Optimal. Leaf size=16 \[ -\frac {x^3}{2 a \tan ^{-1}(a x)^2} \]
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Rubi [A] time = 0.09, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.15, size = 16, normalized size = 1.00 \[ -\frac {x^3}{2 a \tan ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 14, normalized size = 0.88 \[ -\frac {x^{3}}{2 \, a \arctan \left (a x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.44, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (a^{2} x^{2}+1\right ) \arctan \left (a x \right )^{3}}-\frac {3 x^{2}}{2 a \arctan \left (a x \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 14, normalized size = 0.88 \[ -\frac {x^{3}}{2 \, a \arctan \left (a x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 14, normalized size = 0.88 \[ -\frac {x^3}{2\,a\,{\mathrm {atan}\left (a\,x\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \left (- \frac {2 a x^{3}}{a^{2} x^{2} \operatorname {atan}^{3}{\left (a x \right )} + \operatorname {atan}^{3}{\left (a x \right )}}\right )\, dx + \int \frac {3 x^{2} \operatorname {atan}{\left (a x \right )}}{a^{2} x^{2} \operatorname {atan}^{3}{\left (a x \right )} + \operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {3 a^{2} x^{4} \operatorname {atan}{\left (a x \right )}}{a^{2} x^{2} \operatorname {atan}^{3}{\left (a x \right )} + \operatorname {atan}^{3}{\left (a x \right )}}\, dx}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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