Optimal. Leaf size=97 \[ -\frac{3}{2} i \sin ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(a x)}\right )+\frac{3}{2} \sin ^{-1}(a x) \text{PolyLog}\left (3,e^{2 i \sin ^{-1}(a x)}\right )+\frac{3}{4} i \text{PolyLog}\left (4,e^{2 i \sin ^{-1}(a x)}\right )-\frac{1}{4} i \sin ^{-1}(a x)^4+\sin ^{-1}(a x)^3 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.108999, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {4625, 3717, 2190, 2531, 6609, 2282, 6589} \[ -\frac{3}{2} i \sin ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(a x)}\right )+\frac{3}{2} \sin ^{-1}(a x) \text{PolyLog}\left (3,e^{2 i \sin ^{-1}(a x)}\right )+\frac{3}{4} i \text{PolyLog}\left (4,e^{2 i \sin ^{-1}(a x)}\right )-\frac{1}{4} i \sin ^{-1}(a x)^4+\sin ^{-1}(a x)^3 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 4625
Rule 3717
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\sin ^{-1}(a x)^3}{x} \, dx &=\operatorname{Subst}\left (\int x^3 \cot (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac{1}{4} i \sin ^{-1}(a x)^4-2 i \operatorname{Subst}\left (\int \frac{e^{2 i x} x^3}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac{1}{4} i \sin ^{-1}(a x)^4+\sin ^{-1}(a x)^3 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-3 \operatorname{Subst}\left (\int x^2 \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac{1}{4} i \sin ^{-1}(a x)^4+\sin ^{-1}(a x)^3 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-\frac{3}{2} i \sin ^{-1}(a x)^2 \text{Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )+3 i \operatorname{Subst}\left (\int x \text{Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac{1}{4} i \sin ^{-1}(a x)^4+\sin ^{-1}(a x)^3 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-\frac{3}{2} i \sin ^{-1}(a x)^2 \text{Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )+\frac{3}{2} \sin ^{-1}(a x) \text{Li}_3\left (e^{2 i \sin ^{-1}(a x)}\right )-\frac{3}{2} \operatorname{Subst}\left (\int \text{Li}_3\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac{1}{4} i \sin ^{-1}(a x)^4+\sin ^{-1}(a x)^3 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-\frac{3}{2} i \sin ^{-1}(a x)^2 \text{Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )+\frac{3}{2} \sin ^{-1}(a x) \text{Li}_3\left (e^{2 i \sin ^{-1}(a x)}\right )+\frac{3}{4} i \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(a x)}\right )\\ &=-\frac{1}{4} i \sin ^{-1}(a x)^4+\sin ^{-1}(a x)^3 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-\frac{3}{2} i \sin ^{-1}(a x)^2 \text{Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )+\frac{3}{2} \sin ^{-1}(a x) \text{Li}_3\left (e^{2 i \sin ^{-1}(a x)}\right )+\frac{3}{4} i \text{Li}_4\left (e^{2 i \sin ^{-1}(a x)}\right )\\ \end{align*}
Mathematica [A] time = 0.0556827, size = 97, normalized size = 1. \[ -\frac{1}{64} i \left (-96 \sin ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{-2 i \sin ^{-1}(a x)}\right )+96 i \sin ^{-1}(a x) \text{PolyLog}\left (3,e^{-2 i \sin ^{-1}(a x)}\right )+48 \text{PolyLog}\left (4,e^{-2 i \sin ^{-1}(a x)}\right )-16 \sin ^{-1}(a x)^4+64 i \sin ^{-1}(a x)^3 \log \left (1-e^{-2 i \sin ^{-1}(a x)}\right )+\pi ^4\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.046, size = 229, normalized size = 2.4 \begin{align*} -{\frac{i}{4}} \left ( \arcsin \left ( ax \right ) \right ) ^{4}+ \left ( \arcsin \left ( ax \right ) \right ) ^{3}\ln \left ( 1+iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) -3\,i \left ( \arcsin \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) +6\,\arcsin \left ( ax \right ){\it polylog} \left ( 3,-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) +6\,i{\it polylog} \left ( 4,-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) + \left ( \arcsin \left ( ax \right ) \right ) ^{3}\ln \left ( 1-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) -3\,i \left ( \arcsin \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) +6\,\arcsin \left ( ax \right ){\it polylog} \left ( 3,iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) +6\,i{\it polylog} \left ( 4,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 )^{3}}{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{\arcsin \left (a x\right )^{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}^{3}{\left (a x \right )}}{x}\, 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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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