Optimal. Leaf size=43 \[ \frac{1}{4 x^7 \left (1-x^4\right )}-\frac{11}{12 x^3}-\frac{11}{28 x^7}+\frac{11}{8} \tan ^{-1}(x)+\frac{11}{8} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.0120157, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{1}{4 x^7 \left (1-x^4\right )}-\frac{11}{12 x^3}-\frac{11}{28 x^7}+\frac{11}{8} \tan ^{-1}(x)+\frac{11}{8} \tanh ^{-1}(x) \]
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^8 \left (1-2 x^4+x^8\right )} \, dx &=\int \frac{1}{x^8 \left (-1+x^4\right )^2} \, dx\\ &=\frac{1}{4 x^7 \left (1-x^4\right )}-\frac{11}{4} \int \frac{1}{x^8 \left (-1+x^4\right )} \, dx\\ &=-\frac{11}{28 x^7}+\frac{1}{4 x^7 \left (1-x^4\right )}-\frac{11}{4} \int \frac{1}{x^4 \left (-1+x^4\right )} \, dx\\ &=-\frac{11}{28 x^7}-\frac{11}{12 x^3}+\frac{1}{4 x^7 \left (1-x^4\right )}-\frac{11}{4} \int \frac{1}{-1+x^4} \, dx\\ &=-\frac{11}{28 x^7}-\frac{11}{12 x^3}+\frac{1}{4 x^7 \left (1-x^4\right )}+\frac{11}{8} \int \frac{1}{1-x^2} \, dx+\frac{11}{8} \int \frac{1}{1+x^2} \, dx\\ &=-\frac{11}{28 x^7}-\frac{11}{12 x^3}+\frac{1}{4 x^7 \left (1-x^4\right )}+\frac{11}{8} \tan ^{-1}(x)+\frac{11}{8} \tanh ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0194211, size = 43, normalized size = 1. \[ \frac{1}{336} \left (-\frac{84 x}{x^4-1}-\frac{224}{x^3}-\frac{48}{x^7}-231 \log (1-x)+231 \log (x+1)+462 \tan ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 52, normalized size = 1.2 \begin{align*}{\frac{x}{8\,{x}^{2}+8}}+{\frac{11\,\arctan \left ( x \right ) }{8}}-{\frac{1}{7\,{x}^{7}}}-{\frac{2}{3\,{x}^{3}}}-{\frac{1}{16+16\,x}}+{\frac{11\,\ln \left ( 1+x \right ) }{16}}-{\frac{1}{16\,x-16}}-{\frac{11\,\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 = 1.57545, size = 57, normalized size = 1.33 \begin{align*} -\frac{77 \, x^{8} - 44 \, x^{4} - 12}{84 \,{\left (x^{11} - x^{7}\right )}} + \frac{11}{8} \, \arctan \left (x\right ) + \frac{11}{16} \, \log \left (x + 1\right ) - \frac{11}{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 = 1.51699, size = 182, normalized size = 4.23 \begin{align*} -\frac{308 \, x^{8} - 176 \, x^{4} - 462 \,{\left (x^{11} - x^{7}\right )} \arctan \left (x\right ) - 231 \,{\left (x^{11} - x^{7}\right )} \log \left (x + 1\right ) + 231 \,{\left (x^{11} - x^{7}\right )} \log \left (x - 1\right ) - 48}{336 \,{\left (x^{11} - x^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.229206, size = 44, normalized size = 1.02 \begin{align*} - \frac{11 \log{\left (x - 1 \right )}}{16} + \frac{11 \log{\left (x + 1 \right )}}{16} + \frac{11 \operatorname{atan}{\left (x \right )}}{8} - \frac{77 x^{8} - 44 x^{4} - 12}{84 x^{11} - 84 x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07282, size = 55, normalized size = 1.28 \begin{align*} -\frac{x}{4 \,{\left (x^{4} - 1\right )}} - \frac{14 \, x^{4} + 3}{21 \, x^{7}} + \frac{11}{8} \, \arctan \left (x\right ) + \frac{11}{16} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{11}{16} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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