Optimal. Leaf size=264 \[ \frac {3}{2} i a^2 \sin ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-\frac {3}{2} i a^2 \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-3 a^2 \sin ^{-1}(a x) \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+3 a^2 \sin ^{-1}(a x) \text {Li}_3\left (e^{i \sin ^{-1}(a x)}\right )+3 i a^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-3 i a^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-3 i a^2 \text {Li}_4\left (-e^{i \sin ^{-1}(a x)}\right )+3 i a^2 \text {Li}_4\left (e^{i \sin ^{-1}(a x)}\right )-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{2 x^2}-a^2 \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-6 a^2 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\frac {3 a \sin ^{-1}(a x)^2}{2 x} \]
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Rubi [A] time = 0.36, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4701, 4709, 4183, 2531, 6609, 2282, 6589, 4627, 2279, 2391} \[ \frac {3}{2} i a^2 \sin ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-\frac {3}{2} i a^2 \sin ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-3 a^2 \sin ^{-1}(a x) \text {PolyLog}\left (3,-e^{i \sin ^{-1}(a x)}\right )+3 a^2 \sin ^{-1}(a x) \text {PolyLog}\left (3,e^{i \sin ^{-1}(a x)}\right )+3 i a^2 \text {PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-3 i a^2 \text {PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-3 i a^2 \text {PolyLog}\left (4,-e^{i \sin ^{-1}(a x)}\right )+3 i a^2 \text {PolyLog}\left (4,e^{i \sin ^{-1}(a x)}\right )-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{2 x^2}-a^2 \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-6 a^2 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\frac {3 a \sin ^{-1}(a x)^2}{2 x} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 4183
Rule 4627
Rule 4701
Rule 4709
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx &=-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{2 x^2}+\frac {1}{2} (3 a) \int \frac {\sin ^{-1}(a x)^2}{x^2} \, dx+\frac {1}{2} a^2 \int \frac {\sin ^{-1}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {3 a \sin ^{-1}(a x)^2}{2 x}-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{2 x^2}+\frac {1}{2} a^2 \operatorname {Subst}\left (\int x^3 \csc (x) \, dx,x,\sin ^{-1}(a x)\right )+\left (3 a^2\right ) \int \frac {\sin ^{-1}(a x)}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {3 a \sin ^{-1}(a x)^2}{2 x}-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{2 x^2}-a^2 \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\frac {1}{2} \left (3 a^2\right ) \operatorname {Subst}\left (\int x^2 \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+\frac {1}{2} \left (3 a^2\right ) \operatorname {Subst}\left (\int x^2 \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+\left (3 a^2\right ) \operatorname {Subst}\left (\int x \csc (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {3 a \sin ^{-1}(a x)^2}{2 x}-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{2 x^2}-6 a^2 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-a^2 \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+\frac {3}{2} i a^2 \sin ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-\frac {3}{2} i a^2 \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-\left (3 i a^2\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+\left (3 i a^2\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )-\left (3 a^2\right ) \operatorname {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+\left (3 a^2\right ) \operatorname {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {3 a \sin ^{-1}(a x)^2}{2 x}-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{2 x^2}-6 a^2 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-a^2 \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+\frac {3}{2} i a^2 \sin ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-\frac {3}{2} i a^2 \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-3 a^2 \sin ^{-1}(a x) \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+3 a^2 \sin ^{-1}(a x) \text {Li}_3\left (e^{i \sin ^{-1}(a x)}\right )+\left (3 i a^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )-\left (3 i a^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )+\left (3 a^2\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )-\left (3 a^2\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {3 a \sin ^{-1}(a x)^2}{2 x}-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{2 x^2}-6 a^2 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-a^2 \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+3 i a^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )+\frac {3}{2} i a^2 \sin ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-3 i a^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-\frac {3}{2} i a^2 \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-3 a^2 \sin ^{-1}(a x) \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+3 a^2 \sin ^{-1}(a x) \text {Li}_3\left (e^{i \sin ^{-1}(a x)}\right )-\left (3 i a^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )+\left (3 i a^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )\\ &=-\frac {3 a \sin ^{-1}(a x)^2}{2 x}-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{2 x^2}-6 a^2 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-a^2 \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+3 i a^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )+\frac {3}{2} i a^2 \sin ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-3 i a^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-\frac {3}{2} i a^2 \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-3 a^2 \sin ^{-1}(a x) \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+3 a^2 \sin ^{-1}(a x) \text {Li}_3\left (e^{i \sin ^{-1}(a x)}\right )-3 i a^2 \text {Li}_4\left (-e^{i \sin ^{-1}(a x)}\right )+3 i a^2 \text {Li}_4\left (e^{i \sin ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 5.04, size = 317, normalized size = 1.20 \[ \frac {1}{16} a^2 \left (24 i \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{-i \sin ^{-1}(a x)}\right )+48 \sin ^{-1}(a x) \text {Li}_3\left (e^{-i \sin ^{-1}(a x)}\right )-48 \sin ^{-1}(a x) \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+24 i \left (\sin ^{-1}(a x)^2+2\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-48 i \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-48 i \text {Li}_4\left (e^{-i \sin ^{-1}(a x)}\right )-48 i \text {Li}_4\left (-e^{i \sin ^{-1}(a x)}\right )+2 i \sin ^{-1}(a x)^4+8 \sin ^{-1}(a x)^3 \log \left (1-e^{-i \sin ^{-1}(a x)}\right )-8 \sin ^{-1}(a x)^3 \log \left (1+e^{i \sin ^{-1}(a x)}\right )+48 \sin ^{-1}(a x) \log \left (1-e^{i \sin ^{-1}(a x)}\right )-48 \sin ^{-1}(a x) \log \left (1+e^{i \sin ^{-1}(a x)}\right )-12 \sin ^{-1}(a x)^2 \tan \left (\frac {1}{2} \sin ^{-1}(a x)\right )-12 \sin ^{-1}(a x)^2 \cot \left (\frac {1}{2} \sin ^{-1}(a x)\right )-2 \sin ^{-1}(a x)^3 \csc ^2\left (\frac {1}{2} \sin ^{-1}(a x)\right )+2 \sin ^{-1}(a x)^3 \sec ^2\left (\frac {1}{2} \sin ^{-1}(a x)\right )-i \pi ^4\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{3}}{a^{2} x^{5} - x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arcsin \left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 428, normalized size = 1.62 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right )^{2} \left (a^{2} x^{2} \arcsin \left (a x \right )-3 a x \sqrt {-a^{2} x^{2}+1}-\arcsin \left (a x \right )\right )}{2 \left (a^{2} x^{2}-1\right ) x^{2}}-\frac {\ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right ) \arcsin \left (a x \right )^{3} a^{2}}{2}+\frac {\ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right ) \arcsin \left (a x \right )^{3} a^{2}}{2}-3 a^{2} \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+\frac {3 i a^{2} \arcsin \left (a x \right )^{2} \polylog \left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )}{2}-3 a^{2} \arcsin \left (a x \right ) \polylog \left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+3 a^{2} \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-\frac {3 i a^{2} \arcsin \left (a x \right )^{2} \polylog \left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )}{2}+3 a^{2} \arcsin \left (a x \right ) \polylog \left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )+3 i a^{2} \polylog \left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-3 i a^{2} \polylog \left (4, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-3 i a^{2} \polylog \left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+3 i a^{2} \polylog \left (4, i a x +\sqrt {-a^{2} x^{2}+1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arcsin \left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {asin}\left (a\,x\right )}^3}{x^3\,\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asin}^{3}{\left (a x \right )}}{x^{3} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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