Integrand size = 22, antiderivative size = 52 \[ \int \frac {\arcsin (a x)}{x \sqrt {1-a^2 x^2}} \, dx=-2 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4803, 4268, 2317, 2438} \[ \int \frac {\arcsin (a x)}{x \sqrt {1-a^2 x^2}} \, dx=-2 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right ) \]
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Rule 2317
Rule 2438
Rule 4268
Rule 4803
Rubi steps \begin{align*} \text {integral}& = \text {Subst}(\int x \csc (x) \, dx,x,\arcsin (a x)) \\ & = -2 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )-\text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (a x)\right )+\text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (a x)\right ) \\ & = -2 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )+i \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \arcsin (a x)}\right )-i \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \arcsin (a x)}\right ) \\ & = -2 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.37 \[ \int \frac {\arcsin (a x)}{x \sqrt {1-a^2 x^2}} \, dx=\arcsin (a x) \left (\log \left (1-e^{i \arcsin (a x)}\right )-\log \left (1+e^{i \arcsin (a x)}\right )\right )+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.98
method | result | size |
default | \(\arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-\arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+i \operatorname {dilog}\left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-i \operatorname {dilog}\left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )\) | \(103\) |
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\[ \int \frac {\arcsin (a x)}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arcsin \left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \]
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\[ \int \frac {\arcsin (a x)}{x \sqrt {1-a^2 x^2}} \, dx=\int \frac {\operatorname {asin}{\left (a x \right )}}{x \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
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\[ \int \frac {\arcsin (a x)}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arcsin \left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \]
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\[ \int \frac {\arcsin (a x)}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arcsin \left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \]
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Timed out. \[ \int \frac {\arcsin (a x)}{x \sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathrm {asin}\left (a\,x\right )}{x\,\sqrt {1-a^2\,x^2}} \,d x \]
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