Optimal. Leaf size=124 \[ -\frac{1}{8} b \sinh (a) \text{Chi}(b x)+\frac{3}{16} b \sinh (3 a) \text{Chi}(3 b x)+\frac{5}{16} b \sinh (5 a) \text{Chi}(5 b x)-\frac{1}{8} b \cosh (a) \text{Shi}(b x)+\frac{3}{16} b \cosh (3 a) \text{Shi}(3 b x)+\frac{5}{16} b \cosh (5 a) \text{Shi}(5 b x)+\frac{\cosh (a+b x)}{8 x}-\frac{\cosh (3 a+3 b x)}{16 x}-\frac{\cosh (5 a+5 b x)}{16 x} \]
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Rubi [A] time = 0.263126, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 14, 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} \[ -\frac{1}{8} b \sinh (a) \text{Chi}(b x)+\frac{3}{16} b \sinh (3 a) \text{Chi}(3 b x)+\frac{5}{16} b \sinh (5 a) \text{Chi}(5 b x)-\frac{1}{8} b \cosh (a) \text{Shi}(b x)+\frac{3}{16} b \cosh (3 a) \text{Shi}(3 b x)+\frac{5}{16} b \cosh (5 a) \text{Shi}(5 b x)+\frac{\cosh (a+b x)}{8 x}-\frac{\cosh (3 a+3 b x)}{16 x}-\frac{\cosh (5 a+5 b x)}{16 x} \]
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 ^2(a+b x)}{x^2} \, dx &=\int \left (-\frac{\cosh (a+b x)}{8 x^2}+\frac{\cosh (3 a+3 b x)}{16 x^2}+\frac{\cosh (5 a+5 b x)}{16 x^2}\right ) \, dx\\ &=\frac{1}{16} \int \frac{\cosh (3 a+3 b x)}{x^2} \, dx+\frac{1}{16} \int \frac{\cosh (5 a+5 b x)}{x^2} \, dx-\frac{1}{8} \int \frac{\cosh (a+b x)}{x^2} \, dx\\ &=\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}-\frac{1}{8} b \int \frac{\sinh (a+b x)}{x} \, dx+\frac{1}{16} (3 b) \int \frac{\sinh (3 a+3 b x)}{x} \, dx+\frac{1}{16} (5 b) \int \frac{\sinh (5 a+5 b x)}{x} \, dx\\ &=\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}-\frac{1}{8} (b \cosh (a)) \int \frac{\sinh (b x)}{x} \, dx+\frac{1}{16} (3 b \cosh (3 a)) \int \frac{\sinh (3 b x)}{x} \, dx+\frac{1}{16} (5 b \cosh (5 a)) \int \frac{\sinh (5 b x)}{x} \, dx-\frac{1}{8} (b \sinh (a)) \int \frac{\cosh (b x)}{x} \, dx+\frac{1}{16} (3 b \sinh (3 a)) \int \frac{\cosh (3 b x)}{x} \, dx+\frac{1}{16} (5 b \sinh (5 a)) \int \frac{\cosh (5 b x)}{x} \, dx\\ &=\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}-\frac{1}{8} b \text{Chi}(b x) \sinh (a)+\frac{3}{16} b \text{Chi}(3 b x) \sinh (3 a)+\frac{5}{16} b \text{Chi}(5 b x) \sinh (5 a)-\frac{1}{8} b \cosh (a) \text{Shi}(b x)+\frac{3}{16} b \cosh (3 a) \text{Shi}(3 b x)+\frac{5}{16} b \cosh (5 a) \text{Shi}(5 b x)\\ \end{align*}
Mathematica [A] time = 0.354474, size = 104, normalized size = 0.84 \[ -\frac{2 b x \sinh (a) \text{Chi}(b x)-3 b x \sinh (3 a) \text{Chi}(3 b x)-5 b x \sinh (5 a) \text{Chi}(5 b x)+2 b x \cosh (a) \text{Shi}(b x)-3 b x \cosh (3 a) \text{Shi}(3 b x)-5 b x \cosh (5 a) \text{Shi}(5 b x)-2 \cosh (a+b x)+\cosh (3 (a+b x))+\cosh (5 (a+b x))}{16 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.109, size = 158, normalized size = 1.3 \begin{align*} -{\frac{{{\rm e}^{-5\,bx-5\,a}}}{32\,x}}+{\frac{5\,b{{\rm e}^{-5\,a}}{\it