Optimal. Leaf size=125 \[ \frac{1}{2} b^2 \sinh (2 a) \text{Chi}(2 b x)+b^2 \sinh (4 a) \text{Chi}(4 b x)+\frac{1}{2} b^2 \cosh (2 a) \text{Shi}(2 b x)+b^2 \cosh (4 a) \text{Shi}(4 b x)-\frac{\sinh (2 a+2 b x)}{8 x^2}-\frac{\sinh (4 a+4 b x)}{16 x^2}-\frac{b \cosh (2 a+2 b x)}{4 x}-\frac{b \cosh (4 a+4 b x)}{4 x} \]
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Rubi [A] time = 0.247623, antiderivative size = 125, 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} \[ \frac{1}{2} b^2 \sinh (2 a) \text{Chi}(2 b x)+b^2 \sinh (4 a) \text{Chi}(4 b x)+\frac{1}{2} b^2 \cosh (2 a) \text{Shi}(2 b x)+b^2 \cosh (4 a) \text{Shi}(4 b x)-\frac{\sinh (2 a+2 b x)}{8 x^2}-\frac{\sinh (4 a+4 b x)}{16 x^2}-\frac{b \cosh (2 a+2 b x)}{4 x}-\frac{b \cosh (4 a+4 b x)}{4 x} \]
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 ^3(a+b x) \sinh (a+b x)}{x^3} \, dx &=\int \left (\frac{\sinh (2 a+2 b x)}{4 x^3}+\frac{\sinh (4 a+4 b x)}{8 x^3}\right ) \, dx\\ &=\frac{1}{8} \int \frac{\sinh (4 a+4 b x)}{x^3} \, dx+\frac{1}{4} \int \frac{\sinh (2 a+2 b x)}{x^3} \, dx\\ &=-\frac{\sinh (2 a+2 b x)}{8 x^2}-\frac{\sinh (4 a+4 b x)}{16 x^2}+\frac{1}{4} b \int \frac{\cosh (2 a+2 b x)}{x^2} \, dx+\frac{1}{4} b \int \frac{\cosh (4 a+4 b x)}{x^2} \, dx\\ &=-\frac{b \cosh (2 a+2 b x)}{4 x}-\frac{b \cosh (4 a+4 b x)}{4 x}-\frac{\sinh (2 a+2 b x)}{8 x^2}-\frac{\sinh (4 a+4 b x)}{16 x^2}+\frac{1}{2} b^2 \int \frac{\sinh (2 a+2 b x)}{x} \, dx+b^2 \int \frac{\sinh (4 a+4 b x)}{x} \, dx\\ &=-\frac{b \cosh (2 a+2 b x)}{4 x}-\frac{b \cosh (4 a+4 b x)}{4 x}-\frac{\sinh (2 a+2 b x)}{8 x^2}-\frac{\sinh (4 a+4 b x)}{16 x^2}+\frac{1}{2} \left (b^2 \cosh (2 a)\right ) \int \frac{\sinh (2 b x)}{x} \, dx+\left (b^2 \cosh (4 a)\right ) \int \frac{\sinh (4 b x)}{x} \, dx+\frac{1}{2} \left (b^2 \sinh (2 a)\right ) \int \frac{\cosh (2 b x)}{x} \, dx+\left (b^2 \sinh (4 a)\right ) \int \frac{\cosh (4 b x)}{x} \, dx\\ &=-\frac{b \cosh (2 a+2 b x)}{4 x}-\frac{b \cosh (4 a+4 b x)}{4 x}+\frac{1}{2} b^2 \text{Chi}(2 b x) \sinh (2 a)+b^2 \text{Chi}(4 b x) \sinh (4 a)-\frac{\sinh (2 a+2 b x)}{8 x^2}-\frac{\sinh (4 a+4 b x)}{16 x^2}+\frac{1}{2} b^2 \cosh (2 a) \text{Shi}(2 b x)+b^2 \cosh (4 a) \text{Shi}(4 b x)\\ \end{align*}
Mathematica [A] time = 0.589403, size = 112, normalized size = 0.9 \[ b^2 \sinh (4 a) \text{Chi}(4 b x)+b^2 \sinh (a) \cosh (a) \text{Chi}(2 b x)+\frac{1}{2} b^2 \cosh (2 a) \text{Shi}(2 b x)+b^2 \cosh (4 a) \text{Shi}(4 b x)-\frac{\sinh (2 (a+b x))+2 b x \cosh (2 (a+b x))}{8 x^2}-\frac{\sinh (4 (a+b x))+4 b x \cosh (4 (a+b x))}{16 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 178, normalized size = 1.4 \begin{align*} -{\frac{b{{\rm e}^{-4\,bx-4\,a}}}{8\,x}}+{\frac{{{\rm e}^{-4\,bx-4\,a}}}{32\,{x}^{2}}}+{\frac{{b}^{2}{{\rm e}^{-4\,a}}{\it Ei} \left ( 1,4\,bx \right ) }{2}}-{\frac{b{{\rm