Optimal. Leaf size=99 \[ -3 i a \sin ^{-1}(a x) \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(a x)}\right )+\frac{3}{2} a \text{PolyLog}\left (3,e^{2 i \sin ^{-1}(a x)}\right )-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{x}-i a \sin ^{-1}(a x)^3+3 a \sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.181608, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {4681, 4625, 3717, 2190, 2531, 2282, 6589} \[ -3 i a \sin ^{-1}(a x) \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(a x)}\right )+\frac{3}{2} a \text{PolyLog}\left (3,e^{2 i \sin ^{-1}(a x)}\right )-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{x}-i a \sin ^{-1}(a x)^3+3 a \sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 4681
Rule 4625
Rule 3717
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\sin ^{-1}(a x)^3}{x^2 \sqrt{1-a^2 x^2}} \, dx &=-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{x}+(3 a) \int \frac{\sin ^{-1}(a x)^2}{x} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{x}+(3 a) \operatorname{Subst}\left (\int x^2 \cot (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-i a \sin ^{-1}(a x)^3-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{x}-(6 i a) \operatorname{Subst}\left (\int \frac{e^{2 i x} x^2}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(a x)\right )\\ &=-i a \sin ^{-1}(a x)^3-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{x}+3 a \sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-(6 a) \operatorname{Subst}\left (\int x \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-i a \sin ^{-1}(a x)^3-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{x}+3 a \sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-3 i a \sin ^{-1}(a x) \text{Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )+(3 i a) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-i a \sin ^{-1}(a x)^3-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{x}+3 a \sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-3 i a \sin ^{-1}(a x) \text{Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )+\frac{1}{2} (3 a) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(a x)}\right )\\ &=-i a \sin ^{-1}(a x)^3-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{x}+3 a \sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-3 i a \sin ^{-1}(a x) \text{Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )+\frac{3}{2} a \text{Li}_3\left (e^{2 i \sin ^{-1}(a x)}\right )\\ \end{align*}
Mathematica [A] time = 0.209354, size = 108, normalized size = 1.09 \[ \frac{1}{8} a \left (24 i \sin ^{-1}(a x) \text{PolyLog}\left (2,e^{-2 i \sin ^{-1}(a x)}\right )+12 \text{PolyLog}\left (3,e^{-2 i \sin ^{-1}(a x)}\right )-\frac{8 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a x}+8 i \sin ^{-1}(a x)^3+24 \sin ^{-1}(a x)^2 \log \left (1-e^{-2 i \sin ^{-1}(a x)}\right )-i \pi ^3\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.105, size = 208, normalized size = 2.1 \begin{align*}{\frac{ \left ( \arcsin \left ( ax \right ) \right ) ^{3}}{x} \left ( iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) }-2\,i \left ( \arcsin \left ( ax \right ) \right ) ^{3}a-6\,ia\arcsin \left ( ax \right ){\it polylog} \left ( 2,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 ) +3\,a \left ( \arcsin \left ( ax \right ) \right ) ^{2}\ln \left ( 1-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\,a{\it polylog} \left ( 3,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{\frac{3}{8} \,{\left (x^{2} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{2} + 8 \, \int \frac{\sqrt{a x + 1} \sqrt{-a x + 1} a x^{2} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right ) + 3 \,{\left (a^{2} x^{3} - x\right )} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{2}}{4 \,{\left (a^{2} x^{2} - 1\right )}}\,{d x}\right )} a^{3} x - \sqrt{a x + 1} \sqrt{-a x + 1} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{3}}{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 )^{3}}{a^{2} x^{4} - 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} \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 )^{3}}{\sqrt{-a^{2} x^{2} + 1} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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