Optimal. Leaf size=16 \[ -\frac{x^3}{2 a \tan ^{-1}(a x)^2} \]
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Rubi [A] time = 0.090723, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026, Rules used = {4926} \[ -\frac{x^3}{2 a \tan ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Rule 4926
Rubi steps
\begin{align*} \int \left (\frac{x^3}{\left (1+a^2 x^2\right ) \tan ^{-1}(a x)^3}-\frac{3 x^2}{2 a \tan ^{-1}(a x)^2}\right ) \, dx &=-\frac{3 \int \frac{x^2}{\tan ^{-1}(a x)^2} \, dx}{2 a}+\int \frac{x^3}{\left (1+a^2 x^2\right ) \tan ^{-1}(a x)^3} \, dx\\ &=-\frac{x^3}{2 a \tan ^{-1}(a x)^2}\\ \end{align*}
Mathematica [A] time = 0.151827, size = 16, normalized size = 1. \[ -\frac{x^3}{2 a \tan ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.108, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{ \left ({a}^{2}{x}^{2}+1 \right ) \left ( \arctan \left ( ax \right ) \right ) ^{3}}}-{\frac{3\,{x}^{2}}{2\,a \left ( \arctan \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.36384, size = 19, normalized size = 1.19 \begin{align*} -\frac{x^{3}}{2 \, a \arctan \left (a x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58476, size = 38, normalized size = 2.38 \begin{align*} -\frac{x^{3}}{2 \, a \arctan \left (a x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int - \frac{2 a x^{3}}{a^{2} x^{2} \operatorname{atan}^{3}{\left (a x \right )} + \operatorname{atan}^{3}{\left (a x \right )}}\, dx + \int \frac{3 x^{2} \operatorname{atan}{\left (a x \right )}}{a^{2} x^{2} \operatorname{atan}^{3}{\left (a x \right )} + \operatorname{atan}^{3}{\left (a x \right )}}\, dx + \int \frac{3 a^{2} x^{4} \operatorname{atan}{\left (a x \right )}}{a^{2} x^{2} \operatorname{atan}^{3}{\left (a x \right )} + \operatorname{atan}^{3}{\left (a x \right )}}\, dx}{2 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3}} - \frac{3 \, x^{2}}{2 \, a \arctan \left (a x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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