Optimal. Leaf size=36 \[ -\frac {7}{12 x^3}+\frac {1}{4 x^3 \left (1-x^4\right )}+\frac {7}{8} \tan ^{-1}(x)+\frac {7}{8} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {28, 290, 325, 212, 206, 203} \[ \frac {1}{4 x^3 \left (1-x^4\right )}-\frac {7}{12 x^3}+\frac {7}{8} \tan ^{-1}(x)+\frac {7}{8} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 28
Rule 203
Rule 206
Rule 212
Rule 290
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (1-2 x^4+x^8\right )} \, dx &=\int \frac {1}{x^4 \left (-1+x^4\right )^2} \, dx\\ &=\frac {1}{4 x^3 \left (1-x^4\right )}-\frac {7}{4} \int \frac {1}{x^4 \left (-1+x^4\right )} \, dx\\ &=-\frac {7}{12 x^3}+\frac {1}{4 x^3 \left (1-x^4\right )}-\frac {7}{4} \int \frac {1}{-1+x^4} \, dx\\ &=-\frac {7}{12 x^3}+\frac {1}{4 x^3 \left (1-x^4\right )}+\frac {7}{8} \int \frac {1}{1-x^2} \, dx+\frac {7}{8} \int \frac {1}{1+x^2} \, dx\\ &=-\frac {7}{12 x^3}+\frac {1}{4 x^3 \left (1-x^4\right )}+\frac {7}{8} \tan ^{-1}(x)+\frac {7}{8} \tanh ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 1.06 \[ \frac {1}{48} \left (-\frac {12 x}{x^4-1}-\frac {16}{x^3}-21 \log (1-x)+21 \log (x+1)+42 \tan ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.73, size = 63, normalized size = 1.75 \[ -\frac {28 \, x^{4} - 42 \, {\left (x^{7} - x^{3}\right )} \arctan \relax (x) - 21 \, {\left (x^{7} - x^{3}\right )} \log \left (x + 1\right ) + 21 \, {\left (x^{7} - x^{3}\right )} \log \left (x - 1\right ) - 16}{48 \, {\left (x^{7} - x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 34, normalized size = 0.94 \[ -\frac {x}{4 \, {\left (x^{4} - 1\right )}} - \frac {1}{3 \, x^{3}} + \frac {7}{8} \, \arctan \relax (x) + \frac {7}{16} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {7}{16} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 47, normalized size = 1.31 \[ \frac {x}{8 x^{2}+8}+\frac {7 \arctan \relax (x )}{8}-\frac {7 \ln \left (x -1\right )}{16}+\frac {7 \ln \left (x +1\right )}{16}-\frac {1}{3 x^{3}}-\frac {1}{16 \left (x +1\right )}-\frac {1}{16 \left (x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.98, size = 37, normalized size = 1.03 \[ -\frac {7 \, x^{4} - 4}{12 \, {\left (x^{7} - x^{3}\right )}} + \frac {7}{8} \, \arctan \relax (x) + \frac {7}{16} \, \log \left (x + 1\right ) - \frac {7}{16} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.29, size = 28, normalized size = 0.78 \[ \frac {7\,\mathrm {atan}\relax (x)}{8}+\frac {7\,\mathrm {atanh}\relax (x)}{8}+\frac {\frac {7\,x^4}{12}-\frac {1}{3}}{x^3-x^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 39, normalized size = 1.08 \[ \frac {4 - 7 x^{4}}{12 x^{7} - 12 x^{3}} - \frac {7 \log {\left (x - 1 \right )}}{16} + \frac {7 \log {\left (x + 1 \right )}}{16} + \frac {7 \operatorname {atan}{\relax (x )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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