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 = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 203
Rule 206
Rule 275
Rule 298
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.55, size = 23, normalized size = 1.35 \[ -\frac {1}{4} \, \arctan \left (x^{2}\right ) + \frac {1}{8} \, \log \left (x^{2} + 1\right ) - \frac {1}{8} \, \log \left (x^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 55, normalized size = 3.24 \[ \frac {1}{4} \, \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) - \frac {1}{4} \, \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) + \frac {1}{8} \, \log \left (x^{2} + 1\right ) - \frac {1}{8} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{8} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 28, normalized size = 1.65 \[ -\frac {\arctan \left (x^{2}\right )}{4}-\frac {\ln \left (x -1\right )}{8}-\frac {\ln \left (x +1\right )}{8}+\frac {\ln \left (x^{2}+1\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.41, size = 23, normalized size = 1.35 \[ -\frac {1}{4} \, \arctan \left (x^{2}\right ) + \frac {1}{8} \, \log \left (x^{2} + 1\right ) - \frac {1}{8} \, \log \left (x^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 13, normalized size = 0.76 \[ \frac {\mathrm {atanh}\left (x^2\right )}{4}-\frac {\mathrm {atan}\left (x^2\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 22, normalized size = 1.29 \[ - \frac {\log {\left (x^{2} - 1 \right )}}{8} + \frac {\log {\left (x^{2} + 1 \right )}}{8} - \frac {\operatorname {atan}{\left (x^{2} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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