Optimal. Leaf size=36 \[ -\frac {5}{4 x}+\frac {1}{4 x \left (1-x^4\right )}-\frac {5}{8} \tan ^{-1}(x)+\frac {5}{8} \tanh ^{-1}(x) \]
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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, 296, 331,
304, 209, 212} \begin {gather*} -\frac {5 \text {ArcTan}(x)}{8}+\frac {1}{4 x \left (1-x^4\right )}-\frac {5}{4 x}+\frac {5}{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^2 \left (1-2 x^4+x^8\right )} \, dx &=\int \frac {1}{x^2 \left (-1+x^4\right )^2} \, dx\\ &=\frac {1}{4 x \left (1-x^4\right )}-\frac {5}{4} \int \frac {1}{x^2 \left (-1+x^4\right )} \, dx\\ &=-\frac {5}{4 x}+\frac {1}{4 x \left (1-x^4\right )}-\frac {5}{4} \int \frac {x^2}{-1+x^4} \, dx\\ &=-\frac {5}{4 x}+\frac {1}{4 x \left (1-x^4\right )}+\frac {5}{8} \int \frac {1}{1-x^2} \, dx-\frac {5}{8} \int \frac {1}{1+x^2} \, dx\\ &=-\frac {5}{4 x}+\frac {1}{4 x \left (1-x^4\right )}-\frac {5}{8} \tan ^{-1}(x)+\frac {5}{8} \tanh ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 40, normalized size = 1.11 \begin {gather*} \frac {1}{16} \left (-\frac {16}{x}-\frac {4 x^3}{-1+x^4}-10 \tan ^{-1}(x)-5 \log (1-x)+5 \log (1+x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 47, normalized size = 1.31
method | result | size |
risch | \(\frac {-\frac {5 x^{4}}{4}+1}{x \left (x^{4}-1\right )}+\frac {5 \ln \left (1+x \right )}{16}-\frac {5 \arctan \left (x \right )}{8}-\frac {5 \ln \left (-1+x \right )}{16}\) | \(36\) |
default | \(-\frac {1}{16 \left (-1+x \right )}-\frac {5 \ln \left (-1+x \right )}{16}-\frac {x}{8 \left (x^{2}+1\right )}-\frac {5 \arctan \left (x \right )}{8}-\frac {1}{x}-\frac {1}{16 \left (1+x \right )}+\frac {5 \ln \left (1+x \right )}{16}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 35, normalized size = 0.97 \begin {gather*} -\frac {5 \, x^{4} - 4}{4 \, {\left (x^{5} - x\right )}} - \frac {5}{8} \, \arctan \left (x\right ) + \frac {5}{16} \, \log \left (x + 1\right ) - \frac {5}{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. 55 vs.
\(2 (26) = 52\).
time = 0.35, size = 55, normalized size = 1.53 \begin {gather*} -\frac {20 \, x^{4} + 10 \, {\left (x^{5} - x\right )} \arctan \left (x\right ) - 5 \, {\left (x^{5} - x\right )} \log \left (x + 1\right ) + 5 \, {\left (x^{5} - x\right )} \log \left (x - 1\right ) - 16}{16 \, {\left (x^{5} - x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 37, normalized size = 1.03 \begin {gather*} \frac {4 - 5 x^{4}}{4 x^{5} - 4 x} - \frac {5 \log {\left (x - 1 \right )}}{16} + \frac {5 \log {\left (x + 1 \right )}}{16} - \frac {5 \operatorname {atan}{\left (x \right )}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.51, size = 37, normalized size = 1.03 \begin {gather*} -\frac {5 \, x^{4} - 4}{4 \, {\left (x^{5} - x\right )}} - \frac {5}{8} \, \arctan \left (x\right ) + \frac {5}{16} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {5}{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 = 26, normalized size = 0.72 \begin {gather*} \frac {5\,\mathrm {atanh}\left (x\right )}{8}-\frac {5\,\mathrm {atan}\left (x\right )}{8}+\frac {\frac {5\,x^4}{4}-1}{x-x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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