Optimal. Leaf size=129 \[ -\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{2 a^2 c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^{3/2}}{a^2 c \sqrt{a^2 c x^2+c}}+\frac{3 x \sqrt{\tan ^{-1}(a x)}}{2 a c \sqrt{a^2 c x^2+c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.211357, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4930, 4905, 4904, 3296, 3305, 3351} \[ -\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{2 a^2 c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^{3/2}}{a^2 c \sqrt{a^2 c x^2+c}}+\frac{3 x \sqrt{\tan ^{-1}(a x)}}{2 a c \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4930
Rule 4905
Rule 4904
Rule 3296
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \frac{x \tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\frac{\tan ^{-1}(a x)^{3/2}}{a^2 c \sqrt{c+a^2 c x^2}}+\frac{3 \int \frac{\sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{2 a}\\ &=-\frac{\tan ^{-1}(a x)^{3/2}}{a^2 c \sqrt{c+a^2 c x^2}}+\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \int \frac{\sqrt{\tan ^{-1}(a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{2 a c \sqrt{c+a^2 c x^2}}\\ &=-\frac{\tan ^{-1}(a x)^{3/2}}{a^2 c \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 )}{2 a^2 c \sqrt{c+a^2 c x^2}}\\ &=\frac{3 x \sqrt{\tan ^{-1}(a x)}}{2 a c \sqrt{c+a^2 c x^2}}-\frac{\tan ^{-1}(a x)^{3/2}}{a^2 c \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 )}{4 a^2 c \sqrt{c+a^2 c x^2}}\\ &=\frac{3 x \sqrt{\tan ^{-1}(a x)}}{2 a c \sqrt{c+a^2 c x^2}}-\frac{\tan ^{-1}(a x)^{3/2}}{a^2 c \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 )}{2 a^2 c \sqrt{c+a^2 c x^2}}\\ &=\frac{3 x \sqrt{\tan ^{-1}(a x)}}{2 a c \sqrt{c+a^2 c x^2}}-\frac{\tan ^{-1}(a x)^{3/2}}{a^2 c \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 )}{2 a^2 c \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [C] time = 0.172261, size = 128, normalized size = 0.99 \[ \frac{3 \sqrt{a^2 x^2+1} \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-i \tan ^{-1}(a x)\right )+3 \sqrt{a^2 x^2+1} \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},i \tan ^{-1}(a x)\right )+4 \left (3 a x-2 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)}{8 a^2 c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.906, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{3}{2}}} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \arctan \left (a x\right )^{\frac{3}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]