Optimal. Leaf size=50 \[ \frac{\sinh (a+b x) \log \left (\sinh \left (\frac{a}{2}+\frac{b x}{2}\right ) \cosh \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b}-\frac{\sinh (a+b x)}{b} \]
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Rubi [A] time = 0.0286319, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {2637, 2554} \[ \frac{\sinh (a+b x) \log \left (\sinh \left (\frac{a}{2}+\frac{b x}{2}\right ) \cosh \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b}-\frac{\sinh (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2554
Rubi steps
\begin{align*} \int \cosh (a+b x) \log \left (\cosh \left (\frac{a}{2}+\frac{b x}{2}\right ) \sinh \left (\frac{a}{2}+\frac{b x}{2}\right )\right ) \, dx &=\frac{\log \left (\cosh \left (\frac{a}{2}+\frac{b x}{2}\right ) \sinh \left (\frac{a}{2}+\frac{b x}{2}\right )\right ) \sinh (a+b x)}{b}-\int \cosh (a+b x) \, dx\\ &=-\frac{\sinh (a+b x)}{b}+\frac{\log \left (\cosh \left (\frac{a}{2}+\frac{b x}{2}\right ) \sinh \left (\frac{a}{2}+\frac{b x}{2}\right )\right ) \sinh (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0105279, size = 33, normalized size = 0.66 \[ \frac{\sinh (a+b x) \log \left (\frac{1}{2} \sinh (a+b x)\right )}{b}-\frac{\sinh (a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 32, normalized size = 0.6 \begin{align*}{\frac{\sinh \left ( bx+a \right ) }{b}\ln \left ({\frac{\sinh \left ( bx+a \right ) }{2}} \right ) }-{\frac{\sinh \left ( bx+a \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0706, size = 151, normalized size = 3.02 \begin{align*} \frac{\log \left (\cosh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right ) \sinh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )\right ) \sinh \left (b x + a\right )}{b} - \frac{b{\left (\frac{2 \,{\left (b x + a\right )}}{b} + \frac{e^{\left (b x + a\right )}}{b} - \frac{e^{\left (-b x - a\right )}}{b}\right )} - b{\left (\frac{2 \,{\left (b x + a\right )}}{b} - \frac{e^{\left (b x + a\right )}}{b} + \frac{e^{\left (-b x - a\right )}}{b}\right )}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.91609, size = 782, normalized size = 15.64 \begin{align*} -\frac{\cosh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{4} + 4 \, \cosh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{3} \sinh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right ) + 6 \, \cosh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{2} \sinh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{2} + 4 \, \cosh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right ) \sinh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{3} + \sinh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{4} -{\left (\cosh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{4} + 4 \, \cosh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{3} \sinh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right ) + 6 \, \cosh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{2} \sinh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{2} + 4 \, \cosh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right ) \sinh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{3} + \sinh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{4} - 1\right )} \log \left (\cosh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right ) \sinh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )\right ) - 1}{2 \,{\left (b \cosh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{2} + 2 \, b \cosh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right ) \sinh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right ) + b \sinh \left (\frac{1}{2} \, b x + \frac{1}{2} \, 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 (\sinh{\left (\frac{a}{2} + \frac{b x}{2} \right )} \cosh{\left (\frac{a}{2} + \frac{b x}{2} \right )} \right )} \cosh{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.34414, size = 127, normalized size = 2.54 \begin{align*} \frac{1}{2} \,{\left (\frac{e^{\left (b x + a\right )}}{b} - \frac{e^{\left (-b x - a\right )}}{b}\right )} \log \left (\frac{1}{4} \,{\left (e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )} + e^{\left (-\frac{1}{2} \, b x - \frac{1}{2} \, a\right )}\right )}{\left (e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )} - e^{\left (-\frac{1}{2} \, b x - \frac{1}{2} \, a\right )}\right )}\right ) - \frac{e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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