Optimal. Leaf size=43 \[ -\frac {9}{20 x^5}-\frac {9}{4 x}+\frac {1}{4 x^5 \left (1-x^4\right )}-\frac {9}{8} \tan ^{-1}(x)+\frac {9}{8} \tanh ^{-1}(x) \]
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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, 296, 331,
304, 209, 212} \begin {gather*} -\frac {9 \text {ArcTan}(x)}{8}-\frac {9}{20 x^5}+\frac {1}{4 x^5 \left (1-x^4\right )}-\frac {9}{4 x}+\frac {9}{8} \tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 209
Rule 212
Rule 296
Rule 304
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (1-2 x^4+x^8\right )} \, dx &=\int \frac {1}{x^6 \left (-1+x^4\right )^2} \, dx\\ &=\frac {1}{4 x^5 \left (1-x^4\right )}-\frac {9}{4} \int \frac {1}{x^6 \left (-1+x^4\right )} \, dx\\ &=-\frac {9}{20 x^5}+\frac {1}{4 x^5 \left (1-x^4\right )}-\frac {9}{4} \int \frac {1}{x^2 \left (-1+x^4\right )} \, dx\\ &=-\frac {9}{20 x^5}-\frac {9}{4 x}+\frac {1}{4 x^5 \left (1-x^4\right )}-\frac {9}{4} \int \frac {x^2}{-1+x^4} \, dx\\ &=-\frac {9}{20 x^5}-\frac {9}{4 x}+\frac {1}{4 x^5 \left (1-x^4\right )}+\frac {9}{8} \int \frac {1}{1-x^2} \, dx-\frac {9}{8} \int \frac {1}{1+x^2} \, dx\\ &=-\frac {9}{20 x^5}-\frac {9}{4 x}+\frac {1}{4 x^5 \left (1-x^4\right )}-\frac {9}{8} \tan ^{-1}(x)+\frac {9}{8} \tanh ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 51, normalized size = 1.19 \begin {gather*} -\frac {1}{5 x^5}-\frac {2}{x}-\frac {x^3}{4 \left (-1+x^4\right )}-\frac {9}{8} \tan ^{-1}(x)-\frac {9}{16} \log (1-x)+\frac {9}{16} \log (1+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 52, normalized size = 1.21
method | result | size |
risch | \(\frac {-\frac {9}{4} x^{8}+\frac {9}{5} x^{4}+\frac {1}{5}}{x^{5} \left (x^{4}-1\right )}-\frac {9 \arctan \left (x \right )}{8}+\frac {9 \ln \left (1+x \right )}{16}-\frac {9 \ln \left (-1+x \right )}{16}\) | \(41\) |
default | \(-\frac {1}{16 \left (-1+x \right )}-\frac {9 \ln \left (-1+x \right )}{16}-\frac {x}{8 \left (x^{2}+1\right )}-\frac {9 \arctan \left (x \right )}{8}-\frac {1}{5 x^{5}}-\frac {2}{x}-\frac {1}{16 \left (1+x \right )}+\frac {9 \ln \left (1+x \right )}{16}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 42, normalized size = 0.98 \begin {gather*} -\frac {45 \, x^{8} - 36 \, x^{4} - 4}{20 \, {\left (x^{9} - x^{5}\right )}} - \frac {9}{8} \, \arctan \left (x\right ) + \frac {9}{16} \, \log \left (x + 1\right ) - \frac {9}{16} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs.
\(2 (31) = 62\).
time = 0.34, size = 68, normalized size = 1.58 \begin {gather*} -\frac {180 \, x^{8} - 144 \, x^{4} + 90 \, {\left (x^{9} - x^{5}\right )} \arctan \left (x\right ) - 45 \, {\left (x^{9} - x^{5}\right )} \log \left (x + 1\right ) + 45 \, {\left (x^{9} - x^{5}\right )} \log \left (x - 1\right ) - 16}{80 \, {\left (x^{9} - x^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 44, normalized size = 1.02 \begin {gather*} - \frac {9 \log {\left (x - 1 \right )}}{16} + \frac {9 \log {\left (x + 1 \right )}}{16} - \frac {9 \operatorname {atan}{\left (x \right )}}{8} + \frac {- 45 x^{8} + 36 x^{4} + 4}{20 x^{9} - 20 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.41, size = 43, normalized size = 1.00 \begin {gather*} -\frac {x^{3}}{4 \, {\left (x^{4} - 1\right )}} - \frac {10 \, x^{4} + 1}{5 \, x^{5}} - \frac {9}{8} \, \arctan \left (x\right ) + \frac {9}{16} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {9}{16} \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 34, normalized size = 0.79 \begin {gather*} \frac {9\,\mathrm {atanh}\left (x\right )}{8}-\frac {9\,\mathrm {atan}\left (x\right )}{8}-\frac {-\frac {9\,x^8}{4}+\frac {9\,x^4}{5}+\frac {1}{5}}{x^5-x^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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