Optimal. Leaf size=92 \[ 2 i \sin ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-2 i \sin ^{-1}(a x) \text{PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-2 \text{PolyLog}\left (3,-e^{i \sin ^{-1}(a x)}\right )+2 \text{PolyLog}\left (3,e^{i \sin ^{-1}(a x)}\right )-2 \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.142958, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4709, 4183, 2531, 2282, 6589} \[ 2 i \sin ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-2 i \sin ^{-1}(a x) \text{PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-2 \text{PolyLog}\left (3,-e^{i \sin ^{-1}(a x)}\right )+2 \text{PolyLog}\left (3,e^{i \sin ^{-1}(a x)}\right )-2 \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 4709
Rule 4183
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\sin ^{-1}(a x)^2}{x \sqrt{1-a^2 x^2}} \, dx &=\operatorname{Subst}\left (\int x^2 \csc (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-2 \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-2 \operatorname{Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+2 \operatorname{Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-2 \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+2 i \sin ^{-1}(a x) \text{Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-2 i \sin ^{-1}(a x) \text{Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-2 i \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+2 i \operatorname{Subst}\left (\int \text{Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-2 \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+2 i \sin ^{-1}(a x) \text{Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-2 i \sin ^{-1}(a x) \text{Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )+2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )\\ &=-2 \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+2 i \sin ^{-1}(a x) \text{Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-2 i \sin ^{-1}(a x) \text{Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-2 \text{Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+2 \text{Li}_3\left (e^{i \sin ^{-1}(a x)}\right )\\ \end{align*}
Mathematica [A] time = 0.10266, size = 116, normalized size = 1.26 \[ 2 i \sin ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-2 i \sin ^{-1}(a x) \text{PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-2 \text{PolyLog}\left (3,-e^{i \sin ^{-1}(a x)}\right )+2 \text{PolyLog}\left (3,e^{i \sin ^{-1}(a x)}\right )+\sin ^{-1}(a x)^2 \log \left (1-e^{i \sin ^{-1}(a x)}\right )-\sin ^{-1}(a x)^2 \log \left (1+e^{i \sin ^{-1}(a x)}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.06, size = 161, normalized size = 1.8 \begin{align*} - \left ( \arcsin \left ( ax \right ) \right ) ^{2}\ln \left ( 1+iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) +2\,i\arcsin \left ( ax \right ){\it polylog} \left ( 2,-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) -2\,{\it polylog} \left ( 3,-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) + \left ( \arcsin \left ( ax \right ) \right ) ^{2}\ln \left ( 1-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) -2\,i\arcsin \left ( ax \right ){\it polylog} \left ( 2,iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) +2\,{\it polylog} \left ( 3,iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) \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}\,{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^{3} - x}, 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 \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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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