Optimal. Leaf size=32 \[ -\frac {3}{4 x^2}+\frac {1}{4 x^2 \left (1-x^4\right )}+\frac {3}{4} \tanh ^{-1}\left (x^2\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {28, 281, 296,
331, 213} \begin {gather*} -\frac {3}{4 x^2}+\frac {3}{4} \tanh ^{-1}\left (x^2\right )+\frac {1}{4 x^2 \left (1-x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 213
Rule 281
Rule 296
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (1-2 x^4+x^8\right )} \, dx &=\int \frac {1}{x^3 \left (-1+x^4\right )^2} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \left (-1+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {1}{4 x^2 \left (1-x^4\right )}-\frac {3}{4} \text {Subst}\left (\int \frac {1}{x^2 \left (-1+x^2\right )} \, dx,x,x^2\right )\\ &=-\frac {3}{4 x^2}+\frac {1}{4 x^2 \left (1-x^4\right )}-\frac {3}{4} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,x^2\right )\\ &=-\frac {3}{4 x^2}+\frac {1}{4 x^2 \left (1-x^4\right )}+\frac {3}{4} \tanh ^{-1}\left (x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 41, normalized size = 1.28 \begin {gather*} \frac {1}{8} \left (\frac {4-6 x^4}{x^2 \left (-1+x^4\right )}-3 \log \left (1-x^2\right )+3 \log \left (1+x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 50, normalized size = 1.56
method | result | size |
risch | \(\frac {\frac {1}{2}-\frac {3 x^{4}}{4}}{x^{2} \left (x^{4}-1\right )}-\frac {3 \ln \left (x^{2}-1\right )}{8}+\frac {3 \ln \left (x^{2}+1\right )}{8}\) | \(36\) |
norman | \(\frac {\frac {1}{2}-\frac {3 x^{4}}{4}}{x^{2} \left (x^{4}-1\right )}-\frac {3 \ln \left (-1+x \right )}{8}-\frac {3 \ln \left (1+x \right )}{8}+\frac {3 \ln \left (x^{2}+1\right )}{8}\) | \(40\) |
default | \(-\frac {1}{16 \left (-1+x \right )}-\frac {3 \ln \left (-1+x \right )}{8}-\frac {1}{8 \left (x^{2}+1\right )}+\frac {3 \ln \left (x^{2}+1\right )}{8}-\frac {1}{2 x^{2}}+\frac {1}{16+16 x}-\frac {3 \ln \left (1+x \right )}{8}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 37, normalized size = 1.16 \begin {gather*} -\frac {3 \, x^{4} - 2}{4 \, {\left (x^{6} - x^{2}\right )}} + \frac {3}{8} \, \log \left (x^{2} + 1\right ) - \frac {3}{8} \, \log \left (x^{2} - 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. 54 vs.
\(2 (24) = 48\).
time = 0.34, size = 54, normalized size = 1.69 \begin {gather*} -\frac {6 \, x^{4} - 3 \, {\left (x^{6} - x^{2}\right )} \log \left (x^{2} + 1\right ) + 3 \, {\left (x^{6} - x^{2}\right )} \log \left (x^{2} - 1\right ) - 4}{8 \, {\left (x^{6} - x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 36, normalized size = 1.12 \begin {gather*} \frac {2 - 3 x^{4}}{4 x^{6} - 4 x^{2}} - \frac {3 \log {\left (x^{2} - 1 \right )}}{8} + \frac {3 \log {\left (x^{2} + 1 \right )}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.85, size = 38, normalized size = 1.19 \begin {gather*} -\frac {3 \, x^{4} - 2}{4 \, {\left (x^{6} - x^{2}\right )}} + \frac {3}{8} \, \log \left (x^{2} + 1\right ) - \frac {3}{8} \, \log \left ({\left | x^{2} - 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 = 0.81 \begin {gather*} \frac {3\,\mathrm {atanh}\left (x^2\right )}{4}+\frac {\frac {3\,x^4}{4}-\frac {1}{2}}{x^2-x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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