Optimal. Leaf size=102 \[ -\frac {3}{2} i a^2 \text {Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )-\frac {3 a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x}-\frac {3}{2} i a^2 \sin ^{-1}(a x)^2+3 a^2 \sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-\frac {\sin ^{-1}(a x)^3}{2 x^2} \]
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Rubi [A] time = 0.17, antiderivative size = 102, 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 = {4627, 4681, 4625, 3717, 2190, 2279, 2391} \[ -\frac {3}{2} i a^2 \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(a x)}\right )-\frac {3 a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x}-\frac {3}{2} i a^2 \sin ^{-1}(a x)^2+3 a^2 \sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-\frac {\sin ^{-1}(a x)^3}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3717
Rule 4625
Rule 4627
Rule 4681
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a x)^3}{x^3} \, dx &=-\frac {\sin ^{-1}(a x)^3}{2 x^2}+\frac {1}{2} (3 a) \int \frac {\sin ^{-1}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {3 a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x}-\frac {\sin ^{-1}(a x)^3}{2 x^2}+\left (3 a^2\right ) \int \frac {\sin ^{-1}(a x)}{x} \, dx\\ &=-\frac {3 a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x}-\frac {\sin ^{-1}(a x)^3}{2 x^2}+\left (3 a^2\right ) \operatorname {Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {3}{2} i a^2 \sin ^{-1}(a x)^2-\frac {3 a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x}-\frac {\sin ^{-1}(a x)^3}{2 x^2}-\left (6 i a^2\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {3}{2} i a^2 \sin ^{-1}(a x)^2-\frac {3 a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x}-\frac {\sin ^{-1}(a x)^3}{2 x^2}+3 a^2 \sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-\left (3 a^2\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {3}{2} i a^2 \sin ^{-1}(a x)^2-\frac {3 a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x}-\frac {\sin ^{-1}(a x)^3}{2 x^2}+3 a^2 \sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )+\frac {1}{2} \left (3 i a^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(a x)}\right )\\ &=-\frac {3}{2} i a^2 \sin ^{-1}(a x)^2-\frac {3 a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x}-\frac {\sin ^{-1}(a x)^3}{2 x^2}+3 a^2 \sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-\frac {3}{2} i a^2 \text {Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.26, size = 92, normalized size = 0.90 \[ -\frac {3}{2} i a^2 \text {Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )-\frac {\sin ^{-1}(a x) \left (3 a x \left (\sqrt {1-a^2 x^2}+i a x\right ) \sin ^{-1}(a x)-6 a^2 x^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )+\sin ^{-1}(a x)^2\right )}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arcsin \left (a x\right )^{3}}{x^{3}}, x\right ) \]
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 \[ \int \frac {\arcsin \left (a x\right )^{3}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 163, normalized size = 1.60 \[ -\frac {3 i a^{2} \arcsin \left (a x \right )^{2}}{2}-\frac {3 a \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}}{2 x}-\frac {\arcsin \left (a x \right )^{3}}{2 x^{2}}+3 a^{2} \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+3 a^{2} \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-3 i a^{2} \polylog \left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )-3 i a^{2} \polylog \left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right ) \]
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 \[ -\frac {\frac {3}{4} \, {\left (\sqrt {a x + 1} \sqrt {-a x + 1} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{2} + 4 \, x \int \frac {3 \, \sqrt {a x + 1} \sqrt {-a x + 1} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{2} - 2 \, {\left (a^{3} x^{3} - a x\right )} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )}{4 \, {\left (a^{2} x^{4} - x^{2}\right )}}\,{d x}\right )} a x + \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{3}}{2 \, x^{2}} \]
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 \[ \int \frac {{\mathrm {asin}\left (a\,x\right )}^3}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {asin}^{3}{\left (a x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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