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 = 0.07, antiderivative size = 35, normalized size of antiderivative = 1.00, 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 12
Rule 2554
Rule 2637
Rule 3298
Rule 3301
Rule 3303
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*}
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Mathematica [A] time = 0.04, 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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fricas [B] time = 0.45, size = 134, normalized size = 3.83 \[ -\frac {{\left ({\rm Ei}\left (b x\right ) - {\rm Ei}\left (-b x\right )\right )} \cosh \left (b x + a\right ) \cosh \relax (a) - \log \relax (x) \sinh \left (b x + a\right )^{2} + {\left ({\rm Ei}\left (b x\right ) + {\rm Ei}\left (-b x\right )\right )} \cosh \left (b x + a\right ) \sinh \relax (a) - {\left (\cosh \left (b x + a\right )^{2} - 1\right )} \log \relax (x) + {\left ({\left ({\rm Ei}\left (b x\right ) - {\rm Ei}\left (-b x\right )\right )} \cosh \relax (a) - 2 \, \cosh \left (b x + a\right ) \log \relax (x) + {\left ({\rm Ei}\left (b x\right ) + {\rm Ei}\left (-b x\right )\right )} \sinh \relax (a)\right )} \sinh \left (b x + a\right )}{2 \, {\left (b \cosh \left (b x + a\right ) + b \sinh \left (b x + a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 54, normalized size = 1.54 \[ \frac {1}{2} \, {\left (\frac {e^{\left (b x + a\right )}}{b} - \frac {e^{\left (-b x - a\right )}}{b}\right )} \log \relax (x) + \frac {{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - {\rm Ei}\left (b x\right ) e^{a}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.94, size = 58, normalized size = 1.66 \[ -\frac {\Ei \left (1, b x \right ) {\mathrm e}^{-a}}{2 b}+\frac {\Ei \left (1, -b x \right ) {\mathrm e}^{a}}{2 b}+\left (-\frac {{\mathrm e}^{-b x -a}}{2 b}+\frac {{\mathrm e}^{b x +a}}{2 b}\right ) \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 37, normalized size = 1.06 \[ \frac {\log \relax (x) \sinh \left (b x + a\right )}{b} + \frac {{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - {\rm Ei}\left (b x\right ) e^{a}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \mathrm {cosh}\left (a+b\,x\right )\,\ln \relax (x) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \log {\relax (x )} \cosh {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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