Optimal. Leaf size=108 \[ 6 i a \sin ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-6 i a \sin ^{-1}(a x) \text{PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-6 a \text{PolyLog}\left (3,-e^{i \sin ^{-1}(a x)}\right )+6 a \text{PolyLog}\left (3,e^{i \sin ^{-1}(a x)}\right )-\frac{\sin ^{-1}(a x)^3}{x}-6 a \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.1632, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4627, 4709, 4183, 2531, 2282, 6589} \[ 6 i a \sin ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-6 i a \sin ^{-1}(a x) \text{PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-6 a \text{PolyLog}\left (3,-e^{i \sin ^{-1}(a x)}\right )+6 a \text{PolyLog}\left (3,e^{i \sin ^{-1}(a x)}\right )-\frac{\sin ^{-1}(a x)^3}{x}-6 a \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 4627
Rule 4709
Rule 4183
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\sin ^{-1}(a x)^3}{x^2} \, dx &=-\frac{\sin ^{-1}(a x)^3}{x}+(3 a) \int \frac{\sin ^{-1}(a x)^2}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sin ^{-1}(a x)^3}{x}+(3 a) \operatorname{Subst}\left (\int x^2 \csc (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac{\sin ^{-1}(a x)^3}{x}-6 a \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-(6 a) \operatorname{Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+(6 a) \operatorname{Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac{\sin ^{-1}(a x)^3}{x}-6 a \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+6 i a \sin ^{-1}(a x) \text{Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-6 i a \sin ^{-1}(a x) \text{Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-(6 i a) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+(6 i a) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac{\sin ^{-1}(a x)^3}{x}-6 a \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+6 i a \sin ^{-1}(a x) \text{Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-6 i a \sin ^{-1}(a x) \text{Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-(6 a) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )+(6 a) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )\\ &=-\frac{\sin ^{-1}(a x)^3}{x}-6 a \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+6 i a \sin ^{-1}(a x) \text{Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-6 i a \sin ^{-1}(a x) \text{Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-6 a \text{Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+6 a \text{Li}_3\left (e^{i \sin ^{-1}(a x)}\right )\\ \end{align*}
Mathematica [A] time = 0.12395, size = 133, normalized size = 1.23 \[ a \left (6 i \sin ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-6 i \sin ^{-1}(a x) \text{PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-6 \text{PolyLog}\left (3,-e^{i \sin ^{-1}(a x)}\right )+6 \text{PolyLog}\left (3,e^{i \sin ^{-1}(a x)}\right )-\frac{\sin ^{-1}(a x)^3}{a x}+3 \sin ^{-1}(a x)^2 \log \left (1-e^{i \sin ^{-1}(a x)}\right )-3 \sin ^{-1}(a x)^2 \log \left (1+e^{i \sin ^{-1}(a x)}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.074, size = 179, normalized size = 1.7 \begin{align*} -{\frac{ \left ( \arcsin \left ( ax \right ) \right ) ^{3}}{x}}-3\,a \left ( \arcsin \left ( ax \right ) \right ) ^{2}\ln \left ( 1+iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) +6\,ia\arcsin \left ( ax \right ){\it polylog} \left ( 2,-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) -6\,a{\it polylog} \left ( 3,-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) +3\,a \left ( \arcsin \left ( ax \right ) \right ) ^{2}\ln \left ( 1-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) -6\,ia\arcsin \left ( ax \right ){\it polylog} \left ( 2,iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) +6\,a{\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*} -\frac{\arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{3} + 3 \, a x \int \frac{\sqrt{-a x + 1} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{2}}{\sqrt{a x + 1}{\left (a x - 1\right )} x}\,{d x}}{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{\arcsin \left (a x\right )^{3}}{x^{2}}, 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}^{3}{\left (a x \right )}}{x^{2}}\, 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 )^{3}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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