Optimal. Leaf size=52 \[ i \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-i \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-2 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4709, 4183, 2279, 2391} \[ i \text {PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-i \text {PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-2 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4183
Rule 4709
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a x)}{x \sqrt {1-a^2 x^2}} \, dx &=\operatorname {Subst}\left (\int x \csc (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-2 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\operatorname {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+\operatorname {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-2 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+i \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )-i \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )\\ &=-2 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+i \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-i \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.13, size = 71, normalized size = 1.37 \[ i \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-i \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )+\sin ^{-1}(a x) \left (\log \left (1-e^{i \sin ^{-1}(a x)}\right )-\log \left (1+e^{i \sin ^{-1}(a x)}\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 2.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a^{2} x^{3} - x}, x\right ) \]
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 \[ \int \frac {\arcsin \left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 103, normalized size = 1.98 \[ \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-\arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+i \dilog \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-i \dilog \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right ) \]
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 \[ \int \frac {\arcsin \left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {asin}\left (a\,x\right )}{x\,\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {\operatorname {asin}{\left (a x \right )}}{x \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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