e}^{-2\,bx-2\,a}}}{8\,x}}+{\frac{{{\rm e}^{-2\,bx-2\,a}}}{16\,{x}^{2}}}+{\frac{{b}^{2}{{\rm e}^{-2\,a}}{\it Ei} \left ( 1,2\,bx \right ) }{4}}-{\frac{{{\rm e}^{4\,bx+4\,a}}}{32\,{x}^{2}}}-{\frac{b{{\rm e}^{4\,bx+4\,a}}}{8\,x}}-{\frac{{b}^{2}{{\rm e}^{4\,a}}{\it Ei} \left ( 1,-4\,bx \right ) }{2}}-{\frac{{{\rm e}^{2\,bx+2\,a}}}{16\,{x}^{2}}}-{\frac{b{{\rm e}^{2\,bx+2\,a}}}{8\,x}}-{\frac{{b}^{2}{{\rm e}^{2\,a}}{\it Ei} \left ( 1,-2\,bx \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.38813, size = 81, normalized size = 0.65 \begin{align*} b^{2} e^{\left (-4 \, a\right )} \Gamma \left (-2, 4 \, b x\right ) + \frac{1}{2} \, b^{2} e^{\left (-2 \, a\right )} \Gamma \left (-2, 2 \, b x\right ) - \frac{1}{2} \, b^{2} e^{\left (2 \, a\right )} \Gamma \left (-2, -2 \, b x\right ) - b^{2} e^{\left (4 \, a\right )} \Gamma \left (-2, -4 \, b x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87652, size = 570, normalized size = 4.56 \begin{align*} -\frac{b x \cosh \left (b x + a\right )^{4} + b x \sinh \left (b x + a\right )^{4} + b x \cosh \left (b x + a\right )^{2} + \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} +{\left (6 \, b x \cosh \left (b x + a\right )^{2} + b x\right )} \sinh \left (b x + a\right )^{2} - 2 \,{\left (b^{2} x^{2}{\rm Ei}\left (4 \, b x\right ) - b^{2} x^{2}{\rm Ei}\left (-4 \, b x\right )\right )} \cosh \left (4 \, a\right ) -{\left (b^{2} x^{2}{\rm Ei}\left (2 \, b x\right ) - b^{2} x^{2}{\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (2 \, a\right ) +{\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 2 \,{\left (b^{2} x^{2}{\rm Ei}\left (4 \, b x\right ) + b^{2} x^{2}{\rm Ei}\left (-4 \, b x\right )\right )} \sinh \left (4 \, a\right ) -{\left (b^{2} x^{2}{\rm Ei}\left (2 \, b x\right ) + b^{2} x^{2}{\rm Ei}\left (-2 \, b x\right )\right )} \sinh \left (2 \, a\right )}{4 \, 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{\left (a + b x \right )} \cosh ^{3}{\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 = 1.18339, size = 227, normalized size = 1.82 \begin{align*} \frac{16 \, b^{2} x^{2}{\rm Ei}\left (4 \, b x\right ) e^{\left (4 \, a\right )} + 8 \, b^{2} x^{2}{\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - 8 \, b^{2} x^{2}{\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} - 16 \, b^{2} x^{2}{\rm Ei}\left (-4 \, b x\right ) e^{\left (-4 \, a\right )} - 4 \, b x e^{\left (4 \, b x + 4 \, a\right )} - 4 \, b x e^{\left (2 \, b x + 2 \, a\right )} - 4 \, b x e^{\left (-2 \, b x - 2 \, a\right )} - 4 \, b x e^{\left (-4 \, b x - 4 \, a\right )} - e^{\left (4 \, b x + 4 \, a\right )} - 2 \, e^{\left (2 \, b x + 2 \, a\right )} + 2 \, e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-4 \, b x - 4 \, a\right )}}{32 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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