Optimal. Leaf size=55 \[ \frac {1}{40} \left (5-3 \sqrt {5}\right ) \log \left (3-\sqrt {5}-2 x^4\right )+\frac {1}{40} \left (5+3 \sqrt {5}\right ) \log \left (3+\sqrt {5}-2 x^4\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1371, 646, 31}
\begin {gather*} \frac {1}{40} \left (5-3 \sqrt {5}\right ) \log \left (-2 x^4-\sqrt {5}+3\right )+\frac {1}{40} \left (5+3 \sqrt {5}\right ) \log \left (-2 x^4+\sqrt {5}+3\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 646
Rule 1371
Rubi steps
\begin {align*} \int \frac {x^7}{1-3 x^4+x^8} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x}{1-3 x+x^2} \, dx,x,x^4\right )\\ &=\frac {1}{40} \left (5-3 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x} \, dx,x,x^4\right )+\frac {1}{40} \left (5+3 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x} \, dx,x,x^4\right )\\ &=\frac {1}{40} \left (5-3 \sqrt {5}\right ) \log \left (3-\sqrt {5}-2 x^4\right )+\frac {1}{40} \left (5+3 \sqrt {5}\right ) \log \left (3+\sqrt {5}-2 x^4\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 53, normalized size = 0.96 \begin {gather*} \frac {1}{40} \left (5+3 \sqrt {5}\right ) \log \left (3+\sqrt {5}-2 x^4\right )+\frac {1}{40} \left (5-3 \sqrt {5}\right ) \log \left (-3+\sqrt {5}+2 x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 33, normalized size = 0.60
method | result | size |
default | \(\frac {\ln \left (x^{8}-3 x^{4}+1\right )}{8}-\frac {3 \sqrt {5}\, \arctanh \left (\frac {\left (2 x^{4}-3\right ) \sqrt {5}}{5}\right )}{20}\) | \(33\) |
risch | \(\frac {\ln \left (2 x^{4}-\sqrt {5}-3\right )}{8}+\frac {3 \ln \left (2 x^{4}-\sqrt {5}-3\right ) \sqrt {5}}{40}+\frac {\ln \left (2 x^{4}+\sqrt {5}-3\right )}{8}-\frac {3 \ln \left (2 x^{4}+\sqrt {5}-3\right ) \sqrt {5}}{40}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 45, normalized size = 0.82 \begin {gather*} \frac {3}{40} \, \sqrt {5} \log \left (\frac {2 \, x^{4} - \sqrt {5} - 3}{2 \, x^{4} + \sqrt {5} - 3}\right ) + \frac {1}{8} \, \log \left (x^{8} - 3 \, x^{4} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 57, normalized size = 1.04 \begin {gather*} \frac {3}{40} \, \sqrt {5} \log \left (\frac {2 \, x^{8} - 6 \, x^{4} - \sqrt {5} {\left (2 \, x^{4} - 3\right )} + 7}{x^{8} - 3 \, x^{4} + 1}\right ) + \frac {1}{8} \, \log \left (x^{8} - 3 \, x^{4} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 53, normalized size = 0.96 \begin {gather*} \left (\frac {1}{8} + \frac {3 \sqrt {5}}{40}\right ) \log {\left (x^{4} - \frac {3}{2} - \frac {\sqrt {5}}{2} \right )} + \left (\frac {1}{8} - \frac {3 \sqrt {5}}{40}\right ) \log {\left (x^{4} - \frac {3}{2} + \frac {\sqrt {5}}{2} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.88, size = 48, normalized size = 0.87 \begin {gather*} \frac {3}{40} \, \sqrt {5} \log \left (\frac {{\left | 2 \, x^{4} - \sqrt {5} - 3 \right |}}{{\left | 2 \, x^{4} + \sqrt {5} - 3 \right |}}\right ) + \frac {1}{8} \, \log \left ({\left | x^{8} - 3 \, x^{4} + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 59, normalized size = 1.07 \begin {gather*} \frac {\ln \left (x^4-\frac {\sqrt {5}}{2}-\frac {3}{2}\right )}{8}+\frac {\ln \left (x^4+\frac {\sqrt {5}}{2}-\frac {3}{2}\right )}{8}+\frac {3\,\sqrt {5}\,\ln \left (x^4-\frac {\sqrt {5}}{2}-\frac {3}{2}\right )}{40}-\frac {3\,\sqrt {5}\,\ln \left (x^4+\frac {\sqrt {5}}{2}-\frac {3}{2}\right )}{40} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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