Optimal. Leaf size=17 \[ \frac{1}{4} \tanh ^{-1}\left (x^2\right )-\frac{1}{4} \tan ^{-1}\left (x^2\right ) \]
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Rubi [A] time = 0.0088476, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {275, 298, 203, 206} \[ \frac{1}{4} \tanh ^{-1}\left (x^2\right )-\frac{1}{4} \tan ^{-1}\left (x^2\right ) \]
Antiderivative was successfully verified.
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Rule 275
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{x^5}{1-x^8} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{1-x^4} \, dx,x,x^2\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,x^2\right )-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,x^2\right )\\ &=-\frac{1}{4} \tan ^{-1}\left (x^2\right )+\frac{1}{4} \tanh ^{-1}\left (x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0049834, size = 31, normalized size = 1.82 \[ -\frac{1}{8} \log \left (1-x^2\right )+\frac{1}{8} \log \left (x^2+1\right )+\frac{1}{4} \tan ^{-1}\left (\frac{1}{x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.005, size = 28, normalized size = 1.7 \begin{align*} -{\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54891, size = 31, normalized size = 1.82 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26832, size = 76, normalized size = 4.47 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.132457, size = 22, normalized size = 1.29 \begin{align*} - \frac{\log{\left (x^{2} - 1 \right )}}{8} + \frac{\log{\left (x^{2} + 1 \right )}}{8} - \frac{\operatorname{atan}{\left (x^{2} \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13167, size = 74, normalized size = 4.35 \begin{align*} \frac{1}{4} \, \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) - \frac{1}{4} \, \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{1}{8} \, \log \left (x^{2} + 1\right ) - \frac{1}{8} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{8} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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