Optimal. Leaf size=36 \[ \frac{1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac{b \log \left (c^2 x^4+1\right )}{4 c} \]
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Rubi [A] time = 0.01395, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5033, 260} \[ \frac{1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac{b \log \left (c^2 x^4+1\right )}{4 c} \]
Antiderivative was successfully verified.
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Rule 5033
Rule 260
Rubi steps
\begin{align*} \int x \left (a+b \tan ^{-1}\left (c x^2\right )\right ) \, dx &=\frac{1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-(b c) \int \frac{x^3}{1+c^2 x^4} \, dx\\ &=\frac{1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac{b \log \left (1+c^2 x^4\right )}{4 c}\\ \end{align*}
Mathematica [A] time = 0.0064483, size = 41, normalized size = 1.14 \[ \frac{a x^2}{2}-\frac{b \log \left (c^2 x^4+1\right )}{4 c}+\frac{1}{2} b x^2 \tan ^{-1}\left (c x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 36, normalized size = 1. \begin{align*}{\frac{a{x}^{2}}{2}}+{\frac{b{x}^{2}\arctan \left ( c{x}^{2} \right ) }{2}}-{\frac{b\ln \left ({c}^{2}{x}^{4}+1 \right ) }{4\,c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02717, size = 51, normalized size = 1.42 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{{\left (2 \, c x^{2} \arctan \left (c x^{2}\right ) - \log \left (c^{2} x^{4} + 1\right )\right )} b}{4 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.93178, size = 89, normalized size = 2.47 \begin{align*} \frac{2 \, b c x^{2} \arctan \left (c x^{2}\right ) + 2 \, a c x^{2} - b \log \left (c^{2} x^{4} + 1\right )}{4 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 20.8985, size = 70, normalized size = 1.94 \begin{align*} \begin{cases} \frac{a x^{2}}{2} + \frac{b x^{2} \operatorname{atan}{\left (c x^{2} \right )}}{2} - \frac{b \log{\left (x^{2} + i \sqrt{\frac{1}{c^{2}}} \right )}}{2 c} - \frac{i b \operatorname{atan}{\left (c x^{2} \right )}}{2 c^{4} \left (\frac{1}{c^{2}}\right )^{\frac{3}{2}}} & \text{for}\: c \neq 0 \\\frac{a x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15435, size = 54, normalized size = 1.5 \begin{align*} \frac{2 \, a c x^{2} +{\left (2 \, c x^{2} \arctan \left (c x^{2}\right ) - \log \left (c^{2} x^{4} + 1\right )\right )} b}{4 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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