Optimal. Leaf size=252 \[ -\frac{9 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{8 a c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{24 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)^{3/2}}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\tan ^{-1}(a x)}}{a c^2 \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^{3/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{\sqrt{\tan ^{-1}(a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rubi [A] time = 0.349053, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {4900, 4898, 4905, 4904, 3304, 3352, 3312} \[ -\frac{9 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{8 a c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{24 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)^{3/2}}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\tan ^{-1}(a x)}}{a c^2 \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^{3/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{\sqrt{\tan ^{-1}(a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4900
Rule 4898
Rule 4905
Rule 4904
Rule 3304
Rule 3352
Rule 3312
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=\frac{\sqrt{\tan ^{-1}(a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{x \tan ^{-1}(a x)^{3/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{1}{12} \int \frac{1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{2 \int \frac{\tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}\\ &=\frac{\sqrt{\tan ^{-1}(a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\sqrt{\tan ^{-1}(a x)}}{a c^2 \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^{3/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \tan ^{-1}(a x)^{3/2}}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\int \frac{1}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{2 c}-\frac{\sqrt{1+a^2 x^2} \int \frac{1}{\left (1+a^2 x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{12 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{\tan ^{-1}(a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\sqrt{\tan ^{-1}(a x)}}{a c^2 \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^{3/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \tan ^{-1}(a x)^{3/2}}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \int \frac{1}{\left (1+a^2 x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{2 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\cos ^3(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{12 a c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{\tan ^{-1}(a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\sqrt{\tan ^{-1}(a x)}}{a c^2 \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^{3/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \tan ^{-1}(a x)^{3/2}}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \left (\frac{3 \cos (x)}{4 \sqrt{x}}+\frac{\cos (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\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{\cos (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{2 a c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{\tan ^{-1}(a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\sqrt{\tan ^{-1}(a x)}}{a c^2 \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^{3/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \tan ^{-1}(a x)^{3/2}}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{48 a c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{16 a c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{a c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{\tan ^{-1}(a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\sqrt{\tan ^{-1}(a x)}}{a c^2 \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^{3/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \tan ^{-1}(a x)^{3/2}}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{1+a^2 x^2} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\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 \cos \left (3 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{24 a c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{8 a c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{\tan ^{-1}(a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\sqrt{\tan ^{-1}(a x)}}{a c^2 \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^{3/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \tan ^{-1}(a x)^{3/2}}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{9 \sqrt{\frac{\pi }{2}} \sqrt{1+a^2 x^2} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{8 a c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{1+a^2 x^2} C\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{24 a c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.202902, size = 153, normalized size = 0.61 \[ \frac{-81 \sqrt{2 \pi } \left (a^2 x^2+1\right )^{3/2} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )-\sqrt{6 \pi } \left (a^2 x^2+1\right )^{3/2} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )+24 \sqrt{\tan ^{-1}(a x)} \left (6 a^2 x^2+\left (4 a^3 x^3+6 a x\right ) \tan ^{-1}(a x)+7\right )}{144 c^2 \left (a^3 x^2+a\right ) \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.67, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{3}{2}}} \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{\arctan \left (a x\right )^{\frac{3}{2}}}{{\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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