Optimal. Leaf size=179 \[ -\frac{i \sqrt{1-a^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^2}{c \sqrt{c-a^2 c x^2}}-\frac{i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a c \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.128756, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4653, 4675, 3719, 2190, 2279, 2391} \[ -\frac{i \sqrt{1-a^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^2}{c \sqrt{c-a^2 c x^2}}-\frac{i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a c \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4653
Rule 4675
Rule 3719
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\sin ^{-1}(a x)^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac{x \sin ^{-1}(a x)^2}{c \sqrt{c-a^2 c x^2}}-\frac{\left (2 a \sqrt{1-a^2 x^2}\right ) \int \frac{x \sin ^{-1}(a x)}{1-a^2 x^2} \, dx}{c \sqrt{c-a^2 c x^2}}\\ &=\frac{x \sin ^{-1}(a x)^2}{c \sqrt{c-a^2 c x^2}}-\frac{\left (2 \sqrt{1-a^2 x^2}\right ) \operatorname{Subst}\left (\int x \tan (x) \, dx,x,\sin ^{-1}(a x)\right )}{a c \sqrt{c-a^2 c x^2}}\\ &=\frac{x \sin ^{-1}(a x)^2}{c \sqrt{c-a^2 c x^2}}-\frac{i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a c \sqrt{c-a^2 c x^2}}+\frac{\left (4 i \sqrt{1-a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(a x)\right )}{a c \sqrt{c-a^2 c x^2}}\\ &=\frac{x \sin ^{-1}(a x)^2}{c \sqrt{c-a^2 c x^2}}-\frac{i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a c \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}}-\frac{\left (2 \sqrt{1-a^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a c \sqrt{c-a^2 c x^2}}\\ &=\frac{x \sin ^{-1}(a x)^2}{c \sqrt{c-a^2 c x^2}}-\frac{i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a c \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}}+\frac{\left (i \sqrt{1-a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}}\\ &=\frac{x \sin ^{-1}(a x)^2}{c \sqrt{c-a^2 c x^2}}-\frac{i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a c \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}}-\frac{i \sqrt{1-a^2 x^2} \text{Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.221492, size = 108, normalized size = 0.6 \[ \frac{\sin ^{-1}(a x) \left (a x \sin ^{-1}(a x)+\sqrt{1-a^2 x^2} \left (2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )-i \sin ^{-1}(a x)\right )\right )-i \sqrt{1-a^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt{c \left (1-a^2 x^2\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.102, size = 169, normalized size = 0.9 \begin{align*} -{\frac{ \left ( \arcsin \left ( ax \right ) \right ) ^{2}}{a{c}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) }+{\frac{i}{a{c}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ( 2\,i\arcsin \left ( ax \right ) \ln \left ( 1+ \left ( iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) ^{2} \right ) +2\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}+{\it polylog} \left ( 2,- \left ( iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) ^{2} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arcsin \left (a x\right )^{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]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-a^{2} c x^{2} + c} \arcsin \left (a x\right )^{2}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \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 \frac{\operatorname{asin}^{2}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, dx \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{\arcsin \left (a x\right )^{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]