Optimal. Leaf size=27 \[ \text{Shi}(2 x)-\frac{\sinh (2 x)}{4 x^2}-\frac{\cosh (2 x)}{2 x} \]
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Rubi [A] time = 0.0629465, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5448, 12, 3297, 3298} \[ \text{Shi}(2 x)-\frac{\sinh (2 x)}{4 x^2}-\frac{\cosh (2 x)}{2 x} \]
Antiderivative was successfully verified.
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Rule 5448
Rule 12
Rule 3297
Rule 3298
Rubi steps
\begin{align*} \int \frac{\cosh (x) \sinh (x)}{x^3} \, dx &=\int \frac{\sinh (2 x)}{2 x^3} \, dx\\ &=\frac{1}{2} \int \frac{\sinh (2 x)}{x^3} \, dx\\ &=-\frac{\sinh (2 x)}{4 x^2}+\frac{1}{2} \int \frac{\cosh (2 x)}{x^2} \, dx\\ &=-\frac{\cosh (2 x)}{2 x}-\frac{\sinh (2 x)}{4 x^2}+\int \frac{\sinh (2 x)}{x} \, dx\\ &=-\frac{\cosh (2 x)}{2 x}-\frac{\sinh (2 x)}{4 x^2}+\text{Shi}(2 x)\\ \end{align*}
Mathematica [A] time = 0.007349, size = 27, normalized size = 1. \[ \text{Shi}(2 x)-\frac{\sinh (2 x)}{4 x^2}-\frac{\cosh (2 x)}{2 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 24, normalized size = 0.9 \begin{align*} -{\frac{\cosh \left ( 2\,x \right ) }{2\,x}}+{\it Shi} \left ( 2\,x \right ) -{\frac{\sinh \left ( 2\,x \right ) }{4\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20613, size = 18, normalized size = 0.67 \begin{align*} \Gamma \left (-2, 2 \, x\right ) - \Gamma \left (-2, -2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00945, size = 113, normalized size = 4.19 \begin{align*} \frac{x^{2}{\rm Ei}\left (2 \, x\right ) - x^{2}{\rm Ei}\left (-2 \, x\right ) - x \cosh \left (x\right )^{2} - x \sinh \left (x\right )^{2} - \cosh \left (x\right ) \sinh \left (x\right )}{2 \, 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 (x \right )} \cosh{\left (x \right )}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18049, size = 65, normalized size = 2.41 \begin{align*} \frac{4 \, x^{2}{\rm Ei}\left (2 \, x\right ) - 4 \, x^{2}{\rm Ei}\left (-2 \, x\right ) - 2 \, x e^{\left (2 \, x\right )} - 2 \, x e^{\left (-2 \, x\right )} - e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}}{8 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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