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 = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, 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 2554
Rule 2637
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*}
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Mathematica [A] time = 0.01, 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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fricas [B] time = 0.48, size = 258, normalized size = 5.16 \[ -\frac {\cosh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + 4 \, \cosh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{3} \sinh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) + 6 \, \cosh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} \sinh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} + 4 \, \cosh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) \sinh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{3} + \sinh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} - {\left (\cosh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + 4 \, \cosh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{3} \sinh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) + 6 \, \cosh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} \sinh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} + 4 \, \cosh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) \sinh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{3} + \sinh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} - 1\right )} \log \left (\cosh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) \sinh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )\right ) - 1}{2 \, {\left (b \cosh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} + 2 \, b \cosh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) \sinh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) + b \sinh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.47, size = 94, normalized size = 1.88 \[ \frac {1}{2} \, {\left (\frac {e^{\left (b x + a\right )}}{b} - \frac {e^{\left (-b x - a\right )}}{b}\right )} \log \left (\frac {1}{4} \, {\left (e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )} + e^{\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a\right )}\right )} {\left (e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )} - e^{\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a\right )}\right )}\right ) - \frac {e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.89, size = 32, normalized size = 0.64 \[ \frac {\ln \left (\frac {\sinh \left (b x +a \right )}{2}\right ) \sinh \left (b x +a \right )}{b}-\frac {\sinh \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 112, normalized size = 2.24 \[ \frac {\log \left (\cosh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) \sinh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )\right ) \sinh \left (b x + a\right )}{b} - \frac {b {\left (\frac {2 \, {\left (b x + a\right )}}{b} + \frac {e^{\left (b x + a\right )}}{b} - \frac {e^{\left (-b x - a\right )}}{b}\right )} - b {\left (\frac {2 \, {\left (b x + a\right )}}{b} - \frac {e^{\left (b x + a\right )}}{b} + \frac {e^{\left (-b x - a\right )}}{b}\right )}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.51, size = 31, normalized size = 0.62 \[ \frac {\ln \left (\frac {\mathrm {sinh}\left (a+b\,x\right )}{2}\right )\,\mathrm {sinh}\left (a+b\,x\right )}{b}-\frac {\mathrm {sinh}\left (a+b\,x\right )}{b} \]
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 \[ \int \log {\left (\sinh {\left (\frac {a}{2} + \frac {b x}{2} \right )} \cosh {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )} \cosh {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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