Optimal. Leaf size=53 \[ \frac{1}{4} \sinh (2 a) \text{Chi}(2 b x)+\frac{1}{8} \sinh (4 a) \text{Chi}(4 b x)+\frac{1}{4} \cosh (2 a) \text{Shi}(2 b x)+\frac{1}{8} \cosh (4 a) \text{Shi}(4 b x) \]
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Rubi [A] time = 0.14056, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5448, 3303, 3298, 3301} \[ \frac{1}{4} \sinh (2 a) \text{Chi}(2 b x)+\frac{1}{8} \sinh (4 a) \text{Chi}(4 b x)+\frac{1}{4} \cosh (2 a) \text{Shi}(2 b x)+\frac{1}{8} \cosh (4 a) \text{Shi}(4 b x) \]
Antiderivative was successfully verified.
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Rule 5448
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\cosh ^3(a+b x) \sinh (a+b x)}{x} \, dx &=\int \left (\frac{\sinh (2 a+2 b x)}{4 x}+\frac{\sinh (4 a+4 b x)}{8 x}\right ) \, dx\\ &=\frac{1}{8} \int \frac{\sinh (4 a+4 b x)}{x} \, dx+\frac{1}{4} \int \frac{\sinh (2 a+2 b x)}{x} \, dx\\ &=\frac{1}{4} \cosh (2 a) \int \frac{\sinh (2 b x)}{x} \, dx+\frac{1}{8} \cosh (4 a) \int \frac{\sinh (4 b x)}{x} \, dx+\frac{1}{4} \sinh (2 a) \int \frac{\cosh (2 b x)}{x} \, dx+\frac{1}{8} \sinh (4 a) \int \frac{\cosh (4 b x)}{x} \, dx\\ &=\frac{1}{4} \text{Chi}(2 b x) \sinh (2 a)+\frac{1}{8} \text{Chi}(4 b x) \sinh (4 a)+\frac{1}{4} \cosh (2 a) \text{Shi}(2 b x)+\frac{1}{8} \cosh (4 a) \text{Shi}(4 b x)\\ \end{align*}
Mathematica [A] time = 0.0815933, size = 47, normalized size = 0.89 \[ \frac{1}{8} (2 \sinh (2 a) \text{Chi}(2 b x)+\sinh (4 a) \text{Chi}(4 b x)+2 \cosh (2 a) \text{Shi}(2 b x)+\cosh (4 a) \text{Shi}(4 b x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 50, normalized size = 0.9 \begin{align*}{\frac{{{\rm e}^{-4\,a}}{\it Ei} \left ( 1,4\,bx \right ) }{16}}+{\frac{{{\rm e}^{-2\,a}}{\it Ei} \left ( 1,2\,bx \right ) }{8}}-{\frac{{{\rm e}^{4\,a}}{\it Ei} \left ( 1,-4\,bx \right ) }{16}}-{\frac{{{\rm e}^{2\,a}}{\it Ei} \left ( 1,-2\,bx \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.2708, size = 61, normalized size = 1.15 \begin{align*} \frac{1}{16} \,{\rm Ei}\left (4 \, b x\right ) e^{\left (4 \, a\right )} + \frac{1}{8} \,{\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - \frac{1}{8} \,{\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} - \frac{1}{16} \,{\rm Ei}\left (-4 \, b x\right ) e^{\left (-4 \, a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71797, size = 223, normalized size = 4.21 \begin{align*} \frac{1}{16} \,{\left ({\rm Ei}\left (4 \, b x\right ) -{\rm Ei}\left (-4 \, b x\right )\right )} \cosh \left (4 \, a\right ) + \frac{1}{8} \,{\left ({\rm Ei}\left (2 \, b x\right ) -{\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (2 \, a\right ) + \frac{1}{16} \,{\left ({\rm Ei}\left (4 \, b x\right ) +{\rm Ei}\left (-4 \, b x\right )\right )} \sinh \left (4 \, a\right ) + \frac{1}{8} \,{\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.
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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}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14895, size = 61, normalized size = 1.15 \begin{align*} \frac{1}{16} \,{\rm Ei}\left (4 \, b x\right ) e^{\left (4 \, a\right )} + \frac{1}{8} \,{\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - \frac{1}{8} \,{\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} - \frac{1}{16} \,{\rm Ei}\left (-4 \, b x\right ) e^{\left (-4 \, a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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