Optimal. Leaf size=26 \[ \frac{1}{4 \left (1-x^4\right )}+\frac{1}{4} \log \left (1-x^4\right ) \]
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Rubi [A] time = 0.0128445, antiderivative size = 26, 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 = {28, 266, 43} \[ \frac{1}{4 \left (1-x^4\right )}+\frac{1}{4} \log \left (1-x^4\right ) \]
Antiderivative was successfully verified.
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Rule 28
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^7}{1-2 x^4+x^8} \, dx &=\int \frac{x^7}{\left (-1+x^4\right )^2} \, dx\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x}{(-1+x)^2} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{1}{(-1+x)^2}+\frac{1}{-1+x}\right ) \, dx,x,x^4\right )\\ &=\frac{1}{4 \left (1-x^4\right )}+\frac{1}{4} \log \left (1-x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0068021, size = 22, normalized size = 0.85 \[ \frac{1}{4} \log \left (x^4-1\right )-\frac{1}{4 \left (x^4-1\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 19, normalized size = 0.7 \begin{align*} -{\frac{1}{4\,{x}^{4}-4}}+{\frac{\ln \left ({x}^{4}-1 \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.975217, size = 24, normalized size = 0.92 \begin{align*} -\frac{1}{4 \,{\left (x^{4} - 1\right )}} + \frac{1}{4} \, \log \left (x^{4} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4481, size = 59, normalized size = 2.27 \begin{align*} \frac{{\left (x^{4} - 1\right )} \log \left (x^{4} - 1\right ) - 1}{4 \,{\left (x^{4} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.110643, size = 15, normalized size = 0.58 \begin{align*} \frac{\log{\left (x^{4} - 1 \right )}}{4} - \frac{1}{4 x^{4} - 4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10895, size = 26, normalized size = 1. \begin{align*} -\frac{1}{4 \,{\left (x^{4} - 1\right )}} + \frac{1}{4} \, \log \left ({\left | x^{4} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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