Optimal. Leaf size=67 \[ \frac{\sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a^3 c^3}-\frac{2 x^2}{a c^3 \left (a^2 x^2+1\right )^2 \sqrt{\tan ^{-1}(a x)}} \]
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Rubi [A] time = 0.313015, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4968, 4970, 4406, 3305, 3351} \[ \frac{\sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a^3 c^3}-\frac{2 x^2}{a c^3 \left (a^2 x^2+1\right )^2 \sqrt{\tan ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 4968
Rule 4970
Rule 4406
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \frac{x^2}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac{2 x^2}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+\frac{4 \int \frac{x}{\left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx}{a}-(4 a) \int \frac{x^3}{\left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx\\ &=-\frac{2 x^2}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+\frac{4 \operatorname{Subst}\left (\int \frac{\cos ^3(x) \sin (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}-\frac{4 \operatorname{Subst}\left (\int \frac{\cos (x) \sin ^3(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}\\ &=-\frac{2 x^2}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}-\frac{4 \operatorname{Subst}\left (\int \left (\frac{\sin (2 x)}{4 \sqrt{x}}-\frac{\sin (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}+\frac{4 \operatorname{Subst}\left (\int \left (\frac{\sin (2 x)}{4 \sqrt{x}}+\frac{\sin (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}\\ &=-\frac{2 x^2}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+2 \frac{\operatorname{Subst}\left (\int \frac{\sin (4 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 c^3}\\ &=-\frac{2 x^2}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+2 \frac{\operatorname{Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{a^3 c^3}\\ &=-\frac{2 x^2}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+\frac{\sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a^3 c^3}\\ \end{align*}
Mathematica [C] time = 0.391103, size = 112, normalized size = 1.67 \[ \frac{-\left (a^2 x^2+1\right )^2 \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \tan ^{-1}(a x)\right )-\left (a^2 x^2+1\right )^2 \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \tan ^{-1}(a x)\right )-8 a^2 x^2}{4 a^3 c^3 \left (a^2 x^2+1\right )^2 \sqrt{\tan ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.115, size = 53, normalized size = 0.8 \begin{align*}{\frac{1}{4\,{c}^{3}{a}^{3}} \left ( 2\,\sqrt{2}\sqrt{\arctan \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +\cos \left ( 4\,\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^{6} x^{6} \operatorname{atan}^{\frac{3}{2}}{\left (a x \right )} + 3 a^{4} x^{4} \operatorname{atan}^{\frac{3}{2}}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname{atan}^{\frac{3}{2}}{\left (a x \right )} + \operatorname{atan}^{\frac{3}{2}}{\left (a x \right )}}\, dx}{c^{3}} \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 )}^{3} \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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