Optimal. Leaf size=13 \[ \frac{\text{Ei}(n \sinh (a+b x))}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0228231, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {4340, 2178} \[ \frac{\text{Ei}(n \sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4340
Rule 2178
Rubi steps
\begin{align*} \int e^{n \sinh (a+b x)} \coth (a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^{n x}}{x} \, dx,x,\sinh (a+b x)\right )}{b}\\ &=\frac{\text{Ei}(n \sinh (a+b x))}{b}\\ \end{align*}
Mathematica [A] time = 0.0334604, size = 13, normalized size = 1. \[ \frac{\text{Ei}(n \sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 17, normalized size = 1.3 \begin{align*} -{\frac{{\it Ei} \left ( 1,-n\sinh \left ( bx+a \right ) \right ) }{b}} \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 \coth \left (b x + a\right ) e^{\left (n \sinh \left (b x + a\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.97297, size = 31, normalized size = 2.38 \begin{align*} \frac{{\rm Ei}\left (n \sinh \left (b x + a\right )\right )}{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 e^{n \sinh{\left (a + b x \right )}} \coth{\left (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 \coth \left (b x + a\right ) e^{\left (n \sinh \left (b x + a\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]