Optimal. Leaf size=21 \[ \frac{1}{4} \log \left (x^4+1\right )-\frac{1}{4} \log \left (x^4+2\right ) \]
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Rubi [A] time = 0.0127365, antiderivative size = 21, 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 = {1352, 616, 31} \[ \frac{1}{4} \log \left (x^4+1\right )-\frac{1}{4} \log \left (x^4+2\right ) \]
Antiderivative was successfully verified.
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Rule 1352
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{x^3}{2+3 x^4+x^8} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{2+3 x+x^2} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,x^4\right )-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{2+x} \, dx,x,x^4\right )\\ &=\frac{1}{4} \log \left (1+x^4\right )-\frac{1}{4} \log \left (2+x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0042694, size = 21, normalized size = 1. \[ \frac{1}{4} \log \left (x^4+1\right )-\frac{1}{4} \log \left (x^4+2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 18, normalized size = 0.9 \begin{align*}{\frac{\ln \left ({x}^{4}+1 \right ) }{4}}-{\frac{\ln \left ({x}^{4}+2 \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.969577, size = 23, normalized size = 1.1 \begin{align*} -\frac{1}{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 = 1.69313, size = 51, normalized size = 2.43 \begin{align*} -\frac{1}{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 = 0.113925, size = 15, normalized size = 0.71 \begin{align*} \frac{\log{\left (x^{4} + 1 \right )}}{4} - \frac{\log{\left (x^{4} + 2 \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1405, size = 23, normalized size = 1.1 \begin{align*} -\frac{1}{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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