Optimal. Leaf size=131 \[ -\frac{3}{16} b^2 \sinh (2 a) \text{Chi}(2 b x)+\frac{9}{16} b^2 \sinh (6 a) \text{Chi}(6 b x)-\frac{3}{16} b^2 \cosh (2 a) \text{Shi}(2 b x)+\frac{9}{16} b^2 \cosh (6 a) \text{Shi}(6 b x)+\frac{3 \sinh (2 a+2 b x)}{64 x^2}-\frac{\sinh (6 a+6 b x)}{64 x^2}+\frac{3 b \cosh (2 a+2 b x)}{32 x}-\frac{3 b \cosh (6 a+6 b x)}{32 x} \]
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Rubi [A] time = 0.25697, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 12, 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} \[ -\frac{3}{16} b^2 \sinh (2 a) \text{Chi}(2 b x)+\frac{9}{16} b^2 \sinh (6 a) \text{Chi}(6 b x)-\frac{3}{16} b^2 \cosh (2 a) \text{Shi}(2 b x)+\frac{9}{16} b^2 \cosh (6 a) \text{Shi}(6 b x)+\frac{3 \sinh (2 a+2 b x)}{64 x^2}-\frac{\sinh (6 a+6 b x)}{64 x^2}+\frac{3 b \cosh (2 a+2 b x)}{32 x}-\frac{3 b \cosh (6 a+6 b x)}{32 x} \]
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 ^3(a+b x)}{x^3} \, dx &=\int \left (-\frac{3 \sinh (2 a+2 b x)}{32 x^3}+\frac{\sinh (6 a+6 b x)}{32 x^3}\right ) \, dx\\ &=\frac{1}{32} \int \frac{\sinh (6 a+6 b x)}{x^3} \, dx-\frac{3}{32} \int \frac{\sinh (2 a+2 b x)}{x^3} \, dx\\ &=\frac{3 \sinh (2 a+2 b x)}{64 x^2}-\frac{\sinh (6 a+6 b x)}{64 x^2}-\frac{1}{32} (3 b) \int \frac{\cosh (2 a+2 b x)}{x^2} \, dx+\frac{1}{32} (3 b) \int \frac{\cosh (6 a+6 b x)}{x^2} \, dx\\ &=\frac{3 b \cosh (2 a+2 b x)}{32 x}-\frac{3 b \cosh (6 a+6 b x)}{32 x}+\frac{3 \sinh (2 a+2 b x)}{64 x^2}-\frac{\sinh (6 a+6 b x)}{64 x^2}-\frac{1}{16} \left (3 b^2\right ) \int \frac{\sinh (2 a+2 b x)}{x} \, dx+\frac{1}{16} \left (9 b^2\right ) \int \frac{\sinh (6 a+6 b x)}{x} \, dx\\ &=\frac{3 b \cosh (2 a+2 b x)}{32 x}-\frac{3 b \cosh (6 a+6 b x)}{32 x}+\frac{3 \sinh (2 a+2 b x)}{64 x^2}-\frac{\sinh (6 a+6 b x)}{64 x^2}-\frac{1}{16} \left (3 b^2 \cosh (2 a)\right ) \int \frac{\sinh (2 b x)}{x} \, dx+\frac{1}{16} \left (9 b^2 \cosh (6 a)\right ) \int \frac{\sinh (6 b x)}{x} \, dx-\frac{1}{16} \left (3 b^2 \sinh (2 a)\right ) \int \frac{\cosh (2 b x)}{x} \, dx+\frac{1}{16} \left (9 b^2 \sinh (6 a)\right ) \int \frac{\cosh (6 b x)}{x} \, dx\\ &=\frac{3 b \cosh (2 a+2 b x)}{32 x}-\frac{3 b \cosh (6 a+6 b x)}{32 x}-\frac{3}{16} b^2 \text{Chi}(2 b x) \sinh (2 a)+\frac{9}{16} b^2 \text{Chi}(6 b x) \sinh (6 a)+\frac{3 \sinh (2 a+2 b x)}{64 x^2}-\frac{\sinh (6 a+6 b x)}{64 x^2}-\frac{3}{16} b^2 \cosh (2 a) \text{Shi}(2 b x)+\frac{9}{16} b^2 \cosh (6 a) \text{Shi}(6 b x)\\ \end{align*}
