Optimal. Leaf size=89 \[ \frac{3 \cosh (a) \text{Chi}(b x)}{4 b}-\frac{\cosh (3 a) \text{Chi}(3 b x)}{12 b}+\frac{3 \sinh (a) \text{Shi}(b x)}{4 b}-\frac{\sinh (3 a) \text{Shi}(3 b x)}{12 b}+\frac{\log (x) \cosh ^3(a+b x)}{3 b}-\frac{\log (x) \cosh (a+b x)}{b} \]
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Rubi [A] time = 0.517013, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727, Rules used = {2633, 2554, 12, 6742, 3303, 3298, 3301, 3312} \[ \frac{3 \cosh (a) \text{Chi}(b x)}{4 b}-\frac{\cosh (3 a) \text{Chi}(3 b x)}{12 b}+\frac{3 \sinh (a) \text{Shi}(b x)}{4 b}-\frac{\sinh (3 a) \text{Shi}(3 b x)}{12 b}+\frac{\log (x) \cosh ^3(a+b x)}{3 b}-\frac{\log (x) \cosh (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2633
Rule 2554
Rule 12
Rule 6742
Rule 3303
Rule 3298
Rule 3301
Rule 3312
Rubi steps
\begin{align*} \int \log (x) \sinh ^3(a+b x) \, dx &=-\frac{\cosh (a+b x) \log (x)}{b}+\frac{\cosh ^3(a+b x) \log (x)}{3 b}-\int \frac{\cosh (a+b x) \left (-3+\cosh ^2(a+b x)\right )}{3 b x} \, dx\\ &=-\frac{\cosh (a+b x) \log (x)}{b}+\frac{\cosh ^3(a+b x) \log (x)}{3 b}-\frac{\int \frac{\cosh (a+b x) \left (-3+\cosh ^2(a+b x)\right )}{x} \, dx}{3 b}\\ &=-\frac{\cosh (a+b x) \log (x)}{b}+\frac{\cosh ^3(a+b x) \log (x)}{3 b}-\frac{\int \left (-\frac{3 \cosh (a+b x)}{x}+\frac{\cosh ^3(a+b x)}{x}\right ) \, dx}{3 b}\\ &=-\frac{\cosh (a+b x) \log (x)}{b}+\frac{\cosh ^3(a+b x) \log (x)}{3 b}-\frac{\int \frac{\cosh ^3(a+b x)}{x} \, dx}{3 b}+\frac{\int \frac{\cosh (a+b x)}{x} \, dx}{b}\\ &=-\frac{\cosh (a+b x) \log (x)}{b}+\frac{\cosh ^3(a+b x) \log (x)}{3 b}-\frac{\int \left (\frac{3 \cosh (a+b x)}{4 x}+\frac{\cosh (3 a+3 b x)}{4 x}\right ) \, dx}{3 b}+\frac{\cosh (a) \int \frac{\cosh (b x)}{x} \, dx}{b}+\frac{\sinh (a) \int \frac{\sinh (b x)}{x} \, dx}{b}\\ &=\frac{\cosh (a) \text{Chi}(b x)}{b}-\frac{\cosh (a+b x) \log (x)}{b}+\frac{\cosh ^3(a+b x) \log (x)}{3 b}+\frac{\sinh (a) \text{Shi}(b x)}{b}-\frac{\int \frac{\cosh (3 a+3 b x)}{x} \, dx}{12 b}-\frac{\int \frac{\cosh (a+b x)}{x} \, dx}{4 b}\\ &=\frac{\cosh (a) \text{Chi}(b x)}{b}-\frac{\cosh (a+b x) \log (x)}{b}+\frac{\cosh ^3(a+b x) \log (x)}{3 b}+\frac{\sinh (a) \text{Shi}(b x)}{b}-\frac{\cosh (a) \int \frac{\cosh (b x)}{x} \, dx}{4 b}-\frac{\cosh (3 a) \int \frac{\cosh (3 b x)}{x} \, dx}{12 b}-\frac{\sinh (a) \int \frac{\sinh (b x)}{x} \, dx}{4 b}-\frac{\sinh (3 a) \int \frac{\sinh (3 b x)}{x} \, dx}{12 b}\\ &=\frac{3 \cosh (a) \text{Chi}(b x)}{4 b}-\frac{\cosh (3 a) \text{Chi}(3 b x)}{12 b}-\frac{\cosh (a+b x) \log (x)}{b}+\frac{\cosh ^3(a+b x) \log (x)}{3 b}+\frac{3 \sinh (a) \text{Shi}(b x)}{4 b}-\frac{\sinh (3 a) \text{Shi}(3 b x)}{12 b}\\ \end{align*}
Mathematica [A] time = 0.100467, size = 67, normalized size = 0.75 \[ \frac{9 \cosh (a) \text{Chi}(b x)-\cosh (3 a) \text{Chi}(3 b x)+9 \sinh (a) \text{Shi}(b x)-\sinh (3 a) \text{Shi}(3 b x)-9 \log (x) \cosh (a+b x)+\log (x) \cosh (3 (a+b x))}{12 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 116, normalized size = 1.3 \begin{align*} \left ({\frac{{{\rm e}^{3\,bx+3\,a}}}{24\,b}}-{\frac{3\,{{\rm e}^{bx+a}}}{8\,b}}-{\frac{3\,{{\rm e}^{-bx-a}}}{8\,b}}+{\frac{{{\rm e}^{-3\,bx-3\,a}}}{24\,b}} \right ) \ln \left ( x \right ) +{\frac{{{\rm e}^{-3\,a}}{\it Ei} \left ( 1,3\,bx \right ) }{24\,b}}+{\frac{{{\rm e}^{3\,a}}{\it Ei} \left ( 1,-3\,bx \right ) }{24\,b}}-{\frac{3\,{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) }{8\,b}}-{\frac{3\,{{\rm e}^{a}}{\it Ei} \left ( 1,-bx \right ) }{8\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.24853, size = 149, normalized size = 1.67 \begin{align*} \frac{1}{24} \,{\left (\frac{e^{\left (3 \, b x + 3 \, a\right )}}{b} - \frac{9 \, e^{\left (b x + a\right )}}{b} - \frac{9 \, e^{\left (-b x - a\right )}}{b} + \frac{e^{\left (-3 \, b x - 3 \, a\right )}}{b}\right )} \log \left (x\right ) - \frac{{\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )}}{24 \, b} + \frac{3 \,{\rm Ei}\left (-b x\right ) e^{\left (-a\right )}}{8 \, b} - \frac{{\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )}}{24 \, b} + \frac{3 \,{\rm Ei}\left (b x\right ) e^{a}}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.94033, size = 1669, normalized size = 18.75 \begin{align*} \text{result too large to display} \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 )} \sinh ^{3}{\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 = 1.33102, size = 130, normalized size = 1.46 \begin{align*} -\frac{{\left ({\left (9 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-3 \, b x - 3 \, a\right )} - e^{\left (3 \, b x + 3 \, a\right )} + 9 \, e^{\left (b x + a\right )}\right )} \log \left (x\right )}{24 \, b} - \frac{{\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} - 9 \,{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} +{\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} - 9 \,{\rm Ei}\left (b x\right ) e^{a}}{24 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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