Optimal. Leaf size=60 \[ \frac{2 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a^3 c^2}-\frac{2 x^2}{a c^2 \left (a^2 x^2+1\right ) \sqrt{\tan ^{-1}(a x)}} \]
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Rubi [A] time = 0.144906, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4942, 4970, 4406, 12, 3305, 3351} \[ \frac{2 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a^3 c^2}-\frac{2 x^2}{a c^2 \left (a^2 x^2+1\right ) \sqrt{\tan ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 4942
Rule 4970
Rule 4406
Rule 12
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \frac{x^2}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac{2 x^2}{a c^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}+\frac{4 \int \frac{x}{\left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx}{a}\\ &=-\frac{2 x^2}{a c^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}+\frac{4 \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^2}\\ &=-\frac{2 x^2}{a c^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}+\frac{4 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 \sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^2}\\ &=-\frac{2 x^2}{a c^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}+\frac{2 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^2}\\ &=-\frac{2 x^2}{a c^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}+\frac{4 \operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{a^3 c^2}\\ &=-\frac{2 x^2}{a c^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}+\frac{2 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a^3 c^2}\\ \end{align*}
Mathematica [A] time = 0.182232, size = 60, normalized size = 1. \[ \frac{2 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a^3 c^2}-\frac{2 x^2}{a c^2 \left (a^2 x^2+1\right ) \sqrt{\tan ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.107, size = 46, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{3}{c}^{2}} \left ( 2\,\sqrt{\arctan \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ( 2\,{\frac{\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +\cos \left ( 2\,\arctan \left ( ax \right ) \right ) -1 \right ){\frac{1}{\sqrt{\arctan \left ( ax \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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{x^{2}}{a^{4} x^{4} \operatorname{atan}^{\frac{3}{2}}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname{atan}^{\frac{3}{2}}{\left (a x \right )} + \operatorname{atan}^{\frac{3}{2}}{\left (a x \right )}}\, dx}{c^{2}} \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^{2}}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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