Optimal. Leaf size=139 \[ -\frac{\sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{64 a c^3}-\frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a c^3}+\frac{\tan ^{-1}(a x)^{3/2}}{4 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{4 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{32 a c^3} \]
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Rubi [A] time = 0.144651, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {4904, 3312, 3296, 3305, 3351} \[ -\frac{\sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{64 a c^3}-\frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a c^3}+\frac{\tan ^{-1}(a x)^{3/2}}{4 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{4 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{32 a c^3} \]
Antiderivative was successfully verified.
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Rule 4904
Rule 3312
Rule 3296
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \frac{\sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{x} \cos ^4(x) \, dx,x,\tan ^{-1}(a x)\right )}{a c^3}\\ &=\frac{\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 )}{a c^3}\\ &=\frac{\tan ^{-1}(a x)^{3/2}}{4 a c^3}+\frac{\operatorname{Subst}\left (\int \sqrt{x} \cos (4 x) \, dx,x,\tan ^{-1}(a x)\right )}{8 a c^3}+\frac{\operatorname{Subst}\left (\int \sqrt{x} \cos (2 x) \, dx,x,\tan ^{-1}(a x)\right )}{2 a c^3}\\ &=\frac{\tan ^{-1}(a x)^{3/2}}{4 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{4 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{32 a c^3}-\frac{\operatorname{Subst}\left (\int \frac{\sin (4 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{64 a c^3}-\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{8 a c^3}\\ &=\frac{\tan ^{-1}(a x)^{3/2}}{4 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{4 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{32 a c^3}-\frac{\operatorname{Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{32 a c^3}-\frac{\operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{4 a c^3}\\ &=\frac{\tan ^{-1}(a x)^{3/2}}{4 a c^3}-\frac{\sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{64 a c^3}-\frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{4 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{32 a c^3}\\ \end{align*}
Mathematica [C] time = 0.419533, size = 192, normalized size = 1.38 \[ \frac{\frac{8 \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \tan ^{-1}(a x)\right )}{a}+\frac{8 \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \tan ^{-1}(a x)\right )}{a}+\frac{\sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \tan ^{-1}(a x)\right )}{a}+\frac{\sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \tan ^{-1}(a x)\right )}{a}+\frac{96 a^2 x^3 \tan ^{-1}(a x)}{\left (a^2 x^2+1\right )^2}+\frac{160 x \tan ^{-1}(a x)}{\left (a^2 x^2+1\right )^2}+\frac{64 \tan ^{-1}(a x)^2}{a}}{256 c^3 \sqrt{\tan ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.118, size = 102, normalized size = 0.7 \begin{align*}{\frac{1}{128\,a{c}^{3}} \left ( -\sqrt{2}\sqrt{\arctan \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +32\, \left ( \arctan \left ( ax \right ) \right ) ^{2}+32\,\sin \left ( 2\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) +4\,\sin \left ( 4\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) -16\,\sqrt{\arctan \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \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{\sqrt{\operatorname{atan}{\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{\sqrt{\arctan \left (a x\right )}}{{\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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