Integrand size = 10, antiderivative size = 54 \[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x} \, dx=-\frac {1}{4} \text {arcsinh}\left (a x^2\right )^2+\frac {1}{2} \text {arcsinh}\left (a x^2\right ) \log \left (1-e^{2 \text {arcsinh}\left (a x^2\right )}\right )+\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (a x^2\right )}\right ) \]
[Out]
Time = 0.05 (sec) , antiderivative size = 54, 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} \[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x} \, dx=\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (a x^2\right )}\right )-\frac {1}{4} \text {arcsinh}\left (a x^2\right )^2+\frac {1}{2} \text {arcsinh}\left (a x^2\right ) \log \left (1-e^{2 \text {arcsinh}\left (a x^2\right )}\right ) \]
[In]
[Out]
Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 5869
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x \coth (x) \, dx,x,\text {arcsinh}\left (a x^2\right )\right ) \\ & = -\frac {1}{4} \text {arcsinh}\left (a x^2\right )^2-\text {Subst}\left (\int \frac {e^{2 x} x}{1-e^{2 x}} \, dx,x,\text {arcsinh}\left (a x^2\right )\right ) \\ & = -\frac {1}{4} \text {arcsinh}\left (a x^2\right )^2+\frac {1}{2} \text {arcsinh}\left (a x^2\right ) \log \left (1-e^{2 \text {arcsinh}\left (a x^2\right )}\right )-\frac {1}{2} \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arcsinh}\left (a x^2\right )\right ) \\ & = -\frac {1}{4} \text {arcsinh}\left (a x^2\right )^2+\frac {1}{2} \text {arcsinh}\left (a x^2\right ) \log \left (1-e^{2 \text {arcsinh}\left (a x^2\right )}\right )-\frac {1}{4} \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arcsinh}\left (a x^2\right )}\right ) \\ & = -\frac {1}{4} \text {arcsinh}\left (a x^2\right )^2+\frac {1}{2} \text {arcsinh}\left (a x^2\right ) \log \left (1-e^{2 \text {arcsinh}\left (a x^2\right )}\right )+\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (a x^2\right )}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x} \, dx=-\frac {1}{4} \text {arcsinh}\left (a x^2\right )^2+\frac {1}{2} \text {arcsinh}\left (a x^2\right ) \log \left (1-e^{2 \text {arcsinh}\left (a x^2\right )}\right )+\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (a x^2\right )}\right ) \]
[In]
[Out]
\[\int \frac {\operatorname {arcsinh}\left (a \,x^{2}\right )}{x}d x\]
[In]
[Out]
\[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (a x^{2}\right )}{x} \,d x } \]
[In]
[Out]
\[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x} \, dx=\int \frac {\operatorname {asinh}{\left (a x^{2} \right )}}{x}\, dx \]
[In]
[Out]
\[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (a x^{2}\right )}{x} \,d x } \]
[In]
[Out]
\[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (a x^{2}\right )}{x} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\text {arcsinh}\left (a x^2\right )}{x} \, dx=\int \frac {\mathrm {asinh}\left (a\,x^2\right )}{x} \,d x \]
[In]
[Out]