\ d}+{{b}^{2}}-{2 \ b}\right)}\ {Ei \left({\frac{{b \ d}+ b}{x + d}}\right)}\ {{e}^{- b}}}}{2}$

(53)

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

fricas

unparse(r2::InputForm)

   (56)
  "(((-1)*x^2+((-1)*b*d+((-1)*b+2))*x+(((-1)*b+1)*d^2+((-1)*b+2)*d))*exp(((-1)*
  b*x+b)/(x+d))+((b^2+(-2)*b)*d^2+(2*b^2+(-4)*b)*d+(b^2+(-2)*b))*Ei((b*d+b)/(x+
  d))*exp((-1)*b))/2"

$\label{eq54}\verb#"(((-1)x^2+((-1)bd+((-1)b+2))x+(((-1)b+1)d^2+((-1)b+2)d))exp(((-1)bx+b)/(x+d))+((b^2+(-2)b)d^2+(2b^2+(-4)b)d+(b^2+(-2)b))Ei((bd+b)/(x+d))exp((-1)b))/2"#$

(54)

Type: String

Test

fricas

ii := integral(sin(x), x)

            x
          ++
   (57)   |   sin(%A)d%A
         ++

$\label{eq55}\int^{ \displaystyle x}{{\sin \left({\%A}\right)}\ {d \%A}}$

(55)

Type: Expression(Integer)

... --Mark, Thu, 30 Dec 2021 17:31:48 +0000 reply

fricas

integrate((7*x^13+10*x^8+4*x^7-7*x^6-4*x^3-4*x^2+3*x+3)/(x^14-2*x^8-2*x^7-2*x^4-4*x^3-x^2+2*x+1),x)

   (58)
            14      8      7      4      3    2
       log(x   - 2 x  - 2 x  - 2 x  - 4 x  - x  + 2 x + 1)
     + 
          +-+
         \|2
      *
         log
                      9      8      3      2        +-+    14      8      7
                (- 2 x  - 2 x  + 2 x  + 4 x  + 2 x)\|2  + x   - 2 x  - 2 x
              + 
                   4      3      2
                2 x  + 4 x  + 3 x  + 2 x + 1
           /
               14      8      7      4      3    2
              x   - 2 x  - 2 x  - 2 x  - 4 x  - x  + 2 x + 1
  /
     2

$\label{eq56}\frac{{\log \left({{{x}^{14}}-{2 \ {{x}^{8}}}-{2 \ {{x}^{7}}}-{2 \ {{x}^{4}}}-{4 \ {{x}^{3}}}-{{x}^{2}}+{2 \ x}+ 1}\right)}+{{\sqrt{2}}\ {\log \left({\frac{{{\left(-{2 \ {{x}^{9}}}-{2 \ {{x}^{8}}}+{2 \ {{x}^{3}}}+{4 \ {{x}^{2}}}+{2 \ x}\right)}\ {\sqrt{2}}}+{{x}^{14}}-{2 \ {{x}^{8}}}-{2 \ {{x}^{7}}}+{2 \ {{x}^{4}}}+{4 \ {{x}^{3}}}+{3 \ {{x}^{2}}}+{2 \ x}+ 1}{{{x}^{14}}-{2 \ {{x}^{8}}}-{2 \ {{x}^{7}}}-{2 \ {{x}^{4}}}-{4 \ {{x}^{3}}}-{{x}^{2}}+{2 \ x}+ 1}}\right)}}}{2}$

(56)

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