Integrand size = 10, antiderivative size = 50 \[ \int \frac {\text {arcsinh}(a x)^2}{x^2} \, dx=-\frac {\text {arcsinh}(a x)^2}{x}-4 a \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-2 a \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+2 a \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5776, 5816, 4267, 2317, 2438} \[ \int \frac {\text {arcsinh}(a x)^2}{x^2} \, dx=-4 a \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-2 a \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+2 a \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-\frac {\text {arcsinh}(a x)^2}{x} \]
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Rule 2317
Rule 2438
Rule 4267
Rule 5776
Rule 5816
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arcsinh}(a x)^2}{x}+(2 a) \int \frac {\text {arcsinh}(a x)}{x \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {\text {arcsinh}(a x)^2}{x}+(2 a) \text {Subst}(\int x \text {csch}(x) \, dx,x,\text {arcsinh}(a x)) \\ & = -\frac {\text {arcsinh}(a x)^2}{x}-4 a \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-(2 a) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )+(2 a) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -\frac {\text {arcsinh}(a x)^2}{x}-4 a \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-(2 a) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(a x)}\right )+(2 a) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(a x)}\right ) \\ & = -\frac {\text {arcsinh}(a x)^2}{x}-4 a \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-2 a \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+2 a \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right ) \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.50 \[ \int \frac {\text {arcsinh}(a x)^2}{x^2} \, dx=a \left (-\text {arcsinh}(a x) \left (\frac {\text {arcsinh}(a x)}{a x}-2 \log \left (1-e^{-\text {arcsinh}(a x)}\right )+2 \log \left (1+e^{-\text {arcsinh}(a x)}\right )\right )+2 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(a x)}\right )-2 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(a x)}\right )\right ) \]
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Time = 0.05 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.16
method | result | size |
derivativedivides | \(a \left (-\frac {\operatorname {arcsinh}\left (a x \right )^{2}}{a x}-2 \,\operatorname {arcsinh}\left (a x \right ) \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )\right )\) | \(108\) |
default | \(a \left (-\frac {\operatorname {arcsinh}\left (a x \right )^{2}}{a x}-2 \,\operatorname {arcsinh}\left (a x \right ) \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )\right )\) | \(108\) |
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\[ \int \frac {\text {arcsinh}(a x)^2}{x^2} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{2}}{x^{2}} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a x)^2}{x^2} \, dx=\int \frac {\operatorname {asinh}^{2}{\left (a x \right )}}{x^{2}}\, dx \]
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\[ \int \frac {\text {arcsinh}(a x)^2}{x^2} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{2}}{x^{2}} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a x)^2}{x^2} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{2}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\text {arcsinh}(a x)^2}{x^2} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^2}{x^2} \,d x \]
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