Optimal. Leaf size=60 \[ \sinh ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(a x)}\right )-\frac{1}{2} \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(a x)}\right )-\frac{1}{3} \sinh ^{-1}(a x)^3+\sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0967535, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5659, 3716, 2190, 2531, 2282, 6589} \[ \sinh ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(a x)}\right )-\frac{1}{2} \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(a x)}\right )-\frac{1}{3} \sinh ^{-1}(a x)^3+\sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5659
Rule 3716
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}(a x)^2}{x} \, dx &=\operatorname{Subst}\left (\int x^2 \coth (x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac{1}{3} \sinh ^{-1}(a x)^3-2 \operatorname{Subst}\left (\int \frac{e^{2 x} x^2}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac{1}{3} \sinh ^{-1}(a x)^3+\sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )-2 \operatorname{Subst}\left (\int x \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac{1}{3} \sinh ^{-1}(a x)^3+\sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+\sinh ^{-1}(a x) \text{Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-\operatorname{Subst}\left (\int \text{Li}_2\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac{1}{3} \sinh ^{-1}(a x)^3+\sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+\sinh ^{-1}(a x) \text{Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 \sinh ^{-1}(a x)}\right )\\ &=-\frac{1}{3} \sinh ^{-1}(a x)^3+\sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+\sinh ^{-1}(a x) \text{Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-\frac{1}{2} \text{Li}_3\left (e^{2 \sinh ^{-1}(a x)}\right )\\ \end{align*}
Mathematica [A] time = 0.0055132, size = 60, normalized size = 1. \[ \sinh ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(a x)}\right )-\frac{1}{2} \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(a x)}\right )-\frac{1}{3} \sinh ^{-1}(a x)^3+\sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.036, size = 151, normalized size = 2.5 \begin{align*} -{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}}{3}}+ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\ln \left ( 1+ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) +2\,{\it Arcsinh} \left ( ax \right ){\it polylog} \left ( 2,-ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) -2\,{\it polylog} \left ( 3,-ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) + \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\ln \left ( 1-ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) +2\,{\it Arcsinh} \left ( ax \right ){\it polylog} \left ( 2,ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) -2\,{\it polylog} \left ( 3,ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsinh}\left (a x\right )^{2}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arsinh}\left (a x\right )^{2}}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asinh}^{2}{\left (a x \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsinh}\left (a x\right )^{2}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]