Optimal. Leaf size=80 \[ \frac{1}{4} b \cosh (a) \text{Chi}(b x)+\frac{3}{4} b \cosh (3 a) \text{Chi}(3 b x)+\frac{1}{4} b \sinh (a) \text{Shi}(b x)+\frac{3}{4} b \sinh (3 a) \text{Shi}(3 b x)-\frac{\sinh (a+b x)}{4 x}-\frac{\sinh (3 a+3 b x)}{4 x} \]
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Rubi [A] time = 0.185551, antiderivative size = 80, 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} \[ \frac{1}{4} b \cosh (a) \text{Chi}(b x)+\frac{3}{4} b \cosh (3 a) \text{Chi}(3 b x)+\frac{1}{4} b \sinh (a) \text{Shi}(b x)+\frac{3}{4} b \sinh (3 a) \text{Shi}(3 b x)-\frac{\sinh (a+b x)}{4 x}-\frac{\sinh (3 a+3 b x)}{4 x} \]
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 ^2(a+b x) \sinh (a+b x)}{x^2} \, dx &=\int \left (\frac{\sinh (a+b x)}{4 x^2}+\frac{\sinh (3 a+3 b x)}{4 x^2}\right ) \, dx\\ &=\frac{1}{4} \int \frac{\sinh (a+b x)}{x^2} \, dx+\frac{1}{4} \int \frac{\sinh (3 a+3 b x)}{x^2} \, dx\\ &=-\frac{\sinh (a+b x)}{4 x}-\frac{\sinh (3 a+3 b x)}{4 x}+\frac{1}{4} b \int \frac{\cosh (a+b x)}{x} \, dx+\frac{1}{4} (3 b) \int \frac{\cosh (3 a+3 b x)}{x} \, dx\\ &=-\frac{\sinh (a+b x)}{4 x}-\frac{\sinh (3 a+3 b x)}{4 x}+\frac{1}{4} (b \cosh (a)) \int \frac{\cosh (b x)}{x} \, dx+\frac{1}{4} (3 b \cosh (3 a)) \int \frac{\cosh (3 b x)}{x} \, dx+\frac{1}{4} (b \sinh (a)) \int \frac{\sinh (b x)}{x} \, dx+\frac{1}{4} (3 b \sinh (3 a)) \int \frac{\sinh (3 b x)}{x} \, dx\\ &=\frac{1}{4} b \cosh (a) \text{Chi}(b x)+\frac{3}{4} b \cosh (3 a) \text{Chi}(3 b x)-\frac{\sinh (a+b x)}{4 x}-\frac{\sinh (3 a+3 b x)}{4 x}+\frac{1}{4} b \sinh (a) \text{Shi}(b x)+\frac{3}{4} b \sinh (3 a) \text{Shi}(3 b x)\\ \end{align*}
Mathematica [A] time = 0.148197, size = 70, normalized size = 0.88 \[ \frac{b x \cosh (a) \text{Chi}(b x)+3 b x \cosh (3 a) \text{Chi}(3 b x)+b x \sinh (a) \text{Shi}(b x)+3 b x \sinh (3 a) \text{Shi}(3 b x)-\sinh (a+b x)-\sinh (3 (a+b x))}{4 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 104, normalized size = 1.3 \begin{align*}{\frac{{{\rm e}^{-3\,bx-3\,a}}}{8\,x}}-{\frac{3\,b{{\rm e}^{-3\,a}}{\it Ei} \left ( 1,3\,bx \right ) }{8}}+{\frac{{{\rm e}^{-bx-a}}}{8\,x}}-{\frac{b{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) }{8}}-{\frac{{{\rm e}^{bx+a}}}{8\,x}}-{\frac{b{{\rm e}^{a}}{\it Ei} \left ( 1,-bx \right ) }{8}}-{\frac{{{\rm e}^{3\,bx+3\,a}}}{8\,x}}-{\frac{3\,b{{\rm e}^{3\,a}}{\it Ei} \left ( 1,-3\,bx \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.38231, size = 68, normalized size = 0.85 \begin{align*} \frac{3}{8} \, b e^{\left (-3 \, a\right )} \Gamma \left (-1, 3 \, b x\right ) + \frac{1}{8} \, b e^{\left (-a\right )} \Gamma \left (-1, b x\right ) + \frac{1}{8} \, b e^{a} \Gamma \left (-1, -b x\right ) + \frac{3}{8} \, b e^{\left (3 \, a\right )} \Gamma \left (-1, -3 \, b x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89148, size = 327, normalized size = 4.09 \begin{align*} -\frac{2 \, \sinh \left (b x + a\right )^{3} - 3 \,{\left (b x{\rm Ei}\left (3 \, b x\right ) + b x{\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (3 \, a\right ) -{\left (b x{\rm Ei}\left (b x\right ) + b x{\rm Ei}\left (-b x\right )\right )} \cosh \left (a\right ) + 2 \,{\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right ) - 3 \,{\left (b x{\rm Ei}\left (3 \, b x\right ) - b x{\rm Ei}\left (-3 \, b x\right )\right )} \sinh \left (3 \, a\right ) -{\left (b x{\rm Ei}\left (b x\right ) - b x{\rm Ei}\left (-b x\right )\right )} \sinh \left (a\right )}{8 \, 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 ^{2}{\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 = 1.13776, size = 122, normalized size = 1.52 \begin{align*} \frac{3 \, b x{\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} + b x{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + 3 \, b x{\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} + b x{\rm Ei}\left (b x\right ) e^{a} - 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 )}}{8 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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