Optimal. Leaf size=77 \[ \frac {\left (a^2 C+2 a A b-b^2 C\right ) \log (a-b \sinh (x)+b \cosh (x))}{2 a^2 b}+\frac {x (2 a A-b C)}{2 a^2}+\frac {C \sinh (x)}{2 a}+\frac {C \cosh (x)}{2 a} \]
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Rubi [A] time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3131} \[ \frac {\left (a^2 C+2 a A b-b^2 C\right ) \log (a-b \sinh (x)+b \cosh (x))}{2 a^2 b}+\frac {x (2 a A-b C)}{2 a^2}+\frac {C \sinh (x)}{2 a}+\frac {C \cosh (x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 3131
Rubi steps
\begin {align*} \int \frac {A+C \sinh (x)}{a+b \cosh (x)-b \sinh (x)} \, dx &=\frac {(2 a A-b C) x}{2 a^2}+\frac {C \cosh (x)}{2 a}+\frac {\left (2 a A b+a^2 C-b^2 C\right ) \log (a+b \cosh (x)-b \sinh (x))}{2 a^2 b}+\frac {C \sinh (x)}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 86, normalized size = 1.12 \[ \frac {\frac {2 \left (a^2 C+2 a A b-b^2 C\right ) \log \left ((a-b) \sinh \left (\frac {x}{2}\right )+(a+b) \cosh \left (\frac {x}{2}\right )\right )}{b}+x \left (-\frac {a^2 C}{b}+2 a A-b C\right )+2 a C \sinh (x)+2 a C \cosh (x)}{4 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 59, normalized size = 0.77 \[ -\frac {C a^{2} x - C a b \cosh \relax (x) - C a b \sinh \relax (x) - {\left (C a^{2} + 2 \, A a b - C b^{2}\right )} \log \left (a \cosh \relax (x) + a \sinh \relax (x) + b\right )}{2 \, a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 49, normalized size = 0.64 \[ -\frac {C x}{2 \, b} + \frac {C e^{x}}{2 \, a} + \frac {{\left (C a^{2} + 2 \, A a b - C b^{2}\right )} \log \left ({\left | a e^{x} + b \right |}\right )}{2 \, a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 125, normalized size = 1.62 \[ -\frac {C}{a \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right ) A}{a}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right ) b C}{2 a^{2}}-\frac {C \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 b}+\frac {\ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b +a +b \right ) A}{a}+\frac {\ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b +a +b \right ) C}{2 b}-\frac {b \ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b +a +b \right ) C}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 65, normalized size = 0.84 \[ A {\left (\frac {x}{a} + \frac {\log \left (b e^{\left (-x\right )} + a\right )}{a}\right )} - \frac {1}{2} \, C {\left (\frac {b x}{a^{2}} - \frac {e^{x}}{a} - \frac {{\left (a^{2} - b^{2}\right )} \log \left (b e^{\left (-x\right )} + a\right )}{a^{2} b}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 48, normalized size = 0.62 \[ \frac {C\,{\mathrm {e}}^x}{2\,a}-\frac {C\,x}{2\,b}+\frac {\ln \left (b+a\,{\mathrm {e}}^x\right )\,\left (C\,a^2+2\,A\,a\,b-C\,b^2\right )}{2\,a^2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.26, size = 852, normalized size = 11.06 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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