Optimal. Leaf size=14 \[ 2 \log \left (x^2+1\right )+x-\tan ^{-1}(x) \]
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Rubi [A] time = 0.0248439, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {1593, 1584, 774, 635, 203, 260} \[ 2 \log \left (x^2+1\right )+x-\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 1593
Rule 1584
Rule 774
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{4 x^2+x^3}{x+x^3} \, dx &=\int \frac{4 x^2+x^3}{x \left (1+x^2\right )} \, dx\\ &=\int \frac{x (4+x)}{1+x^2} \, dx\\ &=x+\int \frac{-1+4 x}{1+x^2} \, dx\\ &=x+4 \int \frac{x}{1+x^2} \, dx-\int \frac{1}{1+x^2} \, dx\\ &=x-\tan ^{-1}(x)+2 \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0040059, size = 14, normalized size = 1. \[ 2 \log \left (x^2+1\right )+x-\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 15, normalized size = 1.1 \begin{align*} x-\arctan \left ( x \right ) +2\,\ln \left ({x}^{2}+1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47022, size = 19, normalized size = 1.36 \begin{align*} x - \arctan \left (x\right ) + 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 = 0.976394, size = 43, normalized size = 3.07 \begin{align*} x - \arctan \left (x\right ) + 2 \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.09272, size = 12, normalized size = 0.86 \begin{align*} x + 2 \log{\left (x^{2} + 1 \right )} - \operatorname{atan}{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13637, size = 19, normalized size = 1.36 \begin{align*} x - \arctan \left (x\right ) + 2 \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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