Optimal. Leaf size=24 \[ -\frac{x^2}{2}+\frac{1}{4} \tan ^{-1}\left (x^2\right )+\frac{1}{4} \tanh ^{-1}\left (x^2\right ) \]
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Rubi [A] time = 0.0111475, 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, Rules used = {275, 321, 212, 206, 203} \[ -\frac{x^2}{2}+\frac{1}{4} \tan ^{-1}\left (x^2\right )+\frac{1}{4} \tanh ^{-1}\left (x^2\right ) \]
Antiderivative was successfully verified.
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Rule 275
Rule 321
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{x^9}{1-x^8} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{1-x^4} \, dx,x,x^2\right )\\ &=-\frac{x^2}{2}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-x^4} \, dx,x,x^2\right )\\ &=-\frac{x^2}{2}+\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{x^2}{2}+\frac{1}{4} \tan ^{-1}\left (x^2\right )+\frac{1}{4} \tanh ^{-1}\left (x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0058992, 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.
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Maple [A] time = 0.007, size = 33, normalized size = 1.4 \begin{align*} -{\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43611, size = 38, normalized size = 1.58 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27257, size = 89, normalized size = 3.71 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.130982, size = 27, normalized size = 1.12 \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1682, size = 39, normalized size = 1.62 \begin{align*} -\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 ({\left | x^{2} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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