Optimal. Leaf size=179 \[ i a^3 \sin ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-i a^3 \sin ^{-1}(a x) \text{PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-a^3 \text{PolyLog}\left (3,-e^{i \sin ^{-1}(a x)}\right )+a^3 \text{PolyLog}\left (3,e^{i \sin ^{-1}(a x)}\right )-\frac{a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x^2}-a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{a^2 \sin ^{-1}(a x)}{x}+a^3 \left (-\sin ^{-1}(a x)^2\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\frac{\sin ^{-1}(a x)^3}{3 x^3} \]
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Rubi [A] time = 0.284581, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {4627, 4701, 4709, 4183, 2531, 2282, 6589, 266, 63, 208} \[ i a^3 \sin ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-i a^3 \sin ^{-1}(a x) \text{PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-a^3 \text{PolyLog}\left (3,-e^{i \sin ^{-1}(a x)}\right )+a^3 \text{PolyLog}\left (3,e^{i \sin ^{-1}(a x)}\right )-\frac{a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x^2}-a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{a^2 \sin ^{-1}(a x)}{x}+a^3 \left (-\sin ^{-1}(a x)^2\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\frac{\sin ^{-1}(a x)^3}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 4627
Rule 4701
Rule 4709
Rule 4183
Rule 2531
Rule 2282
Rule 6589
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sin ^{-1}(a x)^3}{x^4} \, dx &=-\frac{\sin ^{-1}(a x)^3}{3 x^3}+a \int \frac{\sin ^{-1}(a x)^2}{x^3 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x^2}-\frac{\sin ^{-1}(a x)^3}{3 x^3}+a^2 \int \frac{\sin ^{-1}(a x)}{x^2} \, dx+\frac{1}{2} a^3 \int \frac{\sin ^{-1}(a x)^2}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a^2 \sin ^{-1}(a x)}{x}-\frac{a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x^2}-\frac{\sin ^{-1}(a x)^3}{3 x^3}+\frac{1}{2} a^3 \operatorname{Subst}\left (\int x^2 \csc (x) \, dx,x,\sin ^{-1}(a x)\right )+a^3 \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a^2 \sin ^{-1}(a x)}{x}-\frac{a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x^2}-\frac{\sin ^{-1}(a x)^3}{3 x^3}-a^3 \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+\frac{1}{2} a^3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )-a^3 \operatorname{Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+a^3 \operatorname{Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac{a^2 \sin ^{-1}(a x)}{x}-\frac{a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x^2}-\frac{\sin ^{-1}(a x)^3}{3 x^3}-a^3 \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+i a^3 \sin ^{-1}(a x) \text{Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-i a^3 \sin ^{-1}(a x) \text{Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-a \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )-\left (i a^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+\left (i a^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac{a^2 \sin ^{-1}(a x)}{x}-\frac{a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x^2}-\frac{\sin ^{-1}(a x)^3}{3 x^3}-a^3 \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+i a^3 \sin ^{-1}(a x) \text{Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-i a^3 \sin ^{-1}(a x) \text{Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-a^3 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )+a^3 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )\\ &=-\frac{a^2 \sin ^{-1}(a x)}{x}-\frac{a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x^2}-\frac{\sin ^{-1}(a x)^3}{3 x^3}-a^3 \sin ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+i a^3 \sin ^{-1}(a x) \text{Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-i a^3 \sin ^{-1}(a x) \text{Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-a^3 \text{Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+a^3 \text{Li}_3\left (e^{i \sin ^{-1}(a x)}\right )\\ \end{align*}
Mathematica [A] time = 2.79478, size = 284, normalized size = 1.59 \[ \frac{1}{48} a^3 \left (48 i \sin ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-48 i \sin ^{-1}(a x) \text{PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-48 \text{PolyLog}\left (3,-e^{i \sin ^{-1}(a x)}\right )+48 \text{PolyLog}\left (3,e^{i \sin ^{-1}(a x)}\right )-\frac{16 \sin ^{-1}(a x)^3 \sin ^4\left (\frac{1}{2} \sin ^{-1}(a x)\right )}{a^3 x^3}+24 \sin ^{-1}(a x)^2 \log \left (1-e^{i \sin ^{-1}(a x)}\right )-24 \sin ^{-1}(a x)^2 \log \left (1+e^{i \sin ^{-1}(a x)}\right )-4 \sin ^{-1}(a x)^3 \tan \left (\frac{1}{2} \sin ^{-1}(a x)\right )-24 \sin ^{-1}(a x) \tan \left (\frac{1}{2} \sin ^{-1}(a x)\right )-4 \sin ^{-1}(a x)^3 \cot \left (\frac{1}{2} \sin ^{-1}(a x)\right )-24 \sin ^{-1}(a x) \cot \left (\frac{1}{2} \sin ^{-1}(a x)\right )-a x \sin ^{-1}(a x)^3 \csc ^4\left (\frac{1}{2} \sin ^{-1}(a x)\right )-6 \sin ^{-1}(a x)^2 \csc ^2\left (\frac{1}{2} \sin ^{-1}(a x)\right )+6 \sin ^{-1}(a x)^2 \sec ^2\left (\frac{1}{2} \sin ^{-1}(a x)\right )+48 \log \left (\tan \left (\frac{1}{2} \sin ^{-1}(a x)\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.148, size = 250, normalized size = 1.4 \begin{align*} -{\frac{a \left ( \arcsin \left ( ax \right ) \right ) ^{2}}{2\,{x}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{a}^{2}\arcsin \left ( ax \right ) }{x}}-{\frac{ \left ( \arcsin \left ( ax \right ) \right ) ^{3}}{3\,{x}^{3}}}-{\frac{{a}^{3} \left ( \arcsin \left ( ax \right ) \right ) ^{2}}{2}\ln \left ( 1+iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) }+i{a}^{3}\arcsin \left ( ax \right ){\it polylog} \left ( 2,-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) -{a}^{3}{\it polylog} \left ( 3,-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) +{\frac{{a}^{3} \left ( \arcsin \left ( ax \right ) \right ) ^{2}}{2}\ln \left ( 1-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) }-i{a}^{3}\arcsin \left ( ax \right ){\it polylog} \left ( 2,iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) +{a}^{3}{\it polylog} \left ( 3,iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) -2\,{a}^{3}{\it Artanh} \left ( 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{3 \, a x^{3} \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^{3}}\,{d x} + \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{3}}{3 \, x^{3}} \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^{4}}, 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^{4}}\, 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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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