Optimal. Leaf size=27 \[ \frac {x}{4 \left (1-x^4\right )}-\frac {1}{8} \tan ^{-1}(x)-\frac {1}{8} \tanh ^{-1}(x) \]
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Rubi [A]
time = 0.00, antiderivative size = 27, 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, 294, 218,
212, 209} \begin {gather*} -\frac {\text {ArcTan}(x)}{8}+\frac {x}{4 \left (1-x^4\right )}-\frac {1}{8} \tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 209
Rule 212
Rule 218
Rule 294
Rubi steps
\begin {align*} \int \frac {x^4}{1-2 x^4+x^8} \, dx &=\int \frac {x^4}{\left (-1+x^4\right )^2} \, dx\\ &=\frac {x}{4 \left (1-x^4\right )}+\frac {1}{4} \int \frac {1}{-1+x^4} \, dx\\ &=\frac {x}{4 \left (1-x^4\right )}-\frac {1}{8} \int \frac {1}{1-x^2} \, dx-\frac {1}{8} \int \frac {1}{1+x^2} \, dx\\ &=\frac {x}{4 \left (1-x^4\right )}-\frac {1}{8} \tan ^{-1}(x)-\frac {1}{8} \tanh ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 31, normalized size = 1.15 \begin {gather*} \frac {1}{16} \left (-\frac {4 x}{-1+x^4}-2 \tan ^{-1}(x)+\log (1-x)-\log (1+x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 42, normalized size = 1.56
method | result | size |
risch | \(-\frac {x}{4 \left (x^{4}-1\right )}-\frac {\ln \left (1+x \right )}{16}+\frac {\ln \left (-1+x \right )}{16}-\frac {\arctan \left (x \right )}{8}\) | \(28\) |
default | \(-\frac {1}{16 \left (-1+x \right )}+\frac {\ln \left (-1+x \right )}{16}+\frac {x}{8 x^{2}+8}-\frac {\arctan \left (x \right )}{8}-\frac {1}{16 \left (1+x \right )}-\frac {\ln \left (1+x \right )}{16}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 27, normalized size = 1.00 \begin {gather*} -\frac {x}{4 \, {\left (x^{4} - 1\right )}} - \frac {1}{8} \, \arctan \left (x\right ) - \frac {1}{16} \, \log \left (x + 1\right ) + \frac {1}{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. 43 vs.
\(2 (19) = 38\).
time = 0.34, size = 43, normalized size = 1.59 \begin {gather*} -\frac {2 \, {\left (x^{4} - 1\right )} \arctan \left (x\right ) + {\left (x^{4} - 1\right )} \log \left (x + 1\right ) - {\left (x^{4} - 1\right )} \log \left (x - 1\right ) + 4 \, x}{16 \, {\left (x^{4} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 26, normalized size = 0.96 \begin {gather*} - \frac {x}{4 x^{4} - 4} + \frac {\log {\left (x - 1 \right )}}{16} - \frac {\log {\left (x + 1 \right )}}{16} - \frac {\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 = 3.10, size = 29, normalized size = 1.07 \begin {gather*} -\frac {x}{4 \, {\left (x^{4} - 1\right )}} - \frac {1}{8} \, \arctan \left (x\right ) - \frac {1}{16} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{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 = 0.03, size = 21, normalized size = 0.78 \begin {gather*} -\frac {\mathrm {atan}\left (x\right )}{8}-\frac {\mathrm {atanh}\left (x\right )}{8}-\frac {x}{4\,\left (x^4-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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