Optimal. Leaf size=36 \[ \frac{1}{4 x^3 \left (1-x^4\right )}-\frac{7}{12 x^3}+\frac{7}{8} \tan ^{-1}(x)+\frac{7}{8} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.0088647, antiderivative size = 36, normalized size of antiderivative = 1., 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 290
Rule 325
Rule 212
Rule 206
Rule 203
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*}
Mathematica [A] time = 0.0186844, 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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Maple [A] time = 0.015, size = 47, normalized size = 1.3 \begin{align*}{\frac{x}{8\,{x}^{2}+8}}+{\frac{7\,\arctan \left ( x \right ) }{8}}-{\frac{1}{3\,{x}^{3}}}-{\frac{1}{16+16\,x}}+{\frac{7\,\ln \left ( 1+x \right ) }{16}}-{\frac{1}{16\,x-16}}-{\frac{7\,\ln \left ( x-1 \right ) }{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.60184, size = 50, normalized size = 1.39 \begin{align*} -\frac{7 \, x^{4} - 4}{12 \,{\left (x^{7} - x^{3}\right )}} + \frac{7}{8} \, \arctan \left (x\right ) + \frac{7}{16} \, \log \left (x + 1\right ) - \frac{7}{16} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.46941, size = 157, normalized size = 4.36 \begin{align*} -\frac{28 \, x^{4} - 42 \,{\left (x^{7} - x^{3}\right )} \arctan \left (x\right ) - 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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.187099, size = 39, normalized size = 1.08 \begin{align*} - \frac{7 x^{4} - 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}{\left (x \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11398, size = 46, normalized size = 1.28 \begin{align*} -\frac{x}{4 \,{\left (x^{4} - 1\right )}} - \frac{1}{3 \, x^{3}} + \frac{7}{8} \, \arctan \left (x\right ) + \frac{7}{16} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{7}{16} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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