Optimal. Leaf size=55 \[ \frac{1}{40} \left (5-3 \sqrt{5}\right ) \log \left (-2 x^4-\sqrt{5}+3\right )+\frac{1}{40} \left (5+3 \sqrt{5}\right ) \log \left (-2 x^4+\sqrt{5}+3\right ) \]
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Rubi [A] time = 0.0318653, antiderivative size = 55, normalized size of antiderivative = 1., 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 = {1357, 632, 31} \[ \frac{1}{40} \left (5-3 \sqrt{5}\right ) \log \left (-2 x^4-\sqrt{5}+3\right )+\frac{1}{40} \left (5+3 \sqrt{5}\right ) \log \left (-2 x^4+\sqrt{5}+3\right ) \]
Antiderivative was successfully verified.
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Rule 1357
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{x^7}{1-3 x^4+x^8} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x}{1-3 x+x^2} \, dx,x,x^4\right )\\ &=\frac{1}{40} \left (5-3 \sqrt{5}\right ) \operatorname{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 ) \operatorname{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*}
Mathematica [A] time = 0.0208405, size = 53, normalized size = 0.96 \[ \frac{1}{40} \left (5+3 \sqrt{5}\right ) \log \left (-2 x^4+\sqrt{5}+3\right )+\frac{1}{40} \left (5-3 \sqrt{5}\right ) \log \left (2 x^4+\sqrt{5}-3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 33, normalized size = 0.6 \begin{align*}{\frac{\ln \left ({x}^{8}-3\,{x}^{4}+1 \right ) }{8}}-{\frac{3\,\sqrt{5}}{20}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{4}-3 \right ) \sqrt{5}}{5}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47837, size = 61, normalized size = 1.11 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72496, size = 143, normalized size = 2.6 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.137365, size = 53, normalized size = 0.96 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14867, size = 65, normalized size = 1.18 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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