Optimal. Leaf size=89 \[ -\frac{3 \sqrt{\pi } S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a^2}+\frac{3 x \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{8 a}-\frac{\sin ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \sin ^{-1}(a x)^{3/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.182785, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {4629, 4707, 4641, 4635, 4406, 12, 3305, 3351} \[ -\frac{3 \sqrt{\pi } S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a^2}+\frac{3 x \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{8 a}-\frac{\sin ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \sin ^{-1}(a x)^{3/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4629
Rule 4707
Rule 4641
Rule 4635
Rule 4406
Rule 12
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int x \sin ^{-1}(a x)^{3/2} \, dx &=\frac{1}{2} x^2 \sin ^{-1}(a x)^{3/2}-\frac{1}{4} (3 a) \int \frac{x^2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{3 x \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{8 a}+\frac{1}{2} x^2 \sin ^{-1}(a x)^{3/2}-\frac{3}{16} \int \frac{x}{\sqrt{\sin ^{-1}(a x)}} \, dx-\frac{3 \int \frac{\sqrt{\sin ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx}{8 a}\\ &=\frac{3 x \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{8 a}-\frac{\sin ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \sin ^{-1}(a x)^{3/2}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{16 a^2}\\ &=\frac{3 x \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{8 a}-\frac{\sin ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \sin ^{-1}(a x)^{3/2}-\frac{3 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 \sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{16 a^2}\\ &=\frac{3 x \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{8 a}-\frac{\sin ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \sin ^{-1}(a x)^{3/2}-\frac{3 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{32 a^2}\\ &=\frac{3 x \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{8 a}-\frac{\sin ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \sin ^{-1}(a x)^{3/2}-\frac{3 \operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{16 a^2}\\ &=\frac{3 x \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{8 a}-\frac{\sin ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \sin ^{-1}(a x)^{3/2}-\frac{3 \sqrt{\pi } S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a^2}\\ \end{align*}
Mathematica [C] time = 0.0151286, size = 71, normalized size = 0.8 \[ \frac{\sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-2 i \sin ^{-1}(a x)\right )+\sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},2 i \sin ^{-1}(a x)\right )}{16 \sqrt{2} a^2 \sqrt{\sin ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.036, size = 64, normalized size = 0.7 \begin{align*} -{\frac{1}{32\,{a}^{2}} \left ( 8\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\cos \left ( 2\,\arcsin \left ( ax \right ) \right ) +3\,\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ( 2\,{\frac{\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -6\,\arcsin \left ( ax \right ) \sin \left ( 2\,\arcsin \left ( ax \right ) \right ) \right ){\frac{1}{\sqrt{\arcsin \left ( ax \right ) }}}} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{asin}^{\frac{3}{2}}{\left (a x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [C] time = 1.34236, size = 144, normalized size = 1.62 \begin{align*} -\frac{\arcsin \left (a x\right )^{\frac{3}{2}} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{8 \, a^{2}} - \frac{\arcsin \left (a x\right )^{\frac{3}{2}} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{8 \, a^{2}} - \frac{\left (3 i - 3\right ) \, \sqrt{\pi } \operatorname{erf}\left (\left (i - 1\right ) \, \sqrt{\arcsin \left (a x\right )}\right )}{128 \, a^{2}} + \frac{\left (3 i + 3\right ) \, \sqrt{\pi } \operatorname{erf}\left (-\left (i + 1\right ) \, \sqrt{\arcsin \left (a x\right )}\right )}{128 \, a^{2}} - \frac{3 i \, \sqrt{\arcsin \left (a x\right )} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{32 \, a^{2}} + \frac{3 i \, \sqrt{\arcsin \left (a x\right )} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{32 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]