Integrand size = 10, antiderivative size = 60 \[ \int \frac {\text {arcsinh}(a x)^2}{x} \, dx=-\frac {1}{3} \text {arcsinh}(a x)^3+\text {arcsinh}(a x)^2 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )+\text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(a x)}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5775, 3797, 2221, 2611, 2320, 6724} \[ \int \frac {\text {arcsinh}(a x)^2}{x} \, dx=\text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(a x)}\right )-\frac {1}{3} \text {arcsinh}(a x)^3+\text {arcsinh}(a x)^2 \log \left (1-e^{2 \text {arcsinh}(a x)}\right ) \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3797
Rule 5775
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int x^2 \coth (x) \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -\frac {1}{3} \text {arcsinh}(a x)^3-2 \text {Subst}\left (\int \frac {e^{2 x} x^2}{1-e^{2 x}} \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -\frac {1}{3} \text {arcsinh}(a x)^3+\text {arcsinh}(a x)^2 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )-2 \text {Subst}\left (\int x \log \left (1-e^{2 x}\right ) \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -\frac {1}{3} \text {arcsinh}(a x)^3+\text {arcsinh}(a x)^2 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )+\text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )-\text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{2 x}\right ) \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -\frac {1}{3} \text {arcsinh}(a x)^3+\text {arcsinh}(a x)^2 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )+\text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 \text {arcsinh}(a x)}\right ) \\ & = -\frac {1}{3} \text {arcsinh}(a x)^3+\text {arcsinh}(a x)^2 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )+\text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(a x)}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a x)^2}{x} \, dx=-\frac {1}{3} \text {arcsinh}(a x)^3+\text {arcsinh}(a x)^2 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )+\text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(a x)}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.52
method | result | size |
derivativedivides | \(-\frac {\operatorname {arcsinh}\left (a x \right )^{3}}{3}+\operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, a x +\sqrt {a^{2} x^{2}+1}\right )\) | \(151\) |
default | \(-\frac {\operatorname {arcsinh}\left (a x \right )^{3}}{3}+\operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, a x +\sqrt {a^{2} x^{2}+1}\right )\) | \(151\) |
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\[ \int \frac {\text {arcsinh}(a x)^2}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{2}}{x} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a x)^2}{x} \, dx=\int \frac {\operatorname {asinh}^{2}{\left (a x \right )}}{x}\, dx \]
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\[ \int \frac {\text {arcsinh}(a x)^2}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{2}}{x} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a x)^2}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{2}}{x} \,d x } \]
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Timed out. \[ \int \frac {\text {arcsinh}(a x)^2}{x} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^2}{x} \,d x \]
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