Optimal. Leaf size=89 \[ -\frac{3}{16} b \cosh (2 a) \text{Chi}(2 b x)+\frac{3}{16} b \cosh (6 a) \text{Chi}(6 b x)-\frac{3}{16} b \sinh (2 a) \text{Shi}(2 b x)+\frac{3}{16} b \sinh (6 a) \text{Shi}(6 b x)+\frac{3 \sinh (2 a+2 b x)}{32 x}-\frac{\sinh (6 a+6 b x)}{32 x} \]
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Rubi [A] time = 0.19576, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5448, 3297, 3303, 3298, 3301} \[ -\frac{3}{16} b \cosh (2 a) \text{Chi}(2 b x)+\frac{3}{16} b \cosh (6 a) \text{Chi}(6 b x)-\frac{3}{16} b \sinh (2 a) \text{Shi}(2 b x)+\frac{3}{16} b \sinh (6 a) \text{Shi}(6 b x)+\frac{3 \sinh (2 a+2 b x)}{32 x}-\frac{\sinh (6 a+6 b x)}{32 x} \]
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 ^3(a+b x)}{x^2} \, dx &=\int \left (-\frac{3 \sinh (2 a+2 b x)}{32 x^2}+\frac{\sinh (6 a+6 b x)}{32 x^2}\right ) \, dx\\ &=\frac{1}{32} \int \frac{\sinh (6 a+6 b x)}{x^2} \, dx-\frac{3}{32} \int \frac{\sinh (2 a+2 b x)}{x^2} \, dx\\ &=\frac{3 \sinh (2 a+2 b x)}{32 x}-\frac{\sinh (6 a+6 b x)}{32 x}-\frac{1}{16} (3 b) \int \frac{\cosh (2 a+2 b x)}{x} \, dx+\frac{1}{16} (3 b) \int \frac{\cosh (6 a+6 b x)}{x} \, dx\\ &=\frac{3 \sinh (2 a+2 b x)}{32 x}-\frac{\sinh (6 a+6 b x)}{32 x}-\frac{1}{16} (3 b \cosh (2 a)) \int \frac{\cosh (2 b x)}{x} \, dx+\frac{1}{16} (3 b \cosh (6 a)) \int \frac{\cosh (6 b x)}{x} \, dx-\frac{1}{16} (3 b \sinh (2 a)) \int \frac{\sinh (2 b x)}{x} \, dx+\frac{1}{16} (3 b \sinh (6 a)) \int \frac{\sinh (6 b x)}{x} \, dx\\ &=-\frac{3}{16} b \cosh (2 a) \text{Chi}(2 b x)+\frac{3}{16} b \cosh (6 a) \text{Chi}(6 b x)+\frac{3 \sinh (2 a+2 b x)}{32 x}-\frac{\sinh (6 a+6 b x)}{32 x}-\frac{3}{16} b \sinh (2 a) \text{Shi}(2 b x)+\frac{3}{16} b \sinh (6 a) \text{Shi}(6 b x)\\ \end{align*}
Mathematica [A] time = 0.242686, size = 78, normalized size = 0.88 \[ -\frac{6 b x \cosh (2 a) \text{Chi}(2 b x)-6 b x \cosh (6 a) \text{Chi}(6 b x)+6 b x \sinh (2 a) \text{Shi}(2 b x)-6 b x \sinh (6 a) \text{Shi}(6 b x)-3 \sinh (2 (a+b x))+\sinh (6 (a+b x))}{32 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.095, size = 110, normalized size = 1.2 \begin{align*}{\frac{{{\rm e}^{-6\,bx-6\,a}}}{64\,x}}-{\frac{3\,b{{\rm e}^{-6\,a}}{\it Ei} \left ( 1,6\,bx \right ) }{32}}-{\frac{3\,{{\rm e}^{-2\,bx-2\,a}}}{64\,x}}+{\frac{3\,b{{\rm e}^{-2\,a}}{\it Ei} \left ( 1,2\,bx \right ) }{32}}-{\frac{{{\rm e}^{6\,bx+6\,a}}}{64\,x}}-{\frac{3\,b{{\rm e}^{6\,a}}{\it Ei} \left ( 1,-6\,bx \right ) }{32}}+{\frac{3\,{{\rm e}^{2\,bx+2\,a}}}{64\,x}}+{\frac{3\,b{{\rm e}^{2\,a}}{\it Ei} \left ( 1,-2\,bx \right ) }{32}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.24542, size = 72, normalized size = 0.81 \begin{align*} \frac{3}{32} \, b e^{\left (-6 \, a\right )} \Gamma \left (-1, 6 \, b x\right ) - \frac{3}{32} \, b e^{\left (-2 \, a\right )} \Gamma \left (-1, 2 \, b x\right ) - \frac{3}{32} \, b e^{\left (2 \, a\right )} \Gamma \left (-1, -2 \, b x\right ) + \frac{3}{32} \, b e^{\left (6 \, a\right )} \Gamma \left (-1, -6 \, b x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.72144, size = 432, normalized size = 4.85 \begin{align*} -\frac{20 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} - 3 \,{\left (b x{\rm Ei}\left (6 \, b x\right ) + b x{\rm Ei}\left (-6 \, b x\right )\right )} \cosh \left (6 \, a\right ) + 3 \,{\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 ) + 6 \,{\left (\cosh \left (b x + a\right )^{5} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 3 \,{\left (b x{\rm Ei}\left (6 \, b x\right ) - b x{\rm Ei}\left (-6 \, b x\right )\right )} \sinh \left (6 \, a\right ) + 3 \,{\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 )}{32 \, 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 ^{3}{\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 = 1.18815, size = 135, normalized size = 1.52 \begin{align*} \frac{6 \, b x{\rm Ei}\left (6 \, b x\right ) e^{\left (6 \, a\right )} - 6 \, b x{\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - 6 \, b x{\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} + 6 \, b x{\rm Ei}\left (-6 \, b x\right ) e^{\left (-6 \, a\right )} - e^{\left (6 \, b x + 6 \, a\right )} + 3 \, e^{\left (2 \, b x + 2 \, a\right )} - 3 \, e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}}{64 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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