Optimal. Leaf size=169 \[ -\frac{1}{8} b^3 \cosh (2 a) \text{Chi}(2 b x)+\frac{9}{8} b^3 \cosh (6 a) \text{Chi}(6 b x)-\frac{1}{8} b^3 \sinh (2 a) \text{Shi}(2 b x)+\frac{9}{8} b^3 \sinh (6 a) \text{Shi}(6 b x)+\frac{b^2 \sinh (2 a+2 b x)}{16 x}-\frac{3 b^2 \sinh (6 a+6 b x)}{16 x}+\frac{\sinh (2 a+2 b x)}{32 x^3}-\frac{\sinh (6 a+6 b x)}{96 x^3}+\frac{b \cosh (2 a+2 b x)}{32 x^2}-\frac{b \cosh (6 a+6 b x)}{32 x^2} \]
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Rubi [A] time = 0.315137, antiderivative size = 169, 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} \[ -\frac{1}{8} b^3 \cosh (2 a) \text{Chi}(2 b x)+\frac{9}{8} b^3 \cosh (6 a) \text{Chi}(6 b x)-\frac{1}{8} b^3 \sinh (2 a) \text{Shi}(2 b x)+\frac{9}{8} b^3 \sinh (6 a) \text{Shi}(6 b x)+\frac{b^2 \sinh (2 a+2 b x)}{16 x}-\frac{3 b^2 \sinh (6 a+6 b x)}{16 x}+\frac{\sinh (2 a+2 b x)}{32 x^3}-\frac{\sinh (6 a+6 b x)}{96 x^3}+\frac{b \cosh (2 a+2 b x)}{32 x^2}-\frac{b \cosh (6 a+6 b x)}{32 x^2} \]
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^4} \, dx &=\int \left (-\frac{3 \sinh (2 a+2 b x)}{32 x^4}+\frac{\sinh (6 a+6 b x)}{32 x^4}\right ) \, dx\\ &=\frac{1}{32} \int \frac{\sinh (6 a+6 b x)}{x^4} \, dx-\frac{3}{32} \int \frac{\sinh (2 a+2 b x)}{x^4} \, dx\\ &=\frac{\sinh (2 a+2 b x)}{32 x^3}-\frac{\sinh (6 a+6 b x)}{96 x^3}-\frac{1}{16} b \int \frac{\cosh (2 a+2 b x)}{x^3} \, dx+\frac{1}{16} b \int \frac{\cosh (6 a+6 b x)}{x^3} \, dx\\ &=\frac{b \cosh (2 a+2 b x)}{32 x^2}-\frac{b \cosh (6 a+6 b x)}{32 x^2}+\frac{\sinh (2 a+2 b x)}{32 x^3}-\frac{\sinh (6 a+6 b x)}{96 x^3}-\frac{1}{16} b^2 \int \frac{\sinh (2 a+2 b x)}{x^2} \, dx+\frac{1}{16} \left (3 b^2\right ) \int \frac{\sinh (6 a+6 b x)}{x^2} \, dx\\ &=\frac{b \cosh (2 a+2 b x)}{32 x^2}-\frac{b \cosh (6 a+6 b x)}{32 x^2}+\frac{\sinh (2 a+2 b x)}{32 x^3}+\frac{b^2 \sinh (2 a+2 b x)}{16 x}-\frac{\sinh (6 a+6 b x)}{96 x^3}-\frac{3 b^2 \sinh (6 a+6 b x)}{16 x}-\frac{1}{8} b^3 \int \frac{\cosh (2 a+2 b x)}{x} \, dx+\frac{1}{8} \left (9 b^3\right ) \int \frac{\cosh (6 a+6 b x)}{x} \, dx\\ &=\frac{b \cosh (2 a+2 b x)}{32 x^2}-\frac{b \cosh (6 a+6 b x)}{32 x^2}+\frac{\sinh (2 a+2 b x)}{32 x^3}+\frac{b^2 \sinh (2 a+2 b x)}{16 x}-\frac{\sinh (6 a+6 b x)}{96 x^3}-\frac{3 b^2 \sinh (6 a+6 b x)}{16 x}-\frac{1}{8} \left (b^3 \cosh (2 a)\right ) \int \frac{\cosh (2 b x)}{x} \, dx+\frac{1}{8} \left (9 b^3 \cosh (6 a)\right ) \int \frac{\cosh (6 b x)}{x} \, dx-\frac{1}{8} \left (b^3 \sinh (2 a)\right ) \int \frac{\sinh (2 b x)}{x} \, dx+\frac{1}{8} \left (9 b^3 \sinh (6 a)\right ) \int \frac{\sinh (6 b x)}{x} \, dx\\ &=\frac{b \cosh (2 a+2 b x)}{32 x^2}-\frac{b \cosh (6 a+6 b x)}{32 x^2}-\frac{1}{8} b^3 \cosh (2 a) \text{Chi}(2 b x)+\frac{9}{8} b^3 \cosh (6 a) \text{Chi}(6 b x)+\frac{\sinh (2 a+2 b x)}{32 x^3}+\frac{b^2 \sinh (2 a+2 b x)}{16 x}-\frac{\sinh (6 a+6 b x)}{96 x^3}-\frac{3 b^2 \sinh (6 a+6 b x)}{16 x}-\frac{1}{8} b^3 \sinh (2 a) \text{Shi}(2 b x)+\frac{9}{8} b^3 \sinh (6 a) \text{Shi}(6 b x)\\ \end{align*}
Mathematica [A] time = 0.3308, size = 150, normalized size = 0.89 \[ -\frac{12 b^3 x^3 \cosh (2 a) \text{Chi}(2 b x)-108 b^3 x^3 \cosh (6 a) \text{Chi}(6 b x)+12 b^3 x^3 \sinh (2 a) \text{Shi}(2 b x)-108 b^3 x^3 \sinh (6 a) \text{Shi}(6 b x)-6 b^2 x^2 \sinh (2 (a+b x))+18 b^2 x^2 \sinh (6 (a+b x))-3 \sinh (2 (a+b x))+\sinh (6 (a+b x))-3 b x \cosh (2 (a+b x))+3 b x \cosh (6 (a+b x))}{96 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.097, size = 246, normalized size = 1.5 \begin{align*}{\frac{3\,{b}^{2}{{\rm e}^{-6\,bx-6\,a}}}{32\,x}}-{\frac{b{{\rm