Optimal. Leaf size=16 \[ \frac {1}{4} \tanh ^{-1}\left (x^4\right )-\frac {1}{4 x^4} \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {275, 325, 206} \[ \frac {1}{4} \tanh ^{-1}\left (x^4\right )-\frac {1}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 206
Rule 275
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (1-x^8\right )} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-x^2\right )} \, dx,x,x^4\right )\\ &=-\frac {1}{4 x^4}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,x^4\right )\\ &=-\frac {1}{4 x^4}+\frac {1}{4} \tanh ^{-1}\left (x^4\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 30, normalized size = 1.88 \[ -\frac {1}{4 x^4}-\frac {1}{8} \log \left (1-x^4\right )+\frac {1}{8} \log \left (x^4+1\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 28, normalized size = 1.75 \[ \frac {x^{4} \log \left (x^{4} + 1\right ) - x^{4} \log \left (x^{4} - 1\right ) - 2}{8 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 23, normalized size = 1.44 \[ -\frac {1}{4 \, x^{4}} + \frac {1}{8} \, \log \left (x^{4} + 1\right ) - \frac {1}{8} \, \log \left ({\left | x^{4} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 35, normalized size = 2.19 \[ -\frac {\ln \left (x -1\right )}{8}-\frac {\ln \left (x +1\right )}{8}-\frac {\ln \left (x^{2}+1\right )}{8}+\frac {\ln \left (x^{4}+1\right )}{8}-\frac {1}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 22, normalized size = 1.38 \[ -\frac {1}{4 \, x^{4}} + \frac {1}{8} \, \log \left (x^{4} + 1\right ) - \frac {1}{8} \, \log \left (x^{4} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.05, size = 12, normalized size = 0.75 \[ \frac {\mathrm {atanh}\left (x^4\right )}{4}-\frac {1}{4\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.27, size = 22, normalized size = 1.38 \[ - \frac {\log {\left (x^{4} - 1 \right )}}{8} + \frac {\log {\left (x^{4} + 1 \right )}}{8} - \frac {1}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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