Optimal. Leaf size=163 \[ i a^2 \sin ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-i a^2 \sin ^{-1}(a x) \text{PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-a^2 \text{PolyLog}\left (3,-e^{i \sin ^{-1}(a x)}\right )+a^2 \text{PolyLog}\left (3,e^{i \sin ^{-1}(a x)}\right )-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x^2}-a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+a^2 \left (-\sin ^{-1}(a x)^2\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\frac{a \sin ^{-1}(a x)}{x} \]
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Rubi [A] time = 0.254875, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4701, 4709, 4183, 2531, 2282, 6589, 4627, 266, 63, 208} \[ i a^2 \sin ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-i a^2 \sin ^{-1}(a x) \text{PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-a^2 \text{PolyLog}\left (3,-e^{i \sin ^{-1}(a x)}\right )+a^2 \text{PolyLog}\left (3,e^{i \sin ^{-1}(a x)}\right )-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x^2}-a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+a^2 \left (-\sin ^{-1}(a x)^2\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\frac{a \sin ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Rule 4701
Rule 4709
Rule 4183
Rule 2531
Rule 2282
Rule 6589
Rule 4627
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sin ^{-1}(a x)^2}{x^3 \sqrt{1-a^2 x^2}} \, dx &=-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x^2}+a \int \frac{\sin ^{-1}(a x)}{x^2} \, dx+\frac{1}{2} a^2 \int \frac{\sin ^{-1}(a x)^2}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a \sin ^{-1}(a x)}{x}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x^2}+\frac{1}{2} a^2 \operatorname{Subst}\left (\int x^2 \csc (x) \, dx,x,\sin ^{-1}(a x)\right )+a^2 \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a \sin ^{-1}(a x)}{x}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x^2}-a^2 \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )-a^2 \operatorname{Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+a^2 \operatorname{Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac{a \sin ^{-1}(a x)}{x}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x^2}-a^2 \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+i a^2 \sin ^{-1}(a x) \text{Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-i a^2 \sin ^{-1}(a x) \text{Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-\left (i a^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+\left (i a^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )-\operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )\\ &=-\frac{a \sin ^{-1}(a x)}{x}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x^2}-a^2 \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+i a^2 \sin ^{-1}(a x) \text{Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-i a^2 \sin ^{-1}(a x) \text{Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-a^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )+a^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )\\ &=-\frac{a \sin ^{-1}(a x)}{x}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x^2}-a^2 \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+i a^2 \sin ^{-1}(a x) \text{Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-i a^2 \sin ^{-1}(a x) \text{Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-a^2 \text{Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+a^2 \text{Li}_3\left (e^{i \sin ^{-1}(a x)}\right )\\ \end{align*}
Mathematica [A] time = 1.33626, size = 194, normalized size = 1.19 \[ \frac{1}{8} a^2 \left (8 i \sin ^{-1}(a x) \left (\text{PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-\text{PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )\right )+8 \left (\text{PolyLog}\left (3,e^{i \sin ^{-1}(a x)}\right )-\text{PolyLog}\left (3,-e^{i \sin ^{-1}(a x)}\right )\right )+4 \sin ^{-1}(a x)^2 \left (\log \left (1-e^{i \sin ^{-1}(a x)}\right )-\log \left (1+e^{i \sin ^{-1}(a x)}\right )\right )-4 \sin ^{-1}(a x) \tan \left (\frac{1}{2} \sin ^{-1}(a x)\right )-4 \sin ^{-1}(a x) \cot \left (\frac{1}{2} \sin ^{-1}(a x)\right )+\sin ^{-1}(a x)^2 \left (-\csc ^2\left (\frac{1}{2} \sin ^{-1}(a x)\right )\right )+\sin ^{-1}(a x)^2 \sec ^2\left (\frac{1}{2} \sin ^{-1}(a x)\right )+8 \log \left (\tan \left (\frac{1}{2} \sin ^{-1}(a x)\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.155, size = 269, normalized size = 1.7 \begin{align*} -{\frac{\arcsin \left ( ax \right ) }{ \left ( 2\,{a}^{2}{x}^{2}-2 \right ){x}^{2}}\sqrt{-{a}^{2}{x}^{2}+1} \left ({a}^{2}{x}^{2}\arcsin \left ( ax \right ) -2\,ax\sqrt{-{a}^{2}{x}^{2}+1}-\arcsin \left ( ax \right ) \right ) }+i{a}^{2}\arcsin \left ( ax \right ){\it polylog} \left ( 2,-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) -i{a}^{2}\arcsin \left ( ax \right ){\it polylog} \left ( 2,iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) -{\frac{ \left ( \arcsin \left ( ax \right ) \right ) ^{2}{a}^{2}}{2}\ln \left ( 1+iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) }+{\frac{ \left ( \arcsin \left ( ax \right ) \right ) ^{2}{a}^{2}}{2}\ln \left ( 1-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) }-{a}^{2}{\it polylog} \left ( 3,-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) +{a}^{2}{\it polylog} \left ( 3,iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) -2\,{\it Artanh} \left ( iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ){a}^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arcsin \left (a x\right )^{2}}{\sqrt{-a^{2} x^{2} + 1} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{a^{2} x^{5} - x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asin}^{2}{\left (a x \right )}}{x^{3} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \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{\arcsin \left (a x\right )^{2}}{\sqrt{-a^{2} x^{2} + 1} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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