Optimal. Leaf size=26 \[ \frac{x^4}{4}+\frac{1}{4} \log \left (x^4+1\right )-\log \left (x^4+2\right ) \]
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Rubi [A] time = 0.0189562, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1357, 703, 632, 31} \[ \frac{x^4}{4}+\frac{1}{4} \log \left (x^4+1\right )-\log \left (x^4+2\right ) \]
Antiderivative was successfully verified.
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Rule 1357
Rule 703
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{x^{11}}{2+3 x^4+x^8} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{2+3 x+x^2} \, dx,x,x^4\right )\\ &=\frac{x^4}{4}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{-2-3 x}{2+3 x+x^2} \, dx,x,x^4\right )\\ &=\frac{x^4}{4}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,x^4\right )-\operatorname{Subst}\left (\int \frac{1}{2+x} \, dx,x,x^4\right )\\ &=\frac{x^4}{4}+\frac{1}{4} \log \left (1+x^4\right )-\log \left (2+x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0046948, size = 26, normalized size = 1. \[ \frac{x^4}{4}+\frac{1}{4} \log \left (x^4+1\right )-\log \left (x^4+2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 23, normalized size = 0.9 \begin{align*}{\frac{{x}^{4}}{4}}+{\frac{\ln \left ({x}^{4}+1 \right ) }{4}}-\ln \left ({x}^{4}+2 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0044, size = 30, normalized size = 1.15 \begin{align*} \frac{1}{4} \, x^{4} - \log \left (x^{4} + 2\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 = 1.6815, size = 58, normalized size = 2.23 \begin{align*} \frac{1}{4} \, x^{4} - \log \left (x^{4} + 2\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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Sympy [A] time = 0.125721, size = 19, normalized size = 0.73 \begin{align*} \frac{x^{4}}{4} + \frac{\log{\left (x^{4} + 1 \right )}}{4} - \log{\left (x^{4} + 2 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10572, size = 30, normalized size = 1.15 \begin{align*} \frac{1}{4} \, x^{4} - \log \left (x^{4} + 2\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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