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 = 0.142901, antiderivative size = 66, normalized size of antiderivative = 1., 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 2635
Rule 8
Rule 2554
Rule 12
Rule 5274
Rule 3303
Rule 3298
Rule 3301
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*}
Mathematica [A] time = 0.09926, 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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Maple [A] time = 0.036, size = 97, normalized size = 1.5 \begin{align*} \left ( -{\frac{x}{2}}+{\frac{{{\rm e}^{2\,bx+2\,a}}}{8\,b}}-{\frac{{{\rm e}^{-2\,bx-2\,a}}}{8\,b}} \right ) \ln \left ( x \right ) +{\frac{{{\rm e}^{2\,a}}{\it Ei} \left ( 1,-2\,bx \right ) }{8\,b}}-{\frac{a\ln \left ( bx \right ) }{2\,b}}+{\frac{a\ln \left ( -bx \right ) }{2\,b}}+{\frac{x}{2}}+{\frac{a}{2\,b}}-{\frac{{{\rm e}^{-2\,a}}{\it Ei} \left ( 1,2\,bx \right ) }{8\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15357, size = 90, normalized size = 1.36 \begin{align*} -\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 \left (x\right ) + \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.85988, size = 859, normalized size = 13.02 \begin{align*} \frac{4 \, \cosh \left (b x + a\right ) \log \left (x\right ) \sinh \left (b x + a\right )^{3} + \log \left (x\right ) \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 \left (x\right ) -{\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 \left (x\right ) - 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 \left (x\right )\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 )}} \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 ^{2}{\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 = 1.37262, size = 123, normalized size = 1.86 \begin{align*} -\frac{{\left (4 \, b x -{\left (2 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-2 \, b x - 2 \, a\right )} + 4 \, a - e^{\left (2 \, b x + 2 \, a\right )}\right )} \log \left (x\right )}{8 \, b} + \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 )} + 4 \, a \log \left (x\right ) - 2 \, \log \left (x\right )}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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