Optimal. Leaf size=60 \[ b^2 \sinh (2 a) \text{Chi}(2 b x)+b^2 \cosh (2 a) \text{Shi}(2 b x)-\frac{\sinh (2 a+2 b x)}{4 x^2}-\frac{b \cosh (2 a+2 b x)}{2 x} \]
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Rubi [A] time = 0.123143, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ b^2 \sinh (2 a) \text{Chi}(2 b x)+b^2 \cosh (2 a) \text{Shi}(2 b x)-\frac{\sinh (2 a+2 b x)}{4 x^2}-\frac{b \cosh (2 a+2 b x)}{2 x} \]
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^3} \, dx &=\int \frac{\sinh (2 a+2 b x)}{2 x^3} \, dx\\ &=\frac{1}{2} \int \frac{\sinh (2 a+2 b x)}{x^3} \, dx\\ &=-\frac{\sinh (2 a+2 b x)}{4 x^2}+\frac{1}{2} b \int \frac{\cosh (2 a+2 b x)}{x^2} \, dx\\ &=-\frac{b \cosh (2 a+2 b x)}{2 x}-\frac{\sinh (2 a+2 b x)}{4 x^2}+b^2 \int \frac{\sinh (2 a+2 b x)}{x} \, dx\\ &=-\frac{b \cosh (2 a+2 b x)}{2 x}-\frac{\sinh (2 a+2 b x)}{4 x^2}+\left (b^2 \cosh (2 a)\right ) \int \frac{\sinh (2 b x)}{x} \, dx+\left (b^2 \sinh (2 a)\right ) \int \frac{\cosh (2 b x)}{x} \, dx\\ &=-\frac{b \cosh (2 a+2 b x)}{2 x}+b^2 \text{Chi}(2 b x) \sinh (2 a)-\frac{\sinh (2 a+2 b x)}{4 x^2}+b^2 \cosh (2 a) \text{Shi}(2 b x)\\ \end{align*}
Mathematica [A] time = 0.174611, size = 61, normalized size = 1.02 \[ \frac{1}{2} \left (2 b^2 \sinh (2 a) \text{Chi}(2 b x)+2 b^2 \cosh (2 a) \text{Shi}(2 b x)-\frac{\sinh (2 (a+b x))+2 b x \cosh (2 (a+b x))}{2 x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 90, normalized size = 1.5 \begin{align*} -{\frac{b{{\rm e}^{-2\,bx-2\,a}}}{4\,x}}+{\frac{{{\rm e}^{-2\,bx-2\,a}}}{8\,{x}^{2}}}+{\frac{{b}^{2}{{\rm e}^{-2\,a}}{\it Ei} \left ( 1,2\,bx \right ) }{2}}-{\frac{{{\rm e}^{2\,bx+2\,a}}}{8\,{x}^{2}}}-{\frac{b{{\rm e}^{2\,bx+2\,a}}}{4\,x}}-{\frac{{b}^{2}{{\rm e}^{2\,a}}{\it Ei} \left ( 1,-2\,bx \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.23321, size = 41, normalized size = 0.68 \begin{align*} b^{2} e^{\left (-2 \, a\right )} \Gamma \left (-2, 2 \, b x\right ) - b^{2} e^{\left (2 \, a\right )} \Gamma \left (-2, -2 \, b x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94455, size = 257, normalized size = 4.28 \begin{align*} -\frac{b x \cosh \left (b x + a\right )^{2} + b x \sinh \left (b x + a\right )^{2} -{\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 ) + \cosh \left (b x + a\right ) \sinh \left (b x + 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 )}{2 \, x^{2}} \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 = 1.13612, size = 116, normalized size = 1.93 \begin{align*} \frac{4 \, b^{2} x^{2}{\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - 4 \, b^{2} x^{2}{\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} - 2 \, b x e^{\left (2 \, b x + 2 \, a\right )} - 2 \, 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 )}}{8 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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