Optimal. Leaf size=24 \[ -\frac {1}{2 x^2}-\frac {1}{4} \tan ^{-1}\left (x^2\right )+\frac {1}{4} \tanh ^{-1}\left (x^2\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 = {281, 331, 304,
209, 212} \begin {gather*} -\frac {1}{4} \text {ArcTan}\left (x^2\right )-\frac {1}{2 x^2}+\frac {1}{4} \tanh ^{-1}\left (x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 281
Rule 304
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (1-x^8\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \left (1-x^4\right )} \, dx,x,x^2\right )\\ &=-\frac {1}{2 x^2}+\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,x^2\right )\\ &=-\frac {1}{2 x^2}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,x^2\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,x^2\right )\\ &=-\frac {1}{2 x^2}-\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 = 0.00, size = 38, normalized size = 1.58 \begin {gather*} -\frac {1}{2 x^2}+\frac {1}{4} \tan ^{-1}\left (\frac {1}{x^2}\right )-\frac {1}{8} \log \left (1-x^2\right )+\frac {1}{8} \log \left (1+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 33, normalized size = 1.38
method | result | size |
risch | \(-\frac {1}{2 x^{2}}-\frac {\ln \left (x^{2}-1\right )}{8}-\frac {\arctan \left (x^{2}\right )}{4}+\frac {\ln \left (x^{2}+1\right )}{8}\) | \(29\) |
default | \(-\frac {\ln \left (x +1\right )}{8}-\frac {\arctan \left (x^{2}\right )}{4}-\frac {\ln \left (x -1\right )}{8}-\frac {1}{2 x^{2}}+\frac {\ln \left (x^{2}+1\right )}{8}\) | \(33\) |
meijerg | \(\frac {\left (-1\right )^{\frac {1}{4}} \left (\frac {4 \left (-1\right )^{\frac {3}{4}}}{x^{2}}+\frac {x^{6} \left (-1\right )^{\frac {3}{4}} \left (\ln \left (1-\left (x^{8}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (x^{8}\right )^{\frac {1}{4}}\right )+2 \arctan \left (\left (x^{8}\right )^{\frac {1}{4}}\right )\right )}{\left (x^{8}\right )^{\frac {3}{4}}}\right )}{8}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 28, normalized size = 1.17 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs.
\(2 (18) = 36\).
time = 0.36, size = 37, normalized size = 1.54 \begin {gather*} -\frac {2 \, x^{2} \arctan \left (x^{2}\right ) - x^{2} \log \left (x^{2} + 1\right ) + x^{2} \log \left (x^{2} - 1\right ) + 4}{8 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 29, normalized size = 1.21 \begin {gather*} - \frac {\log {\left (x^{2} - 1 \right )}}{8} + \frac {\log {\left (x^{2} + 1 \right )}}{8} - \frac {\operatorname {atan}{\left (x^{2} \right )}}{4} - \frac {1}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.97, size = 29, normalized size = 1.21 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.05, size = 18, normalized size = 0.75 \begin {gather*} \frac {\mathrm {atanh}\left (x^2\right )}{4}-\frac {\mathrm {atan}\left (x^2\right )}{4}-\frac {1}{2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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