Optimal. Leaf size=17 \[ -\frac {1}{4} \tan ^{-1}\left (x^2\right )+\frac {1}{4} \tanh ^{-1}\left (x^2\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 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 = {281, 304, 209,
212} \begin {gather*} \frac {1}{4} \tanh ^{-1}\left (x^2\right )-\frac {\text {ArcTan}\left (x^2\right )}{4} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 281
Rule 304
Rubi steps
\begin {align*} \int \frac {x^5}{1-x^8} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,x^2\right )\\ &=\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}{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 \begin {gather*} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. \(27\) vs.
\(2(13)=26\).
time = 0.21, size = 28, normalized size = 1.65
method | result | size |
risch | \(-\frac {\ln \left (x^{2}-1\right )}{8}-\frac {\arctan \left (x^{2}\right )}{4}+\frac {\ln \left (x^{2}+1\right )}{8}\) | \(24\) |
default | \(-\frac {\ln \left (x -1\right )}{8}-\frac {\ln \left (x +1\right )}{8}+\frac {\ln \left (x^{2}+1\right )}{8}-\frac {\arctan \left (x^{2}\right )}{4}\) | \(28\) |
meijerg | \(-\frac {x^{6} \left (\ln \left (1-\left (x^{8}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (x^{8}\right )^{\frac {1}{4}}\right )+2 \arctan \left (\left (x^{8}\right )^{\frac {1}{4}}\right )\right )}{8 \left (x^{8}\right )^{\frac {3}{4}}}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 23, normalized size = 1.35 \begin {gather*} -\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 [A]
time = 0.37, size = 23, normalized size = 1.35 \begin {gather*} -\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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Sympy [A]
time = 0.06, size = 22, normalized size = 1.29 \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 55 vs.
\(2 (13) = 26\).
time = 1.43, size = 55, normalized size = 3.24 \begin {gather*} \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 {gather*}
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 \begin {gather*} \frac {\mathrm {atanh}\left (x^2\right )}{4}-\frac {\mathrm {atan}\left (x^2\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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