Optimal. Leaf size=89 \[ \frac{1}{2} b \cosh (2 a) \text{Chi}(2 b x)+\frac{1}{2} b \cosh (4 a) \text{Chi}(4 b x)+\frac{1}{2} b \sinh (2 a) \text{Shi}(2 b x)+\frac{1}{2} b \sinh (4 a) \text{Shi}(4 b x)-\frac{\sinh (2 a+2 b x)}{4 x}-\frac{\sinh (4 a+4 b x)}{8 x} \]
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Rubi [A] time = 0.189175, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {5448, 3297, 3303, 3298, 3301} \[ \frac{1}{2} b \cosh (2 a) \text{Chi}(2 b x)+\frac{1}{2} b \cosh (4 a) \text{Chi}(4 b x)+\frac{1}{2} b \sinh (2 a) \text{Shi}(2 b x)+\frac{1}{2} b \sinh (4 a) \text{Shi}(4 b x)-\frac{\sinh (2 a+2 b x)}{4 x}-\frac{\sinh (4 a+4 b x)}{8 x} \]
Antiderivative was successfully verified.
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Rule 5448
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\cosh ^3(a+b x) \sinh (a+b x)}{x^2} \, dx &=\int \left (\frac{\sinh (2 a+2 b x)}{4 x^2}+\frac{\sinh (4 a+4 b x)}{8 x^2}\right ) \, dx\\ &=\frac{1}{8} \int \frac{\sinh (4 a+4 b x)}{x^2} \, dx+\frac{1}{4} \int \frac{\sinh (2 a+2 b x)}{x^2} \, dx\\ &=-\frac{\sinh (2 a+2 b x)}{4 x}-\frac{\sinh (4 a+4 b x)}{8 x}+\frac{1}{2} b \int \frac{\cosh (2 a+2 b x)}{x} \, dx+\frac{1}{2} b \int \frac{\cosh (4 a+4 b x)}{x} \, dx\\ &=-\frac{\sinh (2 a+2 b x)}{4 x}-\frac{\sinh (4 a+4 b x)}{8 x}+\frac{1}{2} (b \cosh (2 a)) \int \frac{\cosh (2 b x)}{x} \, dx+\frac{1}{2} (b \cosh (4 a)) \int \frac{\cosh (4 b x)}{x} \, dx+\frac{1}{2} (b \sinh (2 a)) \int \frac{\sinh (2 b x)}{x} \, dx+\frac{1}{2} (b \sinh (4 a)) \int \frac{\sinh (4 b x)}{x} \, dx\\ &=\frac{1}{2} b \cosh (2 a) \text{Chi}(2 b x)+\frac{1}{2} b \cosh (4 a) \text{Chi}(4 b x)-\frac{\sinh (2 a+2 b x)}{4 x}-\frac{\sinh (4 a+4 b x)}{8 x}+\frac{1}{2} b \sinh (2 a) \text{Shi}(2 b x)+\frac{1}{2} b \sinh (4 a) \text{Shi}(4 b x)\\ \end{align*}
Mathematica [A] time = 0.204442, size = 80, normalized size = 0.9 \[ \frac{4 b x \cosh (2 a) \text{Chi}(2 b x)+4 b x \cosh (4 a) \text{Chi}(4 b x)+4 b x \sinh (2 a) \text{Shi}(2 b x)+4 b x \sinh (4 a) \text{Shi}(4 b x)-2 \sinh (2 (a+b x))-\sinh (4 (a+b x))}{8 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 110, normalized size = 1.2 \begin{align*}{\frac{{{\rm e}^{-4\,bx-4\,a}}}{16\,x}}-{\frac{b{{\rm e}^{-4\,a}}{\it Ei} \left ( 1,4\,bx \right ) }{4}}+{\frac{{{\rm e}^{-2\,bx-2\,a}}}{8\,x}}-{\frac{b{{\rm e}^{-2\,a}}{\it Ei} \left ( 1,2\,bx \right ) }{4}}-{\frac{{{\rm e}^{4\,bx+4\,a}}}{16\,x}}-{\frac{b{{\rm e}^{4\,a}}{\it Ei} \left ( 1,-4\,bx \right ) }{4}}-{\frac{{{\rm e}^{2\,bx+2\,a}}}{8\,x}}-{\frac{b{{\rm e}^{2\,a}}{\it Ei} \left ( 1,-2\,bx \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.33478, size = 72, normalized size = 0.81 \begin{align*} \frac{1}{4} \, b e^{\left (-4 \, a\right )} \Gamma \left (-1, 4 \, b x\right ) + \frac{1}{4} \, b e^{\left (-2 \, a\right )} \Gamma \left (-1, 2 \, b x\right ) + \frac{1}{4} \, b e^{\left (2 \, a\right )} \Gamma \left (-1, -2 \, b x\right ) + \frac{1}{4} \, b e^{\left (4 \, a\right )} \Gamma \left (-1, -4 \, b x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78677, size = 370, normalized size = 4.16 \begin{align*} -\frac{2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} -{\left (b x{\rm Ei}\left (4 \, b x\right ) + b x{\rm Ei}\left (-4 \, b x\right )\right )} \cosh \left (4 \, a\right ) -{\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 \,{\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) -{\left (b x{\rm Ei}\left (4 \, b x\right ) - b x{\rm Ei}\left (-4 \, b x\right )\right )} \sinh \left (4 \, 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 )}{4 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20084, size = 135, normalized size = 1.52 \begin{align*} \frac{4 \, b x{\rm Ei}\left (4 \, b x\right ) e^{\left (4 \, a\right )} + 4 \, b x{\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} + 4 \, b x{\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} + 4 \, b x{\rm Ei}\left (-4 \, b x\right ) e^{\left (-4 \, a\right )} - e^{\left (4 \, b x + 4 \, a\right )} - 2 \, e^{\left (2 \, b x + 2 \, a\right )} + 2 \, e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-4 \, b x - 4 \, a\right )}}{16 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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