Optimal. Leaf size=41 \[ \frac{\text{PolyLog}\left (2,-e^{-a-b x}\right )}{2 b}-\frac{\text{PolyLog}\left (2,e^{-a-b x}\right )}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0164433, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2282, 5913} \[ \frac{\text{PolyLog}\left (2,-e^{-a-b x}\right )}{2 b}-\frac{\text{PolyLog}\left (2,e^{-a-b x}\right )}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 5913
Rubi steps
\begin{align*} \int \coth ^{-1}\left (e^{a+b x}\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\coth ^{-1}(x)}{x} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\text{Li}_2\left (-e^{-a-b x}\right )}{2 b}-\frac{\text{Li}_2\left (e^{-a-b x}\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0725544, size = 68, normalized size = 1.66 \[ \frac{-\text{PolyLog}\left (2,-e^{a+b x}\right )+\text{PolyLog}\left (2,e^{a+b x}\right )+b x \left (\log \left (1-e^{a+b x}\right )-\log \left (e^{a+b x}+1\right )+2 \coth ^{-1}\left (e^{a+b x}\right )\right )}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.072, size = 67, normalized size = 1.6 \begin{align*}{\frac{\ln \left ({{\rm e}^{bx+a}} \right ){\rm arccoth} \left ({{\rm e}^{bx+a}}\right )}{b}}-{\frac{{\it dilog} \left ({{\rm e}^{bx+a}} \right ) }{2\,b}}-{\frac{{\it dilog} \left ({{\rm e}^{bx+a}}+1 \right ) }{2\,b}}-{\frac{\ln \left ({{\rm e}^{bx+a}} \right ) \ln \left ({{\rm e}^{bx+a}}+1 \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.07206, size = 144, normalized size = 3.51 \begin{align*} \frac{{\left (b x + a\right )} \operatorname{arcoth}\left (e^{\left (b x + a\right )}\right )}{b} - \frac{{\left (b x + a\right )}{\left (\log \left (e^{\left (b x + a\right )} + 1\right ) - \log \left (e^{\left (b x + a\right )} - 1\right )\right )} - \log \left (-e^{\left (b x + a\right )}\right ) \log \left (e^{\left (b x + a\right )} + 1\right ) +{\left (b x + a\right )} \log \left (e^{\left (b x + a\right )} - 1\right ) -{\rm Li}_2\left (e^{\left (b x + a\right )} + 1\right ) +{\rm Li}_2\left (-e^{\left (b x + a\right )} + 1\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.67866, size = 417, normalized size = 10.17 \begin{align*} \frac{b x \log \left (\frac{\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1}{\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1}\right ) - b x \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - a \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) +{\left (b x + a\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) +{\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) -{\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{2 \, b} \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 \operatorname{acoth}{\left (e^{a + b x} \right )}\, 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 \operatorname{arcoth}\left (e^{\left (b x + a\right )}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]