Optimal. Leaf size=71 \[ \frac {x (2 a A+b C)}{2 a^2}-\frac {1}{2} \left (\frac {b C}{a^2}+\frac {2 A}{a}-\frac {C}{b}\right ) \log (a+b \sinh (x)+b \cosh (x))-\frac {C \sinh (x)}{2 a}+\frac {C \cosh (x)}{2 a} \]
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Rubi [A] time = 0.06, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {3131} \[ \frac {x (2 a A+b C)}{2 a^2}-\frac {1}{2} \left (\frac {b C}{a^2}+\frac {2 A}{a}-\frac {C}{b}\right ) \log (a+b \sinh (x)+b \cosh (x))-\frac {C \sinh (x)}{2 a}+\frac {C \cosh (x)}{2 a} \]
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 {1}{2} \left (\frac {2 A}{a}-\frac {C}{b}+\frac {b C}{a^2}\right ) \log (a+b \cosh (x)+b \sinh (x))-\frac {C \sinh (x)}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 86, normalized size = 1.21 \[ \frac {x \left (a^2 C+2 a A b+b^2 C\right )+2 \left (a^2 C-2 a A b-b^2 C\right ) \log \left ((b-a) \sinh \left (\frac {x}{2}\right )+(a+b) \cosh \left (\frac {x}{2}\right )\right )-2 a b C \sinh (x)+2 a b C \cosh (x)}{4 a^2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 107, normalized size = 1.51 \[ \frac {C a b + {\left (2 \, A a b + C b^{2}\right )} x \cosh \relax (x) + {\left (2 \, A a b + C b^{2}\right )} x \sinh \relax (x) + {\left ({\left (C a^{2} - 2 \, A a b - C b^{2}\right )} \cosh \relax (x) + {\left (C a^{2} - 2 \, A a b - C b^{2}\right )} \sinh \relax (x)\right )} \log \left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}{2 \, {\left (a^{2} b \cosh \relax (x) + a^{2} b \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 58, normalized size = 0.82 \[ \frac {C e^{\left (-x\right )}}{2 \, a} + \frac {{\left (2 \, A a + C b\right )} x}{2 \, a^{2}} + \frac {{\left (C a^{2} - 2 \, A a b - C b^{2}\right )} \log \left ({\left | b e^{x} + a \right |}\right )}{2 \, a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 136, normalized size = 1.92 \[ -\frac {C \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 b}-\frac {\ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b -a -b \right ) A}{a}+\frac {\ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b -a -b \right ) C}{2 b}-\frac {b \ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b -a -b \right ) C}{2 a^{2}}+\frac {C}{a \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) A}{a}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) b C}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 58, normalized size = 0.82 \[ \frac {1}{2} \, C {\left (\frac {x}{b} + \frac {e^{\left (-x\right )}}{a} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (a e^{\left (-x\right )} + b\right )}{a^{2} b}\right )} - \frac {A \log \left (a e^{\left (-x\right )} + b\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.66, size = 57, normalized size = 0.80 \[ \frac {C\,{\mathrm {e}}^{-x}}{2\,a}+\frac {x\,\left (2\,A\,a+C\,b\right )}{2\,a^2}-\frac {\ln \left (a+b\,{\mathrm {e}}^x\right )\,\left (-C\,a^2+2\,A\,a\,b+C\,b^2\right )}{2\,a^2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.11, size = 753, normalized size = 10.61 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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