Optimal. Leaf size=77 \[ \frac{x \sqrt{\tan ^{-1}(a x)}}{2 c^2 \left (a^2 x^2+1\right )}-\frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a c^2}+\frac{\tan ^{-1}(a x)^{3/2}}{3 a c^2} \]
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Rubi [A] time = 0.104373, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4892, 4970, 4406, 12, 3305, 3351} \[ \frac{x \sqrt{\tan ^{-1}(a x)}}{2 c^2 \left (a^2 x^2+1\right )}-\frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a c^2}+\frac{\tan ^{-1}(a x)^{3/2}}{3 a c^2} \]
Antiderivative was successfully verified.
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Rule 4892
Rule 4970
Rule 4406
Rule 12
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \frac{\sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx &=\frac{x \sqrt{\tan ^{-1}(a x)}}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^{3/2}}{3 a c^2}-\frac{1}{4} a \int \frac{x}{\left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx\\ &=\frac{x \sqrt{\tan ^{-1}(a x)}}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^{3/2}}{3 a c^2}-\frac{\operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{4 a c^2}\\ &=\frac{x \sqrt{\tan ^{-1}(a x)}}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^{3/2}}{3 a c^2}-\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 \sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{4 a c^2}\\ &=\frac{x \sqrt{\tan ^{-1}(a x)}}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^{3/2}}{3 a c^2}-\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{8 a c^2}\\ &=\frac{x \sqrt{\tan ^{-1}(a x)}}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^{3/2}}{3 a c^2}-\frac{\operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{4 a c^2}\\ &=\frac{x \sqrt{\tan ^{-1}(a x)}}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^{3/2}}{3 a c^2}-\frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a c^2}\\ \end{align*}
Mathematica [C] time = 0.1591, size = 89, normalized size = 1.16 \[ \frac{3 \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-2 i \tan ^{-1}(a x)\right )+3 \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},2 i \tan ^{-1}(a x)\right )+16 \tan ^{-1}(a x)^2}{48 a c^2 \sqrt{\tan ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.108, size = 60, normalized size = 0.8 \begin{align*}{\frac{1}{24\,a{c}^{2}} \left ( 8\, \left ( \arctan \left ( ax \right ) \right ) ^{2}+6\,\sin \left ( 2\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) -3\,\sqrt{\arctan \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ( 2\,{\frac{\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \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{\sqrt{\operatorname{atan}{\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{\sqrt{\arctan \left (a x\right )}}{{\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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