Optimal. Leaf size=43 \[ -\frac {11}{28 x^7}-\frac {11}{12 x^3}+\frac {1}{4 x^7 \left (1-x^4\right )}+\frac {11}{8} \tan ^{-1}(x)+\frac {11}{8} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 203
Rule 206
Rule 212
Rule 290
Rule 325
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*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 1.00 \[ \frac {1}{336} \left (-\frac {48}{x^7}-\frac {84 x}{x^4-1}-\frac {224}{x^3}-231 \log (1-x)+231 \log (x+1)+462 \tan ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.90, size = 68, normalized size = 1.58 \[ -\frac {308 \, x^{8} - 176 \, x^{4} - 462 \, {\left (x^{11} - x^{7}\right )} \arctan \relax (x) - 231 \, {\left (x^{11} - x^{7}\right )} \log \left (x + 1\right ) + 231 \, {\left (x^{11} - x^{7}\right )} \log \left (x - 1\right ) - 48}{336 \, {\left (x^{11} - x^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 41, normalized size = 0.95 \[ -\frac {x}{4 \, {\left (x^{4} - 1\right )}} - \frac {14 \, x^{4} + 3}{21 \, x^{7}} + \frac {11}{8} \, \arctan \relax (x) + \frac {11}{16} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {11}{16} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 52, normalized size = 1.21 \[ \frac {x}{8 x^{2}+8}+\frac {11 \arctan \relax (x )}{8}-\frac {11 \ln \left (x -1\right )}{16}+\frac {11 \ln \left (x +1\right )}{16}-\frac {2}{3 x^{3}}-\frac {1}{7 x^{7}}-\frac {1}{16 \left (x +1\right )}-\frac {1}{16 \left (x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.94, size = 42, normalized size = 0.98 \[ -\frac {77 \, x^{8} - 44 \, x^{4} - 12}{84 \, {\left (x^{11} - x^{7}\right )}} + \frac {11}{8} \, \arctan \relax (x) + \frac {11}{16} \, \log \left (x + 1\right ) - \frac {11}{16} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 34, normalized size = 0.79 \[ \frac {11\,\mathrm {atan}\relax (x)}{8}+\frac {11\,\mathrm {atanh}\relax (x)}{8}-\frac {-\frac {11\,x^8}{12}+\frac {11\,x^4}{21}+\frac {1}{7}}{x^7-x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 44, normalized size = 1.02 \[ - \frac {11 \log {\left (x - 1 \right )}}{16} + \frac {11 \log {\left (x + 1 \right )}}{16} + \frac {11 \operatorname {atan}{\relax (x )}}{8} + \frac {- 77 x^{8} + 44 x^{4} + 12}{84 x^{11} - 84 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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