Optimal. Leaf size=124 \[ \frac{x \tan ^{-1}(a x)^{3/2}}{2 c^2 \left (a^2 x^2+1\right )}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{8 a c^2 \left (a^2 x^2+1\right )}-\frac{3 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a c^2}+\frac{\tan ^{-1}(a x)^{5/2}}{5 a c^2}-\frac{3 \sqrt{\tan ^{-1}(a x)}}{16 a c^2} \]
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Rubi [A] time = 0.147686, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4892, 4930, 4904, 3312, 3304, 3352} \[ \frac{x \tan ^{-1}(a x)^{3/2}}{2 c^2 \left (a^2 x^2+1\right )}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{8 a c^2 \left (a^2 x^2+1\right )}-\frac{3 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a c^2}+\frac{\tan ^{-1}(a x)^{5/2}}{5 a c^2}-\frac{3 \sqrt{\tan ^{-1}(a x)}}{16 a c^2} \]
Antiderivative was successfully verified.
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Rule 4892
Rule 4930
Rule 4904
Rule 3312
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx &=\frac{x \tan ^{-1}(a x)^{3/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^{5/2}}{5 a c^2}-\frac{1}{4} (3 a) \int \frac{x \sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx\\ &=\frac{3 \sqrt{\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^{3/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^{5/2}}{5 a c^2}-\frac{3}{16} \int \frac{1}{\left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx\\ &=\frac{3 \sqrt{\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^{3/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^{5/2}}{5 a c^2}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos ^2(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{16 a c^2}\\ &=\frac{3 \sqrt{\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^{3/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^{5/2}}{5 a c^2}-\frac{3 \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}+\frac{\cos (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{16 a c^2}\\ &=-\frac{3 \sqrt{\tan ^{-1}(a x)}}{16 a c^2}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^{3/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^{5/2}}{5 a c^2}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{32 a c^2}\\ &=-\frac{3 \sqrt{\tan ^{-1}(a x)}}{16 a c^2}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^{3/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^{5/2}}{5 a c^2}-\frac{3 \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{16 a c^2}\\ &=-\frac{3 \sqrt{\tan ^{-1}(a x)}}{16 a c^2}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^{3/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^{5/2}}{5 a c^2}-\frac{3 \sqrt{\pi } C\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a c^2}\\ \end{align*}
Mathematica [A] time = 0.167113, size = 90, normalized size = 0.73 \[ \frac{\frac{2 \sqrt{\tan ^{-1}(a x)} \left (-15 a^2 x^2+16 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2+40 a x \tan ^{-1}(a x)+15\right )}{a^2 x^2+1}-15 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{160 a c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.109, size = 75, normalized size = 0.6 \begin{align*}{\frac{1}{160\,a{c}^{2}} \left ( 32\, \left ( \arctan \left ( ax \right ) \right ) ^{3}+40\, \left ( \arctan \left ( ax \right ) \right ) ^{2}\sin \left ( 2\,\arctan \left ( ax \right ) \right ) +30\,\cos \left ( 2\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) -15\,\sqrt{\arctan \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \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{\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{\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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