Ei} \left ( 1,5\,bx \right ) }{32}}-{\frac{{{\rm e}^{-3\,bx-3\,a}}}{32\,x}}+{\frac{3\,b{{\rm e}^{-3\,a}}{\it Ei} \left ( 1,3\,bx \right ) }{32}}+{\frac{{{\rm e}^{-bx-a}}}{16\,x}}-{\frac{b{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) }{16}}+{\frac{{{\rm e}^{bx+a}}}{16\,x}}+{\frac{b{{\rm e}^{a}}{\it Ei} \left ( 1,-bx \right ) }{16}}-{\frac{{{\rm e}^{3\,bx+3\,a}}}{32\,x}}-{\frac{3\,b{{\rm e}^{3\,a}}{\it Ei} \left ( 1,-3\,bx \right ) }{32}}-{\frac{{{\rm e}^{5\,bx+5\,a}}}{32\,x}}-{\frac{5\,b{{\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 = 1.26234, size = 103, normalized size = 0.83 \begin{align*} -\frac{5}{32} \, b e^{\left (-5 \, a\right )} \Gamma \left (-1, 5 \, b x\right ) - \frac{3}{32} \, b e^{\left (-3 \, a\right )} \Gamma \left (-1, 3 \, b x\right ) + \frac{1}{16} \, b e^{\left (-a\right )} \Gamma \left (-1, b x\right ) - \frac{1}{16} \, b e^{a} \Gamma \left (-1, -b x\right ) + \frac{3}{32} \, b e^{\left (3 \, a\right )} \Gamma \left (-1, -3 \, b x\right ) + \frac{5}{32} \, b e^{\left (5 \, a\right )} \Gamma \left (-1, -5 \, b x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.76466, size = 582, normalized size = 4.69 \begin{align*} -\frac{2 \, \cosh \left (b x + a\right )^{5} + 10 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + 2 \, \cosh \left (b x + a\right )^{3} + 2 \,{\left (10 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} - 5 \,{\left (b x{\rm Ei}\left (5 \, b x\right ) - b x{\rm Ei}\left (-5 \, b x\right )\right )} \cosh \left (5 \, a\right ) - 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 ) + 2 \,{\left (b x{\rm Ei}\left (b x\right ) - b x{\rm Ei}\left (-b x\right )\right )} \cosh \left (a\right ) - 5 \,{\left (b x{\rm Ei}\left (5 \, b x\right ) + b x{\rm Ei}\left (-5 \, b x\right )\right )} \sinh \left (5 \, 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 ) + 2 \,{\left (b x{\rm Ei}\left (b x\right ) + b x{\rm Ei}\left (-b x\right )\right )} \sinh \left (a\right ) - 4 \, \cosh \left (b x + 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 ^{2}{\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 = 1.21371, size = 194, normalized size = 1.56 \begin{align*} \frac{5 \, b x{\rm Ei}\left (5 \, b x\right ) e^{\left (5 \, a\right )} + 3 \, b x{\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} + 2 \, 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 )} - 5 \, b x{\rm Ei}\left (-5 \, b x\right ) e^{\left (-5 \, a\right )} - 2 \, b x{\rm Ei}\left (b x\right ) e^{a} - e^{\left (5 \, b x + 5 \, a\right )} - e^{\left (3 \, b x + 3 \, a\right )} + 2 \, e^{\left (b x + a\right )} + 2 \, e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )} - e^{\left (-5 \, b x - 5 \, a\right )}}{32 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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