e}^{-6\,bx-6\,a}}}{64\,{x}^{2}}}+{\frac{{{\rm e}^{-6\,bx-6\,a}}}{192\,{x}^{3}}}-{\frac{9\,{b}^{3}{{\rm e}^{-6\,a}}{\it Ei} \left ( 1,6\,bx \right ) }{16}}-{\frac{{b}^{2}{{\rm e}^{-2\,bx-2\,a}}}{32\,x}}+{\frac{b{{\rm e}^{-2\,bx-2\,a}}}{64\,{x}^{2}}}-{\frac{{{\rm e}^{-2\,bx-2\,a}}}{64\,{x}^{3}}}+{\frac{{b}^{3}{{\rm e}^{-2\,a}}{\it Ei} \left ( 1,2\,bx \right ) }{16}}+{\frac{{{\rm e}^{2\,bx+2\,a}}}{64\,{x}^{3}}}+{\frac{b{{\rm e}^{2\,bx+2\,a}}}{64\,{x}^{2}}}+{\frac{{b}^{2}{{\rm e}^{2\,bx+2\,a}}}{32\,x}}+{\frac{{b}^{3}{{\rm e}^{2\,a}}{\it Ei} \left ( 1,-2\,bx \right ) }{16}}-{\frac{{{\rm e}^{6\,bx+6\,a}}}{192\,{x}^{3}}}-{\frac{b{{\rm e}^{6\,bx+6\,a}}}{64\,{x}^{2}}}-{\frac{3\,{b}^{2}{{\rm e}^{6\,bx+6\,a}}}{32\,x}}-{\frac{9\,{b}^{3}{{\rm e}^{6\,a}}{\it Ei} \left ( 1,-6\,bx \right ) }{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.37818, size = 82, normalized size = 0.49 \begin{align*} \frac{27}{8} \, b^{3} e^{\left (-6 \, a\right )} \Gamma \left (-3, 6 \, b x\right ) - \frac{3}{8} \, b^{3} e^{\left (-2 \, a\right )} \Gamma \left (-3, 2 \, b x\right ) - \frac{3}{8} \, b^{3} e^{\left (2 \, a\right )} \Gamma \left (-3, -2 \, b x\right ) + \frac{27}{8} \, b^{3} e^{\left (6 \, a\right )} \Gamma \left (-3, -6 \, b x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.79641, size = 792, normalized size = 4.69 \begin{align*} -\frac{3 \, b x \cosh \left (b x + a\right )^{6} + 45 \, b x \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} + 3 \, b x \sinh \left (b x + a\right )^{6} + 20 \,{\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3} + 6 \,{\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} - 3 \, b x \cosh \left (b x + a\right )^{2} + 3 \,{\left (15 \, b x \cosh \left (b x + a\right )^{4} - b x\right )} \sinh \left (b x + a\right )^{2} - 54 \,{\left (b^{3} x^{3}{\rm Ei}\left (6 \, b x\right ) + b^{3} x^{3}{\rm Ei}\left (-6 \, b x\right )\right )} \cosh \left (6 \, a\right ) + 6 \,{\left (b^{3} x^{3}{\rm Ei}\left (2 \, b x\right ) + b^{3} x^{3}{\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (2 \, a\right ) + 6 \,{\left ({\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{5} -{\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 54 \,{\left (b^{3} x^{3}{\rm Ei}\left (6 \, b x\right ) - b^{3} x^{3}{\rm Ei}\left (-6 \, b x\right )\right )} \sinh \left (6 \, a\right ) + 6 \,{\left (b^{3} x^{3}{\rm Ei}\left (2 \, b x\right ) - b^{3} x^{3}{\rm Ei}\left (-2 \, b x\right )\right )} \sinh \left (2 \, a\right )}{96 \, x^{3}} \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^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19145, size = 319, normalized size = 1.89 \begin{align*} \frac{108 \, b^{3} x^{3}{\rm Ei}\left (6 \, b x\right ) e^{\left (6 \, a\right )} - 12 \, b^{3} x^{3}{\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - 12 \, b^{3} x^{3}{\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} + 108 \, b^{3} x^{3}{\rm Ei}\left (-6 \, b x\right ) e^{\left (-6 \, a\right )} - 18 \, b^{2} x^{2} e^{\left (6 \, b x + 6 \, a\right )} + 6 \, b^{2} x^{2} e^{\left (2 \, b x + 2 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (-2 \, b x - 2 \, a\right )} + 18 \, b^{2} x^{2} e^{\left (-6 \, b x - 6 \, a\right )} - 3 \, b x e^{\left (6 \, b x + 6 \, a\right )} + 3 \, b x e^{\left (2 \, b x + 2 \, a\right )} + 3 \, b x e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, b x e^{\left (-6 \, b x - 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 )}}{192 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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