Optimal. Leaf size=56 \[ -i \text {Li}_2\left (e^{2 i \sin ^{-1}\left (\sqrt {x}\right )}\right )-i \sin ^{-1}\left (\sqrt {x}\right )^2+2 \sin ^{-1}\left (\sqrt {x}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\sqrt {x}\right )}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4830, 3717, 2190, 2279, 2391} \[ -i \text {PolyLog}\left (2,e^{2 i \sin ^{-1}\left (\sqrt {x}\right )}\right )-i \sin ^{-1}\left (\sqrt {x}\right )^2+2 \sin ^{-1}\left (\sqrt {x}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\sqrt {x}\right )}\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3717
Rule 4830
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}\left (\sqrt {x}\right )}{x} \, dx &=2 \operatorname {Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}\left (\sqrt {x}\right )\right )\\ &=-i \sin ^{-1}\left (\sqrt {x}\right )^2-4 i \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}\left (\sqrt {x}\right )\right )\\ &=-i \sin ^{-1}\left (\sqrt {x}\right )^2+2 \sin ^{-1}\left (\sqrt {x}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\sqrt {x}\right )}\right )-2 \operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}\left (\sqrt {x}\right )\right )\\ &=-i \sin ^{-1}\left (\sqrt {x}\right )^2+2 \sin ^{-1}\left (\sqrt {x}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\sqrt {x}\right )}\right )+i \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}\left (\sqrt {x}\right )}\right )\\ &=-i \sin ^{-1}\left (\sqrt {x}\right )^2+2 \sin ^{-1}\left (\sqrt {x}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\sqrt {x}\right )}\right )-i \text {Li}_2\left (e^{2 i \sin ^{-1}\left (\sqrt {x}\right )}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 53, normalized size = 0.95 \[ 2 \sin ^{-1}\left (\sqrt {x}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\sqrt {x}\right )}\right )-i \left (\sin ^{-1}\left (\sqrt {x}\right )^2+\text {Li}_2\left (e^{2 i \sin ^{-1}\left (\sqrt {x}\right )}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arcsin \left (\sqrt {x}\right )}{x}, x\right ) \]
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 \[ \int \frac {\arcsin \left (\sqrt {x}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 97, normalized size = 1.73 \[ -i \arcsin \left (\sqrt {x}\right )^{2}+2 \arcsin \left (\sqrt {x}\right ) \ln \left (1+i \sqrt {x}+\sqrt {1-x}\right )+2 \arcsin \left (\sqrt {x}\right ) \ln \left (1-i \sqrt {x}-\sqrt {1-x}\right )-2 i \polylog \left (2, -i \sqrt {x}-\sqrt {1-x}\right )-2 i \polylog \left (2, i \sqrt {x}+\sqrt {1-x}\right ) \]
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 \[ \int \frac {\arcsin \left (\sqrt {x}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.53, size = 42, normalized size = 0.75 \[ -\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {asin}\left (\sqrt {x}\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}-{\mathrm {asin}\left (\sqrt {x}\right )}^2\,1{}\mathrm {i}+2\,\ln \left (1-{\mathrm {e}}^{\mathrm {asin}\left (\sqrt {x}\right )\,2{}\mathrm {i}}\right )\,\mathrm {asin}\left (\sqrt {x}\right ) \]
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 \[ \int \frac {\operatorname {asin}{\left (\sqrt {x} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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