Optimal. Leaf size=85 \[ \frac{2}{3} b^3 \cosh (2 a) \text{Chi}(2 b x)+\frac{2}{3} b^3 \sinh (2 a) \text{Shi}(2 b x)-\frac{b^2 \sinh (2 a+2 b x)}{3 x}-\frac{\sinh (2 a+2 b x)}{6 x^3}-\frac{b \cosh (2 a+2 b x)}{6 x^2} \]
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Rubi [A] time = 0.150576, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5448, 12, 3297, 3303, 3298, 3301} \[ \frac{2}{3} b^3 \cosh (2 a) \text{Chi}(2 b x)+\frac{2}{3} b^3 \sinh (2 a) \text{Shi}(2 b x)-\frac{b^2 \sinh (2 a+2 b x)}{3 x}-\frac{\sinh (2 a+2 b x)}{6 x^3}-\frac{b \cosh (2 a+2 b x)}{6 x^2} \]
Antiderivative was successfully verified.
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Rule 5448
Rule 12
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\cosh (a+b x) \sinh (a+b x)}{x^4} \, dx &=\int \frac{\sinh (2 a+2 b x)}{2 x^4} \, dx\\ &=\frac{1}{2} \int \frac{\sinh (2 a+2 b x)}{x^4} \, dx\\ &=-\frac{\sinh (2 a+2 b x)}{6 x^3}+\frac{1}{3} b \int \frac{\cosh (2 a+2 b x)}{x^3} \, dx\\ &=-\frac{b \cosh (2 a+2 b x)}{6 x^2}-\frac{\sinh (2 a+2 b x)}{6 x^3}+\frac{1}{3} b^2 \int \frac{\sinh (2 a+2 b x)}{x^2} \, dx\\ &=-\frac{b \cosh (2 a+2 b x)}{6 x^2}-\frac{\sinh (2 a+2 b x)}{6 x^3}-\frac{b^2 \sinh (2 a+2 b x)}{3 x}+\frac{1}{3} \left (2 b^3\right ) \int \frac{\cosh (2 a+2 b x)}{x} \, dx\\ &=-\frac{b \cosh (2 a+2 b x)}{6 x^2}-\frac{\sinh (2 a+2 b x)}{6 x^3}-\frac{b^2 \sinh (2 a+2 b x)}{3 x}+\frac{1}{3} \left (2 b^3 \cosh (2 a)\right ) \int \frac{\cosh (2 b x)}{x} \, dx+\frac{1}{3} \left (2 b^3 \sinh (2 a)\right ) \int \frac{\sinh (2 b x)}{x} \, dx\\ &=-\frac{b \cosh (2 a+2 b x)}{6 x^2}+\frac{2}{3} b^3 \cosh (2 a) \text{Chi}(2 b x)-\frac{\sinh (2 a+2 b x)}{6 x^3}-\frac{b^2 \sinh (2 a+2 b x)}{3 x}+\frac{2}{3} b^3 \sinh (2 a) \text{Shi}(2 b x)\\ \end{align*}
Mathematica [A] time = 0.155667, size = 77, normalized size = 0.91 \[ -\frac{-4 b^3 x^3 \cosh (2 a) \text{Chi}(2 b x)-4 b^3 x^3 \sinh (2 a) \text{Shi}(2 b x)+2 b^2 x^2 \sinh (2 (a+b x))+\sinh (2 (a+b x))+b x \cosh (2 (a+b x))}{6 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 124, normalized size = 1.5 \begin{align*}{\frac{{b}^{2}{{\rm e}^{-2\,bx-2\,a}}}{6\,x}}-{\frac{b{{\rm e}^{-2\,bx-2\,a}}}{12\,{x}^{2}}}+{\frac{{{\rm e}^{-2\,bx-2\,a}}}{12\,{x}^{3}}}-{\frac{{b}^{3}{{\rm e}^{-2\,a}}{\it Ei} \left ( 1,2\,bx \right ) }{3}}-{\frac{{{\rm e}^{2\,bx+2\,a}}}{12\,{x}^{3}}}-{\frac{b{{\rm e}^{2\,bx+2\,a}}}{12\,{x}^{2}}}-{\frac{{b}^{2}{{\rm e}^{2\,bx+2\,a}}}{6\,x}}-{\frac{{b}^{3}{{\rm e}^{2\,a}}{\it Ei} \left ( 1,-2\,bx \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.39122, size = 42, normalized size = 0.49 \begin{align*} 2 \, b^{3} e^{\left (-2 \, a\right )} \Gamma \left (-3, 2 \, b x\right ) + 2 \, b^{3} e^{\left (2 \, a\right )} \Gamma \left (-3, -2 \, b x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96662, size = 286, normalized size = 3.36 \begin{align*} -\frac{b x \cosh \left (b x + a\right )^{2} + b x \sinh \left (b x + a\right )^{2} + 2 \,{\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - 2 \,{\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 ) - 2 \,{\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 )}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11924, size = 162, normalized size = 1.91 \begin{align*} \frac{4 \, b^{3} x^{3}{\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} + 4 \, b^{3} x^{3}{\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} - 2 \, b^{2} x^{2} e^{\left (2 \, b x + 2 \, a\right )} + 2 \, b^{2} x^{2} e^{\left (-2 \, b x - 2 \, a\right )} - b x e^{\left (2 \, b x + 2 \, a\right )} - b x e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )}}{12 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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