Optimal. Leaf size=97 \[ -\frac {3}{2} i \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )+\frac {3}{2} \sin ^{-1}(a x) \text {Li}_3\left (e^{2 i \sin ^{-1}(a x)}\right )+\frac {3}{4} i \text {Li}_4\left (e^{2 i \sin ^{-1}(a x)}\right )-\frac {1}{4} i \sin ^{-1}(a x)^4+\sin ^{-1}(a x)^3 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, 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 2190
Rule 2282
Rule 2531
Rule 3717
Rule 4625
Rule 6589
Rule 6609
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*}
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Mathematica [A] time = 0.06, size = 97, normalized size = 1.00 \[ -\frac {1}{64} i \left (-96 \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{-2 i \sin ^{-1}(a x)}\right )+96 i \sin ^{-1}(a x) \text {Li}_3\left (e^{-2 i \sin ^{-1}(a x)}\right )+48 \text {Li}_4\left (e^{-2 i \sin ^{-1}(a x)}\right )-16 \sin ^{-1}(a x)^4+64 i \sin ^{-1}(a x)^3 \log \left (1-e^{-2 i \sin ^{-1}(a x)}\right )+\pi ^4\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arcsin \left (a x\right )^{3}}{x}, x\right ) \]
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 \[ \int \frac {\arcsin \left (a x\right )^{3}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 229, normalized size = 2.36 \[ -\frac {i \arcsin \left (a x \right )^{4}}{4}+\arcsin \left (a x \right )^{3} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-3 i \arcsin \left (a x \right )^{2} \polylog \left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+6 \arcsin \left (a x \right ) \polylog \left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+6 i \polylog \left (4, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+\arcsin \left (a x \right )^{3} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-3 i \arcsin \left (a x \right )^{2} \polylog \left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+6 \arcsin \left (a x \right ) \polylog \left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )+6 i \polylog \left (4, i a x +\sqrt {-a^{2} x^{2}+1}\right ) \]
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 \[ \int \frac {\arcsin \left (a x\right )^{3}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {asin}\left (a\,x\right )}^3}{x} \,d x \]
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 \[ \int \frac {\operatorname {asin}^{3}{\left (a x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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