Optimal. Leaf size=34 \[ \frac {5 x}{4}+\frac {x^5}{4 \left (1-x^4\right )}-\frac {5}{8} \tan ^{-1}(x)-\frac {5}{8} \tanh ^{-1}(x) \]
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Rubi [A]
time = 0.01, antiderivative size = 34, 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, 294, 327,
218, 212, 209} \begin {gather*} -\frac {5 \text {ArcTan}(x)}{8}+\frac {x^5}{4 \left (1-x^4\right )}+\frac {5 x}{4}-\frac {5}{8} \tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 209
Rule 212
Rule 218
Rule 294
Rule 327
Rubi steps
\begin {align*} \int \frac {x^8}{1-2 x^4+x^8} \, dx &=\int \frac {x^8}{\left (-1+x^4\right )^2} \, dx\\ &=\frac {x^5}{4 \left (1-x^4\right )}+\frac {5}{4} \int \frac {x^4}{-1+x^4} \, dx\\ &=\frac {5 x}{4}+\frac {x^5}{4 \left (1-x^4\right )}+\frac {5}{4} \int \frac {1}{-1+x^4} \, dx\\ &=\frac {5 x}{4}+\frac {x^5}{4 \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 x}{4}+\frac {x^5}{4 \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 = 38, normalized size = 1.12 \begin {gather*} x-\frac {x}{4 \left (-1+x^4\right )}-\frac {5}{8} \tan ^{-1}(x)+\frac {5}{16} \log (1-x)-\frac {5}{16} \log (1+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 43, normalized size = 1.26
method | result | size |
risch | \(x -\frac {x}{4 \left (x^{4}-1\right )}+\frac {5 \ln \left (-1+x \right )}{16}-\frac {5 \arctan \left (x \right )}{8}-\frac {5 \ln \left (1+x \right )}{16}\) | \(29\) |
default | \(x -\frac {1}{16 \left (-1+x \right )}+\frac {5 \ln \left (-1+x \right )}{16}+\frac {x}{8 x^{2}+8}-\frac {5 \arctan \left (x \right )}{8}-\frac {1}{16 \left (1+x \right )}-\frac {5 \ln \left (1+x \right )}{16}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 28, normalized size = 0.82 \begin {gather*} x - \frac {x}{4 \, {\left (x^{4} - 1\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. 49 vs.
\(2 (24) = 48\).
time = 0.33, size = 49, normalized size = 1.44 \begin {gather*} \frac {16 \, x^{5} - 10 \, {\left (x^{4} - 1\right )} \arctan \left (x\right ) - 5 \, {\left (x^{4} - 1\right )} \log \left (x + 1\right ) + 5 \, {\left (x^{4} - 1\right )} \log \left (x - 1\right ) - 20 \, 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 = 32, normalized size = 0.94 \begin {gather*} x - \frac {x}{4 x^{4} - 4} + \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 = 3.65, size = 30, normalized size = 0.88 \begin {gather*} x - \frac {x}{4 \, {\left (x^{4} - 1\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 = 1.29, size = 26, normalized size = 0.76 \begin {gather*} x-\frac {5\,\mathrm {atan}\left (x\right )}{8}-\frac {x}{4\,\left (x^4-1\right )}+\frac {\mathrm {atan}\left (x\,1{}\mathrm {i}\right )\,5{}\mathrm {i}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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