Integrand size = 8, antiderivative size = 51 \[ \int \frac {\arcsin (a x)}{x} \, dx=-\frac {1}{2} i \arcsin (a x)^2+\arcsin (a x) \log \left (1-e^{2 i \arcsin (a x)}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4721, 3798, 2221, 2317, 2438} \[ \int \frac {\arcsin (a x)}{x} \, dx=-\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )-\frac {1}{2} i \arcsin (a x)^2+\arcsin (a x) \log \left (1-e^{2 i \arcsin (a x)}\right ) \]
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Rule 2221
Rule 2317
Rule 2438
Rule 3798
Rule 4721
Rubi steps \begin{align*} \text {integral}& = \text {Subst}(\int x \cot (x) \, dx,x,\arcsin (a x)) \\ & = -\frac {1}{2} i \arcsin (a x)^2-2 i \text {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {1}{2} i \arcsin (a x)^2+\arcsin (a x) \log \left (1-e^{2 i \arcsin (a x)}\right )-\text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {1}{2} i \arcsin (a x)^2+\arcsin (a x) \log \left (1-e^{2 i \arcsin (a x)}\right )+\frac {1}{2} i \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \arcsin (a x)}\right ) \\ & = -\frac {1}{2} i \arcsin (a x)^2+\arcsin (a x) \log \left (1-e^{2 i \arcsin (a x)}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.90 \[ \int \frac {\arcsin (a x)}{x} \, dx=\arcsin (a x) \log \left (1-e^{2 i \arcsin (a x)}\right )-\frac {1}{2} i \left (\arcsin (a x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )\right ) \]
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Time = 0.12 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.18
method | result | size |
derivativedivides | \(-\frac {i \arcsin \left (a x \right )^{2}}{2}+\arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+\arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\) | \(111\) |
default | \(-\frac {i \arcsin \left (a x \right )^{2}}{2}+\arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+\arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\) | \(111\) |
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\[ \int \frac {\arcsin (a x)}{x} \, dx=\int { \frac {\arcsin \left (a x\right )}{x} \,d x } \]
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\[ \int \frac {\arcsin (a x)}{x} \, dx=\int \frac {\operatorname {asin}{\left (a x \right )}}{x}\, dx \]
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\[ \int \frac {\arcsin (a x)}{x} \, dx=\int { \frac {\arcsin \left (a x\right )}{x} \,d x } \]
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\[ \int \frac {\arcsin (a x)}{x} \, dx=\int { \frac {\arcsin \left (a x\right )}{x} \,d x } \]
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Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.80 \[ \int \frac {\arcsin (a x)}{x} \, dx=\ln \left (1-{\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )\,2{}\mathrm {i}}\right )\,\mathrm {asin}\left (a\,x\right )-\frac {\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}-\frac {{\mathrm {asin}\left (a\,x\right )}^2\,1{}\mathrm {i}}{2} \]
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