Optimal. Leaf size=57 \[ -\frac{2}{a c^2 \left (a^2 x^2+1\right ) \sqrt{\tan ^{-1}(a x)}}-\frac{2 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a c^2} \]
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Rubi [A] time = 0.103588, antiderivative size = 57, 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 = {4902, 4970, 4406, 12, 3305, 3351} \[ -\frac{2}{a c^2 \left (a^2 x^2+1\right ) \sqrt{\tan ^{-1}(a x)}}-\frac{2 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a c^2} \]
Antiderivative was successfully verified.
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Rule 4902
Rule 4970
Rule 4406
Rule 12
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \frac{1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac{2}{a c^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}-(4 a) \int \frac{x}{\left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx\\ &=-\frac{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 c^2}\\ &=-\frac{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 c^2}\\ &=-\frac{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 c^2}\\ &=-\frac{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 c^2}\\ &=-\frac{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 c^2}\\ \end{align*}
Mathematica [A] time = 0.149662, size = 52, normalized size = 0.91 \[ \frac{-\frac{2}{\left (a^2 x^2+1\right ) \sqrt{\tan ^{-1}(a x)}}-2 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.102, size = 47, normalized size = 0.8 \begin{align*} -{\frac{1}{a{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{1}{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{1}{{\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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