3.1469 \(\int \frac{x^9}{1-x^8} \, dx\)

Optimal. Leaf size=24 \[ -\frac{x^2}{2}+\frac{1}{4} \tan ^{-1}\left (x^2\right )+\frac{1}{4} \tanh ^{-1}\left (x^2\right ) \]

[Out]

-x^2/2 + ArcTan[x^2]/4 + ArcTanh[x^2]/4

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Rubi [A]  time = 0.033891, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ -\frac{x^2}{2}+\frac{1}{4} \tan ^{-1}\left (x^2\right )+\frac{1}{4} \tanh ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^9/(1 - x^8),x]

[Out]

-x^2/2 + ArcTan[x^2]/4 + ArcTanh[x^2]/4

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Rubi in Sympy [A]  time = 5.43591, size = 17, normalized size = 0.71 \[ - \frac{x^{2}}{2} + \frac{\operatorname{atan}{\left (x^{2} \right )}}{4} + \frac{\operatorname{atanh}{\left (x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(-x**8+1),x)

[Out]

-x**2/2 + atan(x**2)/4 + atanh(x**2)/4

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Mathematica [A]  time = 0.0105908, size = 38, normalized size = 1.58 \[ -\frac{x^2}{2}-\frac{1}{8} \log \left (1-x^2\right )+\frac{1}{8} \log \left (x^2+1\right )-\frac{1}{4} \tan ^{-1}\left (\frac{1}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(1 - x^8),x]

[Out]

-x^2/2 - ArcTan[x^(-2)]/4 - Log[1 - x^2]/8 + Log[1 + x^2]/8

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Maple [A]  time = 0.009, size = 33, normalized size = 1.4 \[ -{\frac{{x}^{2}}{2}}-{\frac{\ln \left ( -1+x \right ) }{8}}-{\frac{\ln \left ( 1+x \right ) }{8}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{8}}+{\frac{\arctan \left ({x}^{2} \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(-x^8+1),x)

[Out]

-1/2*x^2-1/8*ln(-1+x)-1/8*ln(1+x)+1/8*ln(x^2+1)+1/4*arctan(x^2)

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Maxima [A]  time = 1.58964, size = 38, normalized size = 1.58 \[ -\frac{1}{2} \, x^{2} + \frac{1}{4} \, \arctan \left (x^{2}\right ) + \frac{1}{8} \, \log \left (x^{2} + 1\right ) - \frac{1}{8} \, \log \left (x^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^9/(x^8 - 1),x, algorithm="maxima")

[Out]

-1/2*x^2 + 1/4*arctan(x^2) + 1/8*log(x^2 + 1) - 1/8*log(x^2 - 1)

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Fricas [A]  time = 0.229462, size = 38, normalized size = 1.58 \[ -\frac{1}{2} \, x^{2} + \frac{1}{4} \, \arctan \left (x^{2}\right ) + \frac{1}{8} \, \log \left (x^{2} + 1\right ) - \frac{1}{8} \, \log \left (x^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^9/(x^8 - 1),x, algorithm="fricas")

[Out]

-1/2*x^2 + 1/4*arctan(x^2) + 1/8*log(x^2 + 1) - 1/8*log(x^2 - 1)

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Sympy [A]  time = 0.411876, size = 27, normalized size = 1.12 \[ - \frac{x^{2}}{2} - \frac{\log{\left (x^{2} - 1 \right )}}{8} + \frac{\log{\left (x^{2} + 1 \right )}}{8} + \frac{\operatorname{atan}{\left (x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(-x**8+1),x)

[Out]

-x**2/2 - log(x**2 - 1)/8 + log(x**2 + 1)/8 + atan(x**2)/4

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GIAC/XCAS [A]  time = 0.221796, size = 39, normalized size = 1.62 \[ -\frac{1}{2} \, x^{2} + \frac{1}{4} \, \arctan \left (x^{2}\right ) + \frac{1}{8} \,{\rm ln}\left (x^{2} + 1\right ) - \frac{1}{8} \,{\rm ln}\left ({\left | x^{2} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^9/(x^8 - 1),x, algorithm="giac")

[Out]

-1/2*x^2 + 1/4*arctan(x^2) + 1/8*ln(x^2 + 1) - 1/8*ln(abs(x^2 - 1))