Optimal. Leaf size=35 \[ -\frac{\sinh (a) \text{Chi}(b x)}{b}-\frac{\cosh (a) \text{Shi}(b x)}{b}+\frac{\log (x) \sinh (a+b x)}{b} \]
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Rubi [A] time = 0.070019, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {2637, 2554, 12, 3303, 3298, 3301} \[ -\frac{\sinh (a) \text{Chi}(b x)}{b}-\frac{\cosh (a) \text{Shi}(b x)}{b}+\frac{\log (x) \sinh (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2554
Rule 12
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \cosh (a+b x) \log (x) \, dx &=\frac{\log (x) \sinh (a+b x)}{b}-\int \frac{\sinh (a+b x)}{b x} \, dx\\ &=\frac{\log (x) \sinh (a+b x)}{b}-\frac{\int \frac{\sinh (a+b x)}{x} \, dx}{b}\\ &=\frac{\log (x) \sinh (a+b x)}{b}-\frac{\cosh (a) \int \frac{\sinh (b x)}{x} \, dx}{b}-\frac{\sinh (a) \int \frac{\cosh (b x)}{x} \, dx}{b}\\ &=-\frac{\text{Chi}(b x) \sinh (a)}{b}+\frac{\log (x) \sinh (a+b x)}{b}-\frac{\cosh (a) \text{Shi}(b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0379113, size = 30, normalized size = 0.86 \[ -\frac{\sinh (a) \text{Chi}(b x)+\cosh (a) \text{Shi}(b x)-\log (x) \sinh (a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 58, normalized size = 1.7 \begin{align*} \left ({\frac{{{\rm e}^{bx+a}}}{2\,b}}-{\frac{{{\rm e}^{-bx-a}}}{2\,b}} \right ) \ln \left ( x \right ) +{\frac{{{\rm e}^{a}}{\it Ei} \left ( 1,-bx \right ) }{2\,b}}-{\frac{{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.189, size = 50, normalized size = 1.43 \begin{align*} \frac{\log \left (x\right ) \sinh \left (b x + a\right )}{b} + \frac{{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} -{\rm Ei}\left (b x\right ) e^{a}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.78763, size = 393, normalized size = 11.23 \begin{align*} -\frac{{\left ({\rm Ei}\left (b x\right ) -{\rm Ei}\left (-b x\right )\right )} \cosh \left (b x + a\right ) \cosh \left (a\right ) - \log \left (x\right ) \sinh \left (b x + a\right )^{2} +{\left ({\rm Ei}\left (b x\right ) +{\rm Ei}\left (-b x\right )\right )} \cosh \left (b x + a\right ) \sinh \left (a\right ) -{\left (\cosh \left (b x + a\right )^{2} - 1\right )} \log \left (x\right ) +{\left ({\left ({\rm Ei}\left (b x\right ) -{\rm Ei}\left (-b x\right )\right )} \cosh \left (a\right ) - 2 \, \cosh \left (b x + a\right ) \log \left (x\right ) +{\left ({\rm Ei}\left (b x\right ) +{\rm Ei}\left (-b x\right )\right )} \sinh \left (a\right )\right )} \sinh \left (b x + a\right )}{2 \,{\left (b \cosh \left (b x + a\right ) + b \sinh \left (b x + a\right )\right )}} \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 \log{\left (x \right )} \cosh{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26758, size = 73, normalized size = 2.09 \begin{align*} \frac{1}{2} \,{\left (\frac{e^{\left (b x + a\right )}}{b} - \frac{e^{\left (-b x - a\right )}}{b}\right )} \log \left (x\right ) + \frac{{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} -{\rm Ei}\left (b x\right ) e^{a}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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