Optimal. Leaf size=37 \[ \frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )+\frac{b \log \left (1-c^2 x^4\right )}{4 c} \]
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Rubi [A] time = 0.0149314, antiderivative size = 37, 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 = {6097, 260} \[ \frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )+\frac{b \log \left (1-c^2 x^4\right )}{4 c} \]
Antiderivative was successfully verified.
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Rule 6097
Rule 260
Rubi steps
\begin{align*} \int x \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \, dx &=\frac{1}{2} x^2 \left (a+b \tanh ^{-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 \tanh ^{-1}\left (c x^2\right )\right )+\frac{b \log \left (1-c^2 x^4\right )}{4 c}\\ \end{align*}
Mathematica [A] time = 0.0069701, size = 42, normalized size = 1.14 \[ \frac{a x^2}{2}+\frac{b \log \left (1-c^2 x^4\right )}{4 c}+\frac{1}{2} b x^2 \tanh ^{-1}\left (c x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 37, normalized size = 1. \begin{align*}{\frac{a{x}^{2}}{2}}+{\frac{b{x}^{2}{\it Artanh} \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 = 0.955621, size = 50, normalized size = 1.35 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{{\left (2 \, c x^{2} \operatorname{artanh}\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 = 2.06531, size = 108, normalized size = 2.92 \begin{align*} \frac{b c x^{2} \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\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 = 10.0682, size = 71, normalized size = 1.92 \begin{align*} \begin{cases} \frac{a x^{2}}{2} + \frac{b x^{2} \operatorname{atanh}{\left (c x^{2} \right )}}{2} + \frac{b \log{\left (x - i \sqrt{\frac{1}{c}} \right )}}{2 c} + \frac{b \log{\left (x + i \sqrt{\frac{1}{c}} \right )}}{2 c} - \frac{b \operatorname{atanh}{\left (c x^{2} \right )}}{2 c} & \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 = 1.18776, size = 66, normalized size = 1.78 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{1}{4} \,{\left (x^{2} \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right ) + \frac{\log \left ({\left | c^{2} x^{4} - 1 \right |}\right )}{c}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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