Optimal. Leaf size=43 \[ -\frac {11}{28 x^7}-\frac {11}{12 x^3}+\frac {1}{4 x^7 \left (1-x^4\right )}+\frac {11}{8} \tan ^{-1}(x)+\frac {11}{8} \tanh ^{-1}(x) \]
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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,
218, 212, 209} \begin {gather*} \frac {11 \text {ArcTan}(x)}{8}-\frac {11}{28 x^7}-\frac {11}{12 x^3}+\frac {1}{4 x^7 \left (1-x^4\right )}+\frac {11}{8} \tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 209
Rule 212
Rule 218
Rule 296
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^8 \left (1-2 x^4+x^8\right )} \, dx &=\int \frac {1}{x^8 \left (-1+x^4\right )^2} \, dx\\ &=\frac {1}{4 x^7 \left (1-x^4\right )}-\frac {11}{4} \int \frac {1}{x^8 \left (-1+x^4\right )} \, dx\\ &=-\frac {11}{28 x^7}+\frac {1}{4 x^7 \left (1-x^4\right )}-\frac {11}{4} \int \frac {1}{x^4 \left (-1+x^4\right )} \, dx\\ &=-\frac {11}{28 x^7}-\frac {11}{12 x^3}+\frac {1}{4 x^7 \left (1-x^4\right )}-\frac {11}{4} \int \frac {1}{-1+x^4} \, dx\\ &=-\frac {11}{28 x^7}-\frac {11}{12 x^3}+\frac {1}{4 x^7 \left (1-x^4\right )}+\frac {11}{8} \int \frac {1}{1-x^2} \, dx+\frac {11}{8} \int \frac {1}{1+x^2} \, dx\\ &=-\frac {11}{28 x^7}-\frac {11}{12 x^3}+\frac {1}{4 x^7 \left (1-x^4\right )}+\frac {11}{8} \tan ^{-1}(x)+\frac {11}{8} \tanh ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 43, normalized size = 1.00 \begin {gather*} \frac {1}{336} \left (-\frac {48}{x^7}-\frac {224}{x^3}-\frac {84 x}{-1+x^4}+462 \tan ^{-1}(x)-231 \log (1-x)+231 \log (1+x)\right ) \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 | \(\frac {-\frac {11}{12} x^{8}+\frac {11}{21} x^{4}+\frac {1}{7}}{x^{7} \left (x^{4}-1\right )}-\frac {11 \ln \left (-1+x \right )}{16}+\frac {11 \arctan \left (x \right )}{8}+\frac {11 \ln \left (1+x \right )}{16}\) | \(41\) |
default | \(-\frac {1}{16 \left (-1+x \right )}-\frac {11 \ln \left (-1+x \right )}{16}+\frac {x}{8 x^{2}+8}+\frac {11 \arctan \left (x \right )}{8}-\frac {1}{7 x^{7}}-\frac {2}{3 x^{3}}-\frac {1}{16 \left (1+x \right )}+\frac {11 \ln \left (1+x \right )}{16}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 42, normalized size = 0.98 \begin {gather*} -\frac {77 \, x^{8} - 44 \, x^{4} - 12}{84 \, {\left (x^{11} - x^{7}\right )}} + \frac {11}{8} \, \arctan \left (x\right ) + \frac {11}{16} \, \log \left (x + 1\right ) - \frac {11}{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 = 0.36, size = 68, normalized size = 1.58 \begin {gather*} -\frac {308 \, x^{8} - 176 \, x^{4} - 462 \, {\left (x^{11} - x^{7}\right )} \arctan \left (x\right ) - 231 \, {\left (x^{11} - x^{7}\right )} \log \left (x + 1\right ) + 231 \, {\left (x^{11} - x^{7}\right )} \log \left (x - 1\right ) - 48}{336 \, {\left (x^{11} - x^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 44, normalized size = 1.02 \begin {gather*} - \frac {11 \log {\left (x - 1 \right )}}{16} + \frac {11 \log {\left (x + 1 \right )}}{16} + \frac {11 \operatorname {atan}{\left (x \right )}}{8} + \frac {- 77 x^{8} + 44 x^{4} + 12}{84 x^{11} - 84 x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.94, size = 41, normalized size = 0.95 \begin {gather*} -\frac {x}{4 \, {\left (x^{4} - 1\right )}} - \frac {14 \, x^{4} + 3}{21 \, x^{7}} + \frac {11}{8} \, \arctan \left (x\right ) + \frac {11}{16} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {11}{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 = 0.05, size = 34, normalized size = 0.79 \begin {gather*} \frac {11\,\mathrm {atan}\left (x\right )}{8}+\frac {11\,\mathrm {atanh}\left (x\right )}{8}-\frac {-\frac {11\,x^8}{12}+\frac {11\,x^4}{21}+\frac {1}{7}}{x^7-x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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