Optimal. Leaf size=39 \[ b \cosh (2 a) \text{Chi}(2 b x)+b \sinh (2 a) \text{Shi}(2 b x)-\frac{\sinh (2 a+2 b x)}{2 x} \]
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Rubi [A] time = 0.0948402, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5448, 12, 3297, 3303, 3298, 3301} \[ b \cosh (2 a) \text{Chi}(2 b x)+b \sinh (2 a) \text{Shi}(2 b x)-\frac{\sinh (2 a+2 b x)}{2 x} \]
Antiderivative was successfully verified.
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Rule 5448
Rule 12
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\cosh (a+b x) \sinh (a+b x)}{x^2} \, dx &=\int \frac{\sinh (2 a+2 b x)}{2 x^2} \, dx\\ &=\frac{1}{2} \int \frac{\sinh (2 a+2 b x)}{x^2} \, dx\\ &=-\frac{\sinh (2 a+2 b x)}{2 x}+b \int \frac{\cosh (2 a+2 b x)}{x} \, dx\\ &=-\frac{\sinh (2 a+2 b x)}{2 x}+(b \cosh (2 a)) \int \frac{\cosh (2 b x)}{x} \, dx+(b \sinh (2 a)) \int \frac{\sinh (2 b x)}{x} \, dx\\ &=b \cosh (2 a) \text{Chi}(2 b x)-\frac{\sinh (2 a+2 b x)}{2 x}+b \sinh (2 a) \text{Shi}(2 b x)\\ \end{align*}
Mathematica [A] time = 0.0709192, size = 42, normalized size = 1.08 \[ \frac{1}{2} \left (2 b \cosh (2 a) \text{Chi}(2 b x)+2 b \sinh (2 a) \text{Shi}(2 b x)-\frac{\sinh (2 (a+b x))}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 56, normalized size = 1.4 \begin{align*}{\frac{{{\rm e}^{-2\,bx-2\,a}}}{4\,x}}-{\frac{b{{\rm e}^{-2\,a}}{\it Ei} \left ( 1,2\,bx \right ) }{2}}-{\frac{{{\rm e}^{2\,bx+2\,a}}}{4\,x}}-{\frac{b{{\rm e}^{2\,a}}{\it Ei} \left ( 1,-2\,bx \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.34828, size = 36, normalized size = 0.92 \begin{align*} \frac{1}{2} \, b e^{\left (-2 \, a\right )} \Gamma \left (-1, 2 \, b x\right ) + \frac{1}{2} \, b e^{\left (2 \, a\right )} \Gamma \left (-1, -2 \, b x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89209, size = 174, normalized size = 4.46 \begin{align*} \frac{{\left (b x{\rm Ei}\left (2 \, b x\right ) + b x{\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (2 \, a\right ) - 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) +{\left (b x{\rm Ei}\left (2 \, b x\right ) - b x{\rm Ei}\left (-2 \, b x\right )\right )} \sinh \left (2 \, a\right )}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13854, size = 70, normalized size = 1.79 \begin{align*} \frac{2 \, b x{\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} + 2 \, b x{\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} - e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )}}{4 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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