Optimal. Leaf size=109 \[ -\frac{3 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a^2 c^2}-\frac{\tan ^{-1}(a x)^{3/2}}{2 a^2 c^2 \left (a^2 x^2+1\right )}+\frac{3 x \sqrt{\tan ^{-1}(a x)}}{8 a c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)^{3/2}}{4 a^2 c^2} \]
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Rubi [A] time = 0.152058, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {4930, 4892, 4970, 4406, 12, 3305, 3351} \[ -\frac{3 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a^2 c^2}-\frac{\tan ^{-1}(a x)^{3/2}}{2 a^2 c^2 \left (a^2 x^2+1\right )}+\frac{3 x \sqrt{\tan ^{-1}(a x)}}{8 a c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)^{3/2}}{4 a^2 c^2} \]
Antiderivative was successfully verified.
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Rule 4930
Rule 4892
Rule 4970
Rule 4406
Rule 12
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \frac{x \tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx &=-\frac{\tan ^{-1}(a x)^{3/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac{3 \int \frac{\sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a}\\ &=\frac{3 x \sqrt{\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^{3/2}}{4 a^2 c^2}-\frac{\tan ^{-1}(a x)^{3/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac{3}{16} \int \frac{x}{\left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx\\ &=\frac{3 x \sqrt{\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^{3/2}}{4 a^2 c^2}-\frac{\tan ^{-1}(a x)^{3/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{16 a^2 c^2}\\ &=\frac{3 x \sqrt{\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^{3/2}}{4 a^2 c^2}-\frac{\tan ^{-1}(a x)^{3/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac{3 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 \sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{16 a^2 c^2}\\ &=\frac{3 x \sqrt{\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^{3/2}}{4 a^2 c^2}-\frac{\tan ^{-1}(a x)^{3/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac{3 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{32 a^2 c^2}\\ &=\frac{3 x \sqrt{\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^{3/2}}{4 a^2 c^2}-\frac{\tan ^{-1}(a x)^{3/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac{3 \operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{16 a^2 c^2}\\ &=\frac{3 x \sqrt{\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^{3/2}}{4 a^2 c^2}-\frac{\tan ^{-1}(a x)^{3/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac{3 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a^2 c^2}\\ \end{align*}
Mathematica [A] time = 0.0923423, size = 75, normalized size = 0.69 \[ \frac{\frac{4 \sqrt{\tan ^{-1}(a x)} \left (2 \left (a^2 x^2-1\right ) \tan ^{-1}(a x)+3 a x\right )}{a^2 x^2+1}-3 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a^2 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.099, size = 67, normalized size = 0.6 \begin{align*} -{\frac{1}{32\,{a}^{2}{c}^{2}} \left ( 8\, \left ( \arctan \left ( ax \right ) \right ) ^{2}\cos \left ( 2\,\arctan \left ( ax \right ) \right ) +3\,\sqrt{\arctan \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ( 2\,{\frac{\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -6\,\sin \left ( 2\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) \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 \operatorname{atan}^{\frac{3}{2}}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, 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 \arctan \left (a x\right )^{\frac{3}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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