Optimal. Leaf size=41 \[ \frac {\text {Li}_2\left (-e^{-a-b x}\right )}{2 b}-\frac {\text {Li}_2\left (e^{-a-b x}\right )}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 = 0.08, size = 68, normalized size = 1.66 \[ \frac {-\text {Li}_2\left (-e^{a+b x}\right )+\text {Li}_2\left (e^{a+b x}\right )+b x \left (\log \left (1-e^{a+b x}\right )-\log \left (e^{a+b x}+1\right )+2 \coth ^{-1}\left (e^{a+b x}\right )\right )}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.63, size = 137, normalized size = 3.34 \[ \frac {b x \log \left (\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1}{\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1}\right ) - b x \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - a \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + {\left (b x + a\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {arcoth}\left (e^{\left (b x + a\right )}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 67, normalized size = 1.63 \[ \frac {\ln \left ({\mathrm e}^{b x +a}\right ) \mathrm {arccoth}\left ({\mathrm e}^{b x +a}\right )}{b}-\frac {\dilog \left ({\mathrm e}^{b x +a}\right )}{2 b}-\frac {\dilog \left ({\mathrm e}^{b x +a}+1\right )}{2 b}-\frac {\ln \left ({\mathrm e}^{b x +a}\right ) \ln \left ({\mathrm e}^{b x +a}+1\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.33, size = 107, normalized size = 2.61 \[ \frac {{\left (b x + a\right )} \operatorname {arcoth}\left (e^{\left (b x + a\right )}\right )}{b} - \frac {{\left (b x + a\right )} {\left (\log \left (e^{\left (b x + a\right )} + 1\right ) - \log \left (e^{\left (b x + a\right )} - 1\right )\right )} - \log \left (-e^{\left (b x + a\right )}\right ) \log \left (e^{\left (b x + a\right )} + 1\right ) + {\left (b x + a\right )} \log \left (e^{\left (b x + a\right )} - 1\right ) - {\rm Li}_2\left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )} + 1\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {acoth}\left ({\mathrm {e}}^{a+b\,x}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {acoth}{\left (e^{a + b x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________