Optimal. Leaf size=119 \[ -\frac{1}{8} b^2 \cosh (a) \text{Chi}(b x)+\frac{9}{8} b^2 \cosh (3 a) \text{Chi}(3 b x)-\frac{1}{8} b^2 \sinh (a) \text{Shi}(b x)+\frac{9}{8} b^2 \sinh (3 a) \text{Shi}(3 b x)+\frac{\cosh (a+b x)}{8 x^2}-\frac{\cosh (3 a+3 b x)}{8 x^2}+\frac{b \sinh (a+b x)}{8 x}-\frac{3 b \sinh (3 a+3 b x)}{8 x} \]
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Rubi [A] time = 0.217915, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 12, 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} \[ -\frac{1}{8} b^2 \cosh (a) \text{Chi}(b x)+\frac{9}{8} b^2 \cosh (3 a) \text{Chi}(3 b x)-\frac{1}{8} b^2 \sinh (a) \text{Shi}(b x)+\frac{9}{8} b^2 \sinh (3 a) \text{Shi}(3 b x)+\frac{\cosh (a+b x)}{8 x^2}-\frac{\cosh (3 a+3 b x)}{8 x^2}+\frac{b \sinh (a+b x)}{8 x}-\frac{3 b \sinh (3 a+3 b x)}{8 x} \]
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 (a+b x) \sinh ^2(a+b x)}{x^3} \, dx &=\int \left (-\frac{\cosh (a+b x)}{4 x^3}+\frac{\cosh (3 a+3 b x)}{4 x^3}\right ) \, dx\\ &=-\left (\frac{1}{4} \int \frac{\cosh (a+b x)}{x^3} \, dx\right )+\frac{1}{4} \int \frac{\cosh (3 a+3 b x)}{x^3} \, dx\\ &=\frac{\cosh (a+b x)}{8 x^2}-\frac{\cosh (3 a+3 b x)}{8 x^2}-\frac{1}{8} b \int \frac{\sinh (a+b x)}{x^2} \, dx+\frac{1}{8} (3 b) \int \frac{\sinh (3 a+3 b x)}{x^2} \, dx\\ &=\frac{\cosh (a+b x)}{8 x^2}-\frac{\cosh (3 a+3 b x)}{8 x^2}+\frac{b \sinh (a+b x)}{8 x}-\frac{3 b \sinh (3 a+3 b x)}{8 x}-\frac{1}{8} b^2 \int \frac{\cosh (a+b x)}{x} \, dx+\frac{1}{8} \left (9 b^2\right ) \int \frac{\cosh (3 a+3 b x)}{x} \, dx\\ &=\frac{\cosh (a+b x)}{8 x^2}-\frac{\cosh (3 a+3 b x)}{8 x^2}+\frac{b \sinh (a+b x)}{8 x}-\frac{3 b \sinh (3 a+3 b x)}{8 x}-\frac{1}{8} \left (b^2 \cosh (a)\right ) \int \frac{\cosh (b x)}{x} \, dx+\frac{1}{8} \left (9 b^2 \cosh (3 a)\right ) \int \frac{\cosh (3 b x)}{x} \, dx-\frac{1}{8} \left (b^2 \sinh (a)\right ) \int \frac{\sinh (b x)}{x} \, dx+\frac{1}{8} \left (9 b^2 \sinh (3 a)\right ) \int \frac{\sinh (3 b x)}{x} \, dx\\ &=\frac{\cosh (a+b x)}{8 x^2}-\frac{\cosh (3 a+3 b x)}{8 x^2}-\frac{1}{8} b^2 \cosh (a) \text{Chi}(b x)+\frac{9}{8} b^2 \cosh (3 a) \text{Chi}(3 b x)+\frac{b \sinh (a+b x)}{8 x}-\frac{3 b \sinh (3 a+3 b x)}{8 x}-\frac{1}{8} b^2 \sinh (a) \text{Shi}(b x)+\frac{9}{8} b^2 \sinh (3 a) \text{Shi}(3 b x)\\ \end{align*}
Mathematica [A] time = 0.255259, size = 107, normalized size = 0.9 \[ \frac{-b^2 x^2 \cosh (a) \text{Chi}(b x)+9 b^2 x^2 \cosh (3 a) \text{Chi}(3 b x)-b^2 x^2 \sinh (a) \text{Shi}(b x)+9 b^2 x^2 \sinh (3 a) \text{Shi}(3 b x)+b x \sinh (a+b x)-3 b x \sinh (3 (a+b x))+\cosh (a+b x)-\cosh (3 (a+b x))}{8 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.075, size = 169, normalized size = 1.4 \begin{align*}{\frac{3\,b{{\rm