Optimal. Leaf size=73 \[ -\frac{1}{8} \cosh (a) \text{Chi}(b x)+\frac{1}{16} \cosh (3 a) \text{Chi}(3 b x)+\frac{1}{16} \cosh (5 a) \text{Chi}(5 b x)-\frac{1}{8} \sinh (a) \text{Shi}(b x)+\frac{1}{16} \sinh (3 a) \text{Shi}(3 b x)+\frac{1}{16} \sinh (5 a) \text{Shi}(5 b x) \]
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Rubi [A] time = 0.190071, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5448, 3303, 3298, 3301} \[ -\frac{1}{8} \cosh (a) \text{Chi}(b x)+\frac{1}{16} \cosh (3 a) \text{Chi}(3 b x)+\frac{1}{16} \cosh (5 a) \text{Chi}(5 b x)-\frac{1}{8} \sinh (a) \text{Shi}(b x)+\frac{1}{16} \sinh (3 a) \text{Shi}(3 b x)+\frac{1}{16} \sinh (5 a) \text{Shi}(5 b x) \]
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 ^2(a+b x)}{x} \, dx &=\int \left (-\frac{\cosh (a+b x)}{8 x}+\frac{\cosh (3 a+3 b x)}{16 x}+\frac{\cosh (5 a+5 b x)}{16 x}\right ) \, dx\\ &=\frac{1}{16} \int \frac{\cosh (3 a+3 b x)}{x} \, dx+\frac{1}{16} \int \frac{\cosh (5 a+5 b x)}{x} \, dx-\frac{1}{8} \int \frac{\cosh (a+b x)}{x} \, dx\\ &=-\left (\frac{1}{8} \cosh (a) \int \frac{\cosh (b x)}{x} \, dx\right )+\frac{1}{16} \cosh (3 a) \int \frac{\cosh (3 b x)}{x} \, dx+\frac{1}{16} \cosh (5 a) \int \frac{\cosh (5 b x)}{x} \, dx-\frac{1}{8} \sinh (a) \int \frac{\sinh (b x)}{x} \, dx+\frac{1}{16} \sinh (3 a) \int \frac{\sinh (3 b x)}{x} \, dx+\frac{1}{16} \sinh (5 a) \int \frac{\sinh (5 b x)}{x} \, dx\\ &=-\frac{1}{8} \cosh (a) \text{Chi}(b x)+\frac{1}{16} \cosh (3 a) \text{Chi}(3 b x)+\frac{1}{16} \cosh (5 a) \text{Chi}(5 b x)-\frac{1}{8} \sinh (a) \text{Shi}(b x)+\frac{1}{16} \sinh (3 a) \text{Shi}(3 b x)+\frac{1}{16} \sinh (5 a) \text{Shi}(5 b x)\\ \end{align*}
Mathematica [A] time = 0.104992, size = 61, normalized size = 0.84 \[ \frac{1}{16} (-2 \cosh (a) \text{Chi}(b x)+\cosh (3 a) \text{Chi}(3 b x)+\cosh (5 a) \text{Chi}(5 b x)-2 \sinh (a) \text{Shi}(b x)+\sinh (3 a) \text{Shi}(3 b x)+\sinh (5 a) \text{Shi}(5 b x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.105, size = 71, normalized size = 1. \begin{align*} -{\frac{{{\rm e}^{-5\,a}}{\it Ei} \left ( 1,5\,bx \right ) }{32}}-{\frac{{{\rm e}^{-3\,a}}{\it Ei} \left ( 1,3\,bx \right ) }{32}}+{\frac{{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) }{16}}+{\frac{{{\rm e}^{a}}{\it Ei} \left ( 1,-bx \right ) }{16}}-{\frac{{{\rm e}^{3\,a}}{\it Ei} \left ( 1,-3\,bx \right ) }{32}}-{\frac{{{\rm e}^{5\,a}}{\it Ei} \left ( 1,-5\,bx \right ) }{32}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.30402, size = 86, normalized size = 1.18 \begin{align*} \frac{1}{32} \,{\rm Ei}\left (5 \, b x\right ) e^{\left (5 \, a\right )} + \frac{1}{32} \,{\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} - \frac{1}{16} \,{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + \frac{1}{32} \,{\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} + \frac{1}{32} \,{\rm Ei}\left (-5 \, b x\right ) e^{\left (-5 \, a\right )} - \frac{1}{16} \,{\rm Ei}\left (b x\right ) e^{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74682, size = 323, normalized size = 4.42 \begin{align*} \frac{1}{32} \,{\left ({\rm Ei}\left (5 \, b x\right ) +{\rm Ei}\left (-5 \, b x\right )\right )} \cosh \left (5 \, a\right ) + \frac{1}{32} \,{\left ({\rm Ei}\left (3 \, b x\right ) +{\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (3 \, a\right ) - \frac{1}{16} \,{\left ({\rm Ei}\left (b x\right ) +{\rm Ei}\left (-b x\right )\right )} \cosh \left (a\right ) + \frac{1}{32} \,{\left ({\rm Ei}\left (5 \, b x\right ) -{\rm Ei}\left (-5 \, b x\right )\right )} \sinh \left (5 \, a\right ) + \frac{1}{32} \,{\left ({\rm Ei}\left (3 \, b x\right ) -{\rm Ei}\left (-3 \, b x\right )\right )} \sinh \left (3 \, a\right ) - \frac{1}{16} \,{\left ({\rm Ei}\left (b x\right ) -{\rm Ei}\left (-b x\right )\right )} \sinh \left (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 ^{2}{\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 = 1.15999, size = 86, normalized size = 1.18 \begin{align*} \frac{1}{32} \,{\rm Ei}\left (5 \, b x\right ) e^{\left (5 \, a\right )} + \frac{1}{32} \,{\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} - \frac{1}{16} \,{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + \frac{1}{32} \,{\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} + \frac{1}{32} \,{\rm Ei}\left (-5 \, b x\right ) e^{\left (-5 \, a\right )} - \frac{1}{16} \,{\rm Ei}\left (b x\right ) e^{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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