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 = 0.52, antiderivative size = 89, normalized size of antiderivative = 1.00, 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 12
Rule 2554
Rule 2633
Rule 3298
Rule 3301
Rule 3303
Rule 3312
Rule 6742
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*}
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Mathematica [A] time = 0.11, 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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fricas [B] time = 0.48, size = 587, normalized size = 6.60 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 102, normalized size = 1.15 \[ \frac {1}{24} \, {\left (\frac {e^{\left (3 \, b x + 3 \, a\right )}}{b} - \frac {9 \, e^{\left (b x + a\right )}}{b} - \frac {9 \, e^{\left (-b x - a\right )}}{b} + \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{b}\right )} \log \relax (x) - \frac {{\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} - 9 \, {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + {\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} - 9 \, {\rm Ei}\left (b x\right ) e^{a}}{24 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.04, size = 116, normalized size = 1.30 \[ -\frac {3 \Ei \left (1, b x \right ) {\mathrm e}^{-a}}{8 b}+\frac {\Ei \left (1, -3 b x \right ) {\mathrm e}^{3 a}}{24 b}-\frac {3 \Ei \left (1, -b x \right ) {\mathrm e}^{a}}{8 b}+\frac {\Ei \left (1, 3 b x \right ) {\mathrm e}^{-3 a}}{24 b}+\left (\frac {{\mathrm e}^{-3 b x -3 a}}{24 b}-\frac {3 \,{\mathrm e}^{-b x -a}}{8 b}-\frac {3 \,{\mathrm e}^{b x +a}}{8 b}+\frac {{\mathrm e}^{3 b x +3 a}}{24 b}\right ) \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.77, size = 110, normalized size = 1.24 \[ \frac {1}{24} \, {\left (\frac {e^{\left (3 \, b x + 3 \, a\right )}}{b} - \frac {9 \, e^{\left (b x + a\right )}}{b} - \frac {9 \, e^{\left (-b x - a\right )}}{b} + \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{b}\right )} \log \relax (x) - \frac {{\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )}}{24 \, b} + \frac {3 \, {\rm Ei}\left (-b x\right ) e^{\left (-a\right )}}{8 \, b} - \frac {{\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )}}{24 \, b} + \frac {3 \, {\rm Ei}\left (b x\right ) e^{a}}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {sinh}\left (a+b\,x\right )}^3\,\ln \relax (x) \,d x \]
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 \[ \int \log {\relax (x )} \sinh ^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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