Integrand size = 10, antiderivative size = 52 \[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x} \, dx=\frac {1}{2} \text {arcsinh}\left (\frac {a}{x}\right )^2-\text {arcsinh}\left (\frac {a}{x}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5869, 3797, 2221, 2317, 2438} \[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x} \, dx=-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}\right )+\frac {1}{2} \text {arcsinh}\left (\frac {a}{x}\right )^2-\text {arcsinh}\left (\frac {a}{x}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}\right ) \]
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Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 5869
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int x \coth (x) \, dx,x,\text {arcsinh}\left (\frac {a}{x}\right )\right ) \\ & = \frac {1}{2} \text {arcsinh}\left (\frac {a}{x}\right )^2+2 \text {Subst}\left (\int \frac {e^{2 x} x}{1-e^{2 x}} \, dx,x,\text {arcsinh}\left (\frac {a}{x}\right )\right ) \\ & = \frac {1}{2} \text {arcsinh}\left (\frac {a}{x}\right )^2-\text {arcsinh}\left (\frac {a}{x}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}\right )+\text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arcsinh}\left (\frac {a}{x}\right )\right ) \\ & = \frac {1}{2} \text {arcsinh}\left (\frac {a}{x}\right )^2-\text {arcsinh}\left (\frac {a}{x}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}\right ) \\ & = \frac {1}{2} \text {arcsinh}\left (\frac {a}{x}\right )^2-\text {arcsinh}\left (\frac {a}{x}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x} \, dx=\frac {1}{2} \text {arcsinh}\left (\frac {a}{x}\right )^2-\text {arcsinh}\left (\frac {a}{x}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.19
method | result | size |
derivativedivides | \(\frac {\operatorname {arcsinh}\left (\frac {a}{x}\right )^{2}}{2}-\operatorname {arcsinh}\left (\frac {a}{x}\right ) \ln \left (1+\frac {a}{x}+\sqrt {\frac {a^{2}}{x^{2}}+1}\right )-\operatorname {polylog}\left (2, -\frac {a}{x}-\sqrt {\frac {a^{2}}{x^{2}}+1}\right )-\operatorname {arcsinh}\left (\frac {a}{x}\right ) \ln \left (1-\frac {a}{x}-\sqrt {\frac {a^{2}}{x^{2}}+1}\right )-\operatorname {polylog}\left (2, \frac {a}{x}+\sqrt {\frac {a^{2}}{x^{2}}+1}\right )\) | \(114\) |
default | \(\frac {\operatorname {arcsinh}\left (\frac {a}{x}\right )^{2}}{2}-\operatorname {arcsinh}\left (\frac {a}{x}\right ) \ln \left (1+\frac {a}{x}+\sqrt {\frac {a^{2}}{x^{2}}+1}\right )-\operatorname {polylog}\left (2, -\frac {a}{x}-\sqrt {\frac {a^{2}}{x^{2}}+1}\right )-\operatorname {arcsinh}\left (\frac {a}{x}\right ) \ln \left (1-\frac {a}{x}-\sqrt {\frac {a^{2}}{x^{2}}+1}\right )-\operatorname {polylog}\left (2, \frac {a}{x}+\sqrt {\frac {a^{2}}{x^{2}}+1}\right )\) | \(114\) |
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\[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (\frac {a}{x}\right )}{x} \,d x } \]
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\[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x} \, dx=\int \frac {\operatorname {asinh}{\left (\frac {a}{x} \right )}}{x}\, dx \]
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\[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (\frac {a}{x}\right )}{x} \,d x } \]
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\[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (\frac {a}{x}\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x} \, dx=\int \frac {\mathrm {asinh}\left (\frac {a}{x}\right )}{x} \,d x \]
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