Optimal. Leaf size=67 \[ a \log (x)+\frac {1}{2} i b \text {Li}_2\left (e^{2 i \sin ^{-1}\left (\frac {c}{x}\right )}\right )+\frac {1}{2} i b \sin ^{-1}\left (\frac {c}{x}\right )^2-b \sin ^{-1}\left (\frac {c}{x}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac {c}{x}\right )}\right ) \]
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Rubi [A] time = 0.09, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6742, 4830, 3717, 2190, 2279, 2391} \[ \frac {1}{2} i b \text {PolyLog}\left (2,e^{2 i \sin ^{-1}\left (\frac {c}{x}\right )}\right )+a \log (x)+\frac {1}{2} i b \sin ^{-1}\left (\frac {c}{x}\right )^2-b \sin ^{-1}\left (\frac {c}{x}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac {c}{x}\right )}\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3717
Rule 4830
Rule 6742
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}\left (\frac {c}{x}\right )}{x} \, dx &=\int \left (\frac {a}{x}+\frac {b \sin ^{-1}\left (\frac {c}{x}\right )}{x}\right ) \, dx\\ &=a \log (x)+b \int \frac {\sin ^{-1}\left (\frac {c}{x}\right )}{x} \, dx\\ &=a \log (x)-b \operatorname {Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}\left (\frac {c}{x}\right )\right )\\ &=\frac {1}{2} i b \sin ^{-1}\left (\frac {c}{x}\right )^2+a \log (x)+(2 i b) \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}\left (\frac {c}{x}\right )\right )\\ &=\frac {1}{2} i b \sin ^{-1}\left (\frac {c}{x}\right )^2-b \sin ^{-1}\left (\frac {c}{x}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac {c}{x}\right )}\right )+a \log (x)+b \operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}\left (\frac {c}{x}\right )\right )\\ &=\frac {1}{2} i b \sin ^{-1}\left (\frac {c}{x}\right )^2-b \sin ^{-1}\left (\frac {c}{x}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac {c}{x}\right )}\right )+a \log (x)-\frac {1}{2} (i b) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}\left (\frac {c}{x}\right )}\right )\\ &=\frac {1}{2} i b \sin ^{-1}\left (\frac {c}{x}\right )^2-b \sin ^{-1}\left (\frac {c}{x}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac {c}{x}\right )}\right )+a \log (x)+\frac {1}{2} i b \text {Li}_2\left (e^{2 i \sin ^{-1}\left (\frac {c}{x}\right )}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 61, normalized size = 0.91 \[ a \log (x)+\frac {1}{2} i b \left (\sin ^{-1}\left (\frac {c}{x}\right )^2+\text {Li}_2\left (e^{2 i \sin ^{-1}\left (\frac {c}{x}\right )}\right )\right )-b \sin ^{-1}\left (\frac {c}{x}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac {c}{x}\right )}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.38, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arcsin \left (\frac {c}{x}\right ) + a}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 141, normalized size = 2.10 \[ -a \ln \left (\frac {c}{x}\right )+\frac {i b \arcsin \left (\frac {c}{x}\right )^{2}}{2}-b \arcsin \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}+\sqrt {1-\frac {c^{2}}{x^{2}}}\right )-b \arcsin \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}-\sqrt {1-\frac {c^{2}}{x^{2}}}\right )+i b \polylog \left (2, -\frac {i c}{x}-\sqrt {1-\frac {c^{2}}{x^{2}}}\right )+i b \polylog \left (2, \frac {i c}{x}+\sqrt {1-\frac {c^{2}}{x^{2}}}\right ) \]
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 \[ {\left (c \int -\frac {\sqrt {c + x} \sqrt {-c + x} \log \relax (x)}{c^{2} x - x^{3}}\,{d x} + \arctan \left (c, \sqrt {c + x} \sqrt {-c + x}\right ) \log \relax (x)\right )} b + a \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.44, size = 57, normalized size = 0.85 \[ \frac {b\,{\mathrm {asin}\left (\frac {c}{x}\right )}^2\,1{}\mathrm {i}}{2}+a\,\ln \relax (x)+\frac {b\,\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {asin}\left (\frac {c}{x}\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}-b\,\ln \left (1-{\mathrm {e}}^{\mathrm {asin}\left (\frac {c}{x}\right )\,2{}\mathrm {i}}\right )\,\mathrm {asin}\left (\frac {c}{x}\right ) \]
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 \[ \int \frac {a + b \operatorname {asin}{\left (\frac {c}{x} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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