Optimal. Leaf size=66 \[ -\frac {\sinh (2 a) \text {Chi}(2 b x)}{4 b}-\frac {\cosh (2 a) \text {Shi}(2 b x)}{4 b}+\frac {\log (x) \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x}{2}-\frac {1}{2} x \log (x) \]
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Rubi [A] time = 0.14, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {2635, 8, 2554, 12, 5274, 3303, 3298, 3301} \[ -\frac {\sinh (2 a) \text {Chi}(2 b x)}{4 b}-\frac {\cosh (2 a) \text {Shi}(2 b x)}{4 b}+\frac {\log (x) \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x}{2}-\frac {1}{2} x \log (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 2554
Rule 2635
Rule 3298
Rule 3301
Rule 3303
Rule 5274
Rubi steps
\begin {align*} \int \log (x) \sinh ^2(a+b x) \, dx &=-\frac {1}{2} x \log (x)+\frac {\cosh (a+b x) \log (x) \sinh (a+b x)}{2 b}-\int \frac {1}{4} \left (-2+\frac {\sinh (2 (a+b x))}{b x}\right ) \, dx\\ &=-\frac {1}{2} x \log (x)+\frac {\cosh (a+b x) \log (x) \sinh (a+b x)}{2 b}-\frac {1}{4} \int \left (-2+\frac {\sinh (2 (a+b x))}{b x}\right ) \, dx\\ &=\frac {x}{2}-\frac {1}{2} x \log (x)+\frac {\cosh (a+b x) \log (x) \sinh (a+b x)}{2 b}-\frac {\int \frac {\sinh (2 (a+b x))}{x} \, dx}{4 b}\\ &=\frac {x}{2}-\frac {1}{2} x \log (x)+\frac {\cosh (a+b x) \log (x) \sinh (a+b x)}{2 b}-\frac {\int \frac {\sinh (2 a+2 b x)}{x} \, dx}{4 b}\\ &=\frac {x}{2}-\frac {1}{2} x \log (x)+\frac {\cosh (a+b x) \log (x) \sinh (a+b x)}{2 b}-\frac {\cosh (2 a) \int \frac {\sinh (2 b x)}{x} \, dx}{4 b}-\frac {\sinh (2 a) \int \frac {\cosh (2 b x)}{x} \, dx}{4 b}\\ &=\frac {x}{2}-\frac {1}{2} x \log (x)-\frac {\text {Chi}(2 b x) \sinh (2 a)}{4 b}+\frac {\cosh (a+b x) \log (x) \sinh (a+b x)}{2 b}-\frac {\cosh (2 a) \text {Shi}(2 b x)}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 50, normalized size = 0.76 \[ -\frac {\sinh (2 a) \text {Chi}(2 b x)+\cosh (2 a) \text {Shi}(2 b x)-\log (x) \sinh (2 (a+b x))-2 b x+2 b x \log (x)}{4 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 313, normalized size = 4.74 \[ \frac {4 \, \cosh \left (b x + a\right ) \log \relax (x) \sinh \left (b x + a\right )^{3} + \log \relax (x) \sinh \left (b x + a\right )^{4} - {\left ({\rm Ei}\left (2 \, b x\right ) + {\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (b x + a\right )^{2} \sinh \left (2 \, a\right ) + {\left (4 \, b x - {\left ({\rm Ei}\left (2 \, b x\right ) - {\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (2 \, a\right )\right )} \cosh \left (b x + a\right )^{2} + {\left (4 \, b x - {\left ({\rm Ei}\left (2 \, b x\right ) - {\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (2 \, a\right ) - 2 \, {\left (2 \, b x - 3 \, \cosh \left (b x + a\right )^{2}\right )} \log \relax (x) - {\left ({\rm Ei}\left (2 \, b x\right ) + {\rm Ei}\left (-2 \, b x\right )\right )} \sinh \left (2 \, a\right )\right )} \sinh \left (b x + a\right )^{2} - {\left (4 \, b x \cosh \left (b x + a\right )^{2} - \cosh \left (b x + a\right )^{4} + 1\right )} \log \relax (x) - 2 \, {\left ({\left ({\rm Ei}\left (2 \, b x\right ) + {\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (b x + a\right ) \sinh \left (2 \, a\right ) - {\left (4 \, b x - {\left ({\rm Ei}\left (2 \, b x\right ) - {\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (2 \, a\right )\right )} \cosh \left (b x + a\right ) + 2 \, {\left (2 \, b x \cosh \left (b x + a\right ) - \cosh \left (b x + a\right )^{3}\right )} \log \relax (x)\right )} \sinh \left (b x + a\right )}{8 \, {\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 67, normalized size = 1.02 \[ -\frac {1}{8} \, {\left (4 \, x - \frac {e^{\left (2 \, b x + 2 \, a\right )}}{b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{b}\right )} \log \relax (x) + \frac {4 \, b x - {\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} + {\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )}}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.30, size = 97, normalized size = 1.47 \[ -\frac {a \ln \left (b x \right )}{2 b}+\frac {a \ln \left (-b x \right )}{2 b}+\frac {\Ei \left (1, -2 b x \right ) {\mathrm e}^{2 a}}{8 b}-\frac {\Ei \left (1, 2 b x \right ) {\mathrm e}^{-2 a}}{8 b}+\frac {x}{2}+\left (-\frac {x}{2}-\frac {{\mathrm e}^{-2 b x -2 a}}{8 b}+\frac {{\mathrm e}^{2 b x +2 a}}{8 b}\right ) \ln \relax (x )+\frac {a}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 67, normalized size = 1.02 \[ -\frac {1}{8} \, {\left (4 \, x - \frac {e^{\left (2 \, b x + 2 \, a\right )}}{b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{b}\right )} \log \relax (x) + \frac {1}{2} \, x - \frac {{\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )}}{8 \, b} + \frac {{\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )}}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {sinh}\left (a+b\,x\right )}^2\,\ln \relax (x) \,d x \]
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 \[ \int \log {\relax (x )} \sinh ^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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