Optimal. Leaf size=27 \[ \frac{1}{2} \sinh (2 a) \text{Chi}(2 b x)+\frac{1}{2} \cosh (2 a) \text{Shi}(2 b x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0745568, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {5448, 12, 3303, 3298, 3301} \[ \frac{1}{2} \sinh (2 a) \text{Chi}(2 b x)+\frac{1}{2} \cosh (2 a) \text{Shi}(2 b x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5448
Rule 12
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\cosh (a+b x) \sinh (a+b x)}{x} \, dx &=\int \frac{\sinh (2 a+2 b x)}{2 x} \, dx\\ &=\frac{1}{2} \int \frac{\sinh (2 a+2 b x)}{x} \, dx\\ &=\frac{1}{2} \cosh (2 a) \int \frac{\sinh (2 b x)}{x} \, dx+\frac{1}{2} \sinh (2 a) \int \frac{\cosh (2 b x)}{x} \, dx\\ &=\frac{1}{2} \text{Chi}(2 b x) \sinh (2 a)+\frac{1}{2} \cosh (2 a) \text{Shi}(2 b x)\\ \end{align*}
Mathematica [A] time = 0.0254978, size = 25, normalized size = 0.93 \[ \frac{1}{2} (\sinh (2 a) \text{Chi}(2 b x)+\cosh (2 a) \text{Shi}(2 b x)) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.021, size = 26, normalized size = 1. \begin{align*}{\frac{{{\rm e}^{-2\,a}}{\it Ei} \left ( 1,2\,bx \right ) }{4}}-{\frac{{{\rm e}^{2\,a}}{\it Ei} \left ( 1,-2\,bx \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.32902, size = 31, normalized size = 1.15 \begin{align*} \frac{1}{4} \,{\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - \frac{1}{4} \,{\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.02355, size = 109, normalized size = 4.04 \begin{align*} \frac{1}{4} \,{\left ({\rm Ei}\left (2 \, b x\right ) -{\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (2 \, a\right ) + \frac{1}{4} \,{\left ({\rm Ei}\left (2 \, b x\right ) +{\rm Ei}\left (-2 \, b x\right )\right )} \sinh \left (2 \, a\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{\sinh{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14183, size = 31, normalized size = 1.15 \begin{align*} \frac{1}{4} \,{\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - \frac{1}{4} \,{\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]