Optimal. Leaf size=62 \[ i \operatorname {PolyLog}\left (2,-e^{i \sin ^{-1}(x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \sin ^{-1}(x)}\right )+\frac {\sin ^{-1}(x)}{\sqrt {1-x^2}}-\tanh ^{-1}(x)-2 \sin ^{-1}(x) \tanh ^{-1}\left (e^{i \sin ^{-1}(x)}\right ) \]
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Rubi [A] time = 0.12, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {4705, 4709, 4183, 2279, 2391, 206} \[ i \text {PolyLog}\left (2,-e^{i \sin ^{-1}(x)}\right )-i \text {PolyLog}\left (2,e^{i \sin ^{-1}(x)}\right )+\frac {\sin ^{-1}(x)}{\sqrt {1-x^2}}-\tanh ^{-1}(x)-2 \sin ^{-1}(x) \tanh ^{-1}\left (e^{i \sin ^{-1}(x)}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 2279
Rule 2391
Rule 4183
Rule 4705
Rule 4709
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(x)}{x \left (1-x^2\right )^{3/2}} \, dx &=\frac {\sin ^{-1}(x)}{\sqrt {1-x^2}}-\int \frac {1}{1-x^2} \, dx+\int \frac {\sin ^{-1}(x)}{x \sqrt {1-x^2}} \, dx\\ &=\frac {\sin ^{-1}(x)}{\sqrt {1-x^2}}-\tanh ^{-1}(x)+\operatorname {Subst}\left (\int x \csc (x) \, dx,x,\sin ^{-1}(x)\right )\\ &=\frac {\sin ^{-1}(x)}{\sqrt {1-x^2}}-2 \sin ^{-1}(x) \tanh ^{-1}\left (e^{i \sin ^{-1}(x)}\right )-\tanh ^{-1}(x)-\operatorname {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(x)\right )+\operatorname {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(x)\right )\\ &=\frac {\sin ^{-1}(x)}{\sqrt {1-x^2}}-2 \sin ^{-1}(x) \tanh ^{-1}\left (e^{i \sin ^{-1}(x)}\right )-\tanh ^{-1}(x)+i \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(x)}\right )-i \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(x)}\right )\\ &=\frac {\sin ^{-1}(x)}{\sqrt {1-x^2}}-2 \sin ^{-1}(x) \tanh ^{-1}\left (e^{i \sin ^{-1}(x)}\right )-\tanh ^{-1}(x)+i \text {Li}_2\left (-e^{i \sin ^{-1}(x)}\right )-i \text {Li}_2\left (e^{i \sin ^{-1}(x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.19, size = 112, normalized size = 1.81 \[ i \operatorname {PolyLog}\left (2,-e^{i \sin ^{-1}(x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \sin ^{-1}(x)}\right )+\frac {\sin ^{-1}(x)}{\sqrt {1-x^2}}+\sin ^{-1}(x) \log \left (1-e^{i \sin ^{-1}(x)}\right )-\sin ^{-1}(x) \log \left (1+e^{i \sin ^{-1}(x)}\right )+\log \left (\cos \left (\frac {1}{2} \sin ^{-1}(x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(x)\right )\right )-\log \left (\sin \left (\frac {1}{2} \sin ^{-1}(x)\right )+\cos \left (\frac {1}{2} \sin ^{-1}(x)\right )\right ) \]
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{-1}(x)}{x \left (1-x^2\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{2} + 1} \arcsin \relax (x)}{x^{5} - 2 \, x^{3} + x}, x\right ) \]
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 \[ \int \frac {\arcsin \relax (x)}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.64, size = 97, normalized size = 1.56
method | result | size |
default | \(-\frac {\sqrt {-x^{2}+1}\, \arcsin \relax (x )}{x^{2}-1}+2 i \arctan \left (i x +\sqrt {-x^{2}+1}\right )+i \dilog \left (i x +\sqrt {-x^{2}+1}+1\right )-\arcsin \relax (x ) \ln \left (i x +\sqrt {-x^{2}+1}+1\right )+i \dilog \left (i x +\sqrt {-x^{2}+1}\right )\) | \(97\) |
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 \[ \int \frac {\arcsin \relax (x)}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}} x}\,{d x} \]
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 \[ \int \frac {\mathrm {asin}\relax (x)}{x\,{\left (1-x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\operatorname {asin}{\relax (x )}}{x \left (- \left (x - 1\right ) \left (x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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