Optimal. Leaf size=22 \[ \frac{1}{2 \left (x^2+1\right )}+\frac{1}{2} \log \left (x^2+1\right ) \]
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Rubi [A] time = 0.0108123, antiderivative size = 22, 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} \[ \frac{1}{2 \left (x^2+1\right )}+\frac{1}{2} \log \left (x^2+1\right ) \]
Antiderivative was successfully verified.
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Rule 28
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{1+2 x^2+x^4} \, dx &=\int \frac{x^3}{\left (1+x^2\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(1+x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{1}{(1+x)^2}+\frac{1}{1+x}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{2 \left (1+x^2\right )}+\frac{1}{2} \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0045505, size = 18, normalized size = 0.82 \[ \frac{1}{2} \left (\frac{1}{x^2+1}+\log \left (x^2+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 19, normalized size = 0.9 \begin{align*}{\frac{1}{2\,{x}^{2}+2}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965903, size = 24, normalized size = 1.09 \begin{align*} \frac{1}{2 \,{\left (x^{2} + 1\right )}} + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43392, size = 59, normalized size = 2.68 \begin{align*} \frac{{\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) + 1}{2 \,{\left (x^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.083989, size = 15, normalized size = 0.68 \begin{align*} \frac{\log{\left (x^{2} + 1 \right )}}{2} + \frac{1}{2 x^{2} + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14005, size = 24, normalized size = 1.09 \begin{align*} \frac{1}{2 \,{\left (x^{2} + 1\right )}} + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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