Optimal. Leaf size=213 \[ -\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4 a c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{12 a c^2 \sqrt{a^2 c x^2+c}}+\frac{3 x \sqrt{\tan ^{-1}(a x)}}{4 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \sqrt{\tan ^{-1}(a x)} \sin \left (3 \tan ^{-1}(a x)\right )}{12 a c^2 \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.187895, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {4905, 4904, 3312, 3296, 3305, 3351} \[ -\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4 a c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{12 a c^2 \sqrt{a^2 c x^2+c}}+\frac{3 x \sqrt{\tan ^{-1}(a x)}}{4 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \sqrt{\tan ^{-1}(a x)} \sin \left (3 \tan ^{-1}(a x)\right )}{12 a c^2 \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 4905
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 )^{5/2}} \, dx &=\frac{\sqrt{1+a^2 x^2} \int \frac{\sqrt{\tan ^{-1}(a x)}}{\left (1+a^2 x^2\right )^{5/2}} \, dx}{c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \sqrt{x} \cos ^3(x) \, dx,x,\tan ^{-1}(a x)\right )}{a c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \left (\frac{3}{4} \sqrt{x} \cos (x)+\frac{1}{4} \sqrt{x} \cos (3 x)\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \sqrt{x} \cos (3 x) \, dx,x,\tan ^{-1}(a x)\right )}{4 a c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \sqrt{x} \cos (x) \, dx,x,\tan ^{-1}(a x)\right )}{4 a c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{3 x \sqrt{\tan ^{-1}(a x)}}{4 c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \sqrt{\tan ^{-1}(a x)} \sin \left (3 \tan ^{-1}(a x)\right )}{12 a c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{24 a c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{8 a c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{3 x \sqrt{\tan ^{-1}(a x)}}{4 c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \sqrt{\tan ^{-1}(a x)} \sin \left (3 \tan ^{-1}(a x)\right )}{12 a c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{12 a c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{4 a c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{3 x \sqrt{\tan ^{-1}(a x)}}{4 c^2 \sqrt{c+a^2 c x^2}}-\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{1+a^2 x^2} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4 a c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{1+a^2 x^2} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{12 a c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \sqrt{\tan ^{-1}(a x)} \sin \left (3 \tan ^{-1}(a x)\right )}{12 a c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.166428, size = 137, normalized size = 0.64 \[ \frac{-27 \sqrt{2 \pi } \left (a^2 x^2+1\right )^{3/2} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )-\sqrt{6 \pi } \left (a^2 x^2+1\right )^{3/2} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )+24 a x \left (2 a^2 x^2+3\right ) \sqrt{\tan ^{-1}(a x)}}{72 c^2 \left (a^3 x^2+a\right ) \sqrt{a^2 c x^2+c}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.709, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{\arctan \left ( ax \right ) } \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}}}\, dx \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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