Optimal. Leaf size=168 \[ -\frac{3 \sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{512 a^2 c^3}-\frac{3 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{64 a^2 c^3}-\frac{\tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac{3 \tan ^{-1}(a x)^{3/2}}{32 a^2 c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{32 a^2 c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{256 a^2 c^3} \]
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Rubi [A] time = 0.185752, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4930, 4904, 3312, 3296, 3305, 3351} \[ -\frac{3 \sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{512 a^2 c^3}-\frac{3 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{64 a^2 c^3}-\frac{\tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac{3 \tan ^{-1}(a x)^{3/2}}{32 a^2 c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{32 a^2 c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{256 a^2 c^3} \]
Antiderivative was successfully verified.
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Rule 4930
Rule 4904
Rule 3312
Rule 3296
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 )^3} \, dx &=-\frac{\tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 \int \frac{\sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^3} \, dx}{8 a}\\ &=-\frac{\tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 \operatorname{Subst}\left (\int \sqrt{x} \cos ^4(x) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^2 c^3}\\ &=-\frac{\tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 \operatorname{Subst}\left (\int \left (\frac{3 \sqrt{x}}{8}+\frac{1}{2} \sqrt{x} \cos (2 x)+\frac{1}{8} \sqrt{x} \cos (4 x)\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^2 c^3}\\ &=\frac{3 \tan ^{-1}(a x)^{3/2}}{32 a^2 c^3}-\frac{\tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 \operatorname{Subst}\left (\int \sqrt{x} \cos (4 x) \, dx,x,\tan ^{-1}(a x)\right )}{64 a^2 c^3}+\frac{3 \operatorname{Subst}\left (\int \sqrt{x} \cos (2 x) \, dx,x,\tan ^{-1}(a x)\right )}{16 a^2 c^3}\\ &=\frac{3 \tan ^{-1}(a x)^{3/2}}{32 a^2 c^3}-\frac{\tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 \sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{32 a^2 c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{256 a^2 c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{\sin (4 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{512 a^2 c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{64 a^2 c^3}\\ &=\frac{3 \tan ^{-1}(a x)^{3/2}}{32 a^2 c^3}-\frac{\tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 \sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{32 a^2 c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{256 a^2 c^3}-\frac{3 \operatorname{Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{256 a^2 c^3}-\frac{3 \operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{32 a^2 c^3}\\ &=\frac{3 \tan ^{-1}(a x)^{3/2}}{32 a^2 c^3}-\frac{\tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 \sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{512 a^2 c^3}-\frac{3 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{64 a^2 c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{32 a^2 c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{256 a^2 c^3}\\ \end{align*}
Mathematica [C] time = 0.234078, size = 347, normalized size = 2.07 \[ \frac{3 a^4 x^4 \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \tan ^{-1}(a x)\right )+3 a^4 x^4 \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \tan ^{-1}(a x)\right )+6 a^2 x^2 \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \tan ^{-1}(a x)\right )+6 a^2 x^2 \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \tan ^{-1}(a x)\right )+24 \sqrt{2} \left (a^2 x^2+1\right )^2 \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \tan ^{-1}(a x)\right )+24 \sqrt{2} \left (a^2 x^2+1\right )^2 \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \tan ^{-1}(a x)\right )+3 \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \tan ^{-1}(a x)\right )+3 \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \tan ^{-1}(a x)\right )+192 a^4 x^4 \tan ^{-1}(a x)^2+288 a^3 x^3 \tan ^{-1}(a x)+384 a^2 x^2 \tan ^{-1}(a x)^2+480 a x \tan ^{-1}(a x)-320 \tan ^{-1}(a x)^2}{2048 c^3 \left (a^3 x^2+a\right )^2 \sqrt{\tan ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.14, size = 124, normalized size = 0.7 \begin{align*} -{\frac{1}{1024\,{c}^{3}{a}^{2}} \left ( 3\,\sqrt{2}\sqrt{\arctan \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +128\, \left ( \arctan \left ( ax \right ) \right ) ^{2}\cos \left ( 2\,\arctan \left ( ax \right ) \right ) +32\, \left ( \arctan \left ( ax \right ) \right ) ^{2}\cos \left ( 4\,\arctan \left ( ax \right ) \right ) +48\,\sqrt{\arctan \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ( 2\,{\frac{\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -96\,\sin \left ( 2\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) -12\,\sin \left ( 4\,\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^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, 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 \arctan \left (a x\right )^{\frac{3}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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