e}^{-3\,bx-3\,a}}}{16\,x}}-{\frac{{{\rm e}^{-3\,bx-3\,a}}}{16\,{x}^{2}}}-{\frac{9\,{b}^{2}{{\rm e}^{-3\,a}}{\it Ei} \left ( 1,3\,bx \right ) }{16}}-{\frac{b{{\rm e}^{-bx-a}}}{16\,x}}+{\frac{{{\rm e}^{-bx-a}}}{16\,{x}^{2}}}+{\frac{{b}^{2}{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) }{16}}+{\frac{{{\rm e}^{bx+a}}}{16\,{x}^{2}}}+{\frac{b{{\rm e}^{bx+a}}}{16\,x}}+{\frac{{b}^{2}{{\rm e}^{a}}{\it Ei} \left ( 1,-bx \right ) }{16}}-{\frac{{{\rm e}^{3\,bx+3\,a}}}{16\,{x}^{2}}}-{\frac{3\,b{{\rm e}^{3\,bx+3\,a}}}{16\,x}}-{\frac{9\,{b}^{2}{{\rm e}^{3\,a}}{\it Ei} \left ( 1,-3\,bx \right ) }{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.33273, size = 78, normalized size = 0.66 \begin{align*} -\frac{9}{8} \, b^{2} e^{\left (-3 \, a\right )} \Gamma \left (-2, 3 \, b x\right ) + \frac{1}{8} \, b^{2} e^{\left (-a\right )} \Gamma \left (-2, b x\right ) + \frac{1}{8} \, b^{2} e^{a} \Gamma \left (-2, -b x\right ) - \frac{9}{8} \, b^{2} e^{\left (3 \, a\right )} \Gamma \left (-2, -3 \, b x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76983, size = 485, normalized size = 4.08 \begin{align*} -\frac{6 \, b x \sinh \left (b x + a\right )^{3} + 2 \, \cosh \left (b x + a\right )^{3} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 9 \,{\left (b^{2} x^{2}{\rm Ei}\left (3 \, b x\right ) + b^{2} x^{2}{\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (3 \, a\right ) +{\left (b^{2} x^{2}{\rm Ei}\left (b x\right ) + b^{2} x^{2}{\rm Ei}\left (-b x\right )\right )} \cosh \left (a\right ) + 2 \,{\left (9 \, b x \cosh \left (b x + a\right )^{2} - b x\right )} \sinh \left (b x + a\right ) - 9 \,{\left (b^{2} x^{2}{\rm Ei}\left (3 \, b x\right ) - b^{2} x^{2}{\rm Ei}\left (-3 \, b x\right )\right )} \sinh \left (3 \, a\right ) +{\left (b^{2} x^{2}{\rm Ei}\left (b x\right ) - b^{2} x^{2}{\rm Ei}\left (-b x\right )\right )} \sinh \left (a\right ) - 2 \, \cosh \left (b x + a\right )}{16 \, x^{2}} \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{\left (a + b x \right )}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18011, size = 211, normalized size = 1.77 \begin{align*} \frac{9 \, b^{2} x^{2}{\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} - b^{2} x^{2}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + 9 \, b^{2} x^{2}{\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} - b^{2} x^{2}{\rm Ei}\left (b x\right ) e^{a} - 3 \, b x e^{\left (3 \, b x + 3 \, a\right )} + b x e^{\left (b x + a\right )} - b x e^{\left (-b x - a\right )} + 3 \, b x e^{\left (-3 \, b x - 3 \, a\right )} - e^{\left (3 \, b x + 3 \, a\right )} + e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )}}{16 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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