Optimal. Leaf size=174 \[ \frac{8 x}{3 c^2 \left (a^2 x^2+1\right ) \sqrt{\tan ^{-1}(a x)}}-\frac{16 \left (1-a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}{3 a c^2 \left (a^2 x^2+1\right )}+\frac{32 \sqrt{\tan ^{-1}(a x)}}{3 a c^2 \left (a^2 x^2+1\right )}-\frac{2}{3 a c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{3 a c^2}-\frac{16 \sqrt{\tan ^{-1}(a x)}}{3 a c^2} \]
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Rubi [A] time = 0.206001, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4902, 4932, 4930, 4904, 3312, 3304, 3352} \[ \frac{8 x}{3 c^2 \left (a^2 x^2+1\right ) \sqrt{\tan ^{-1}(a x)}}-\frac{16 \left (1-a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}{3 a c^2 \left (a^2 x^2+1\right )}+\frac{32 \sqrt{\tan ^{-1}(a x)}}{3 a c^2 \left (a^2 x^2+1\right )}-\frac{2}{3 a c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{3 a c^2}-\frac{16 \sqrt{\tan ^{-1}(a x)}}{3 a c^2} \]
Antiderivative was successfully verified.
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Rule 4902
Rule 4932
Rule 4930
Rule 4904
Rule 3312
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac{2}{3 a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}-\frac{1}{3} (4 a) \int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2}{3 a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac{8 x}{3 c^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}-\frac{16 \left (1-a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac{1}{3} (64 a) \int \frac{x \sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx\\ &=-\frac{2}{3 a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac{8 x}{3 c^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}+\frac{32 \sqrt{\tan ^{-1}(a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac{16 \left (1-a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac{16}{3} \int \frac{1}{\left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx\\ &=-\frac{2}{3 a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac{8 x}{3 c^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}+\frac{32 \sqrt{\tan ^{-1}(a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac{16 \left (1-a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac{16 \operatorname{Subst}\left (\int \frac{\cos ^2(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a c^2}\\ &=-\frac{2}{3 a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac{8 x}{3 c^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}+\frac{32 \sqrt{\tan ^{-1}(a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac{16 \left (1-a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac{16 \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 )}{3 a c^2}\\ &=-\frac{2}{3 a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac{8 x}{3 c^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}-\frac{16 \sqrt{\tan ^{-1}(a x)}}{3 a c^2}+\frac{32 \sqrt{\tan ^{-1}(a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac{16 \left (1-a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac{8 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a c^2}\\ &=-\frac{2}{3 a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac{8 x}{3 c^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}-\frac{16 \sqrt{\tan ^{-1}(a x)}}{3 a c^2}+\frac{32 \sqrt{\tan ^{-1}(a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac{16 \left (1-a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac{16 \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{3 a c^2}\\ &=-\frac{2}{3 a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac{8 x}{3 c^2 \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}-\frac{16 \sqrt{\tan ^{-1}(a x)}}{3 a c^2}+\frac{32 \sqrt{\tan ^{-1}(a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac{16 \left (1-a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac{8 \sqrt{\pi } C\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{3 a c^2}\\ \end{align*}
Mathematica [C] time = 0.416717, size = 170, normalized size = 0.98 \[ \frac{\frac{\sqrt{2} \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2 \text{Gamma}\left (\frac{1}{2},2 i \tan ^{-1}(a x)\right )}{\sqrt{i \tan ^{-1}(a x)}}+\sqrt{2} \left (a^2 x^2+1\right ) \sqrt{i \tan ^{-1}(a x)} \sqrt{\tan ^{-1}(a x)^2} \text{Gamma}\left (\frac{1}{2},-2 i \tan ^{-1}(a x)\right )-4 \sqrt{\pi } \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2} \text{FresnelC}\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )+8 a x \tan ^{-1}(a x)-2}{3 c^2 \left (a^3 x^2+a\right ) \tan ^{-1}(a x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.106, size = 62, normalized size = 0.4 \begin{align*}{\frac{1}{3\,a{c}^{2}} \left ( -8\,\sqrt{\pi }{\it FresnelC} \left ( 2\,{\frac{\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arctan \left ( ax \right ) \right ) ^{3/2}+4\,\sin \left ( 2\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) -\cos \left ( 2\,\arctan \left ( ax \right ) \right ) -1 \right ) \left ( \arctan \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}} \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{5}{2}}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname{atan}^{\frac{5}{2}}{\left (a x \right )} + \operatorname{atan}^{\frac{5}{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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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