Mathematica [A] time = 0.247007, size = 118, normalized size = 0.9 \[ -\frac{12 b^2 x^2 \sinh (2 a) \text{Chi}(2 b x)-36 b^2 x^2 \sinh (6 a) \text{Chi}(6 b x)+12 b^2 x^2 \cosh (2 a) \text{Shi}(2 b x)-36 b^2 x^2 \cosh (6 a) \text{Shi}(6 b x)-3 \sinh (2 (a+b x))+\sinh (6 (a+b x))-6 b x \cosh (2 (a+b x))+6 b x \cosh (6 (a+b x))}{64 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.092, size = 178, normalized size = 1.4 \begin{align*} -{\frac{3\,b{{\rm e}^{-6\,bx-6\,a}}}{64\,x}}+{\frac{{{\rm e}^{-6\,bx-6\,a}}}{128\,{x}^{2}}}+{\frac{9\,{b}^{2}{{\rm e}^{-6\,a}}{\it Ei} \left ( 1,6\,bx \right ) }{32}}+{\frac{3\,b{{\rm e}^{-2\,bx-2\,a}}}{64\,x}}-{\frac{3\,{{\rm e}^{-2\,bx-2\,a}}}{128\,{x}^{2}}}-{\frac{3\,{b}^{2}{{\rm e}^{-2\,a}}{\it Ei} \left ( 1,2\,bx \right ) }{32}}-{\frac{{{\rm e}^{6\,bx+6\,a}}}{128\,{x}^{2}}}-{\frac{3\,b{{\rm e}^{6\,bx+6\,a}}}{64\,x}}-{\frac{9\,{b}^{2}{{\rm e}^{6\,a}}{\it Ei} \left ( 1,-6\,bx \right ) }{32}}+{\frac{3\,{{\rm e}^{2\,bx+2\,a}}}{128\,{x}^{2}}}+{\frac{3\,b{{\rm e}^{2\,bx+2\,a}}}{64\,x}}+{\frac{3\,{b}^{2}{{\rm e}^{2\,a}}{\it Ei} \left ( 1,-2\,bx \right ) }{32}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.26362, size = 82, normalized size = 0.63 \begin{align*} \frac{9}{16} \, b^{2} e^{\left (-6 \, a\right )} \Gamma \left (-2, 6 \, b x\right ) - \frac{3}{16} \, b^{2} e^{\left (-2 \, a\right )} \Gamma \left (-2, 2 \, b x\right ) + \frac{3}{16} \, b^{2} e^{\left (2 \, a\right )} \Gamma \left (-2, -2 \, b x\right ) - \frac{9}{16} \, b^{2} e^{\left (6 \, a\right )} \Gamma \left (-2, -6 \, b x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.89599, size = 699, normalized size = 5.34 \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} + 10 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3} + 3 \, \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} - 9 \,{\left (b^{2} x^{2}{\rm Ei}\left (6 \, b x\right ) - b^{2} x^{2}{\rm Ei}\left (-6 \, b x\right )\right )} \cosh \left (6 \, a\right ) + 3 \,{\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 ) + 3 \,{\left (\cosh \left (b x + a\right )^{5} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 9 \,{\left (b^{2} x^{2}{\rm Ei}\left (6 \, b x\right ) + b^{2} x^{2}{\rm Ei}\left (-6 \, b x\right )\right )} \sinh \left (6 \, a\right ) + 3 \,{\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 )}{32 \, 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 ^{3}{\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 = 1.16758, size = 227, normalized size = 1.73 \begin{align*} \frac{36 \, b^{2} x^{2}{\rm Ei}\left (6 \, b x\right ) e^{\left (6 \, a\right )} - 12 \, b^{2} x^{2}{\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} + 12 \, b^{2} x^{2}{\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} - 36 \, b^{2} x^{2}{\rm Ei}\left (-6 \, b x\right ) e^{\left (-6 \, a\right )} - 6 \, b x e^{\left (6 \, b x + 6 \, a\right )} + 6 \, b x e^{\left (2 \, b x + 2 \, a\right )} + 6 \, b x e^{\left (-2 \, b x - 2 \, a\right )} - 6 \, 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 )}}{128 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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