Integrand size = 20, antiderivative size = 77 \[ \int \frac {A+C \sinh (x)}{a+b \cosh (x)-b \sinh (x)} \, dx=\frac {(2 a A-b C) x}{2 a^2}+\frac {C \cosh (x)}{2 a}+\frac {\left (2 a A b+a^2 C-b^2 C\right ) \log (a+b \cosh (x)-b \sinh (x))}{2 a^2 b}+\frac {C \sinh (x)}{2 a} \]
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Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3210} \[ \int \frac {A+C \sinh (x)}{a+b \cosh (x)-b \sinh (x)} \, dx=\frac {\left (a^2 C+2 a A b-b^2 C\right ) \log (a-b \sinh (x)+b \cosh (x))}{2 a^2 b}+\frac {x (2 a A-b C)}{2 a^2}+\frac {C \sinh (x)}{2 a}+\frac {C \cosh (x)}{2 a} \]
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Rule 3210
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.37 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.12 \[ \int \frac {A+C \sinh (x)}{a+b \cosh (x)-b \sinh (x)} \, dx=\frac {\left (2 a A-\frac {a^2 C}{b}-b C\right ) x+2 a C \cosh (x)+\frac {2 \left (2 a A b+a^2 C-b^2 C\right ) \log \left ((a+b) \cosh \left (\frac {x}{2}\right )+(a-b) \sinh \left (\frac {x}{2}\right )\right )}{b}+2 a C \sinh (x)}{4 a^2} \]
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Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.81
method | result | size |
risch | \(\frac {C \,{\mathrm e}^{x}}{2 a}-\frac {C x}{2 b}+\frac {\ln \left ({\mathrm e}^{x}+\frac {b}{a}\right ) A}{a}+\frac {\ln \left ({\mathrm e}^{x}+\frac {b}{a}\right ) C}{2 b}-\frac {b \ln \left ({\mathrm e}^{x}+\frac {b}{a}\right ) C}{2 a^{2}}\) | \(62\) |
default | \(\frac {\left (2 A a b +C \,a^{2}-C \,b^{2}\right ) \ln \left (a \tanh \left (\frac {x}{2}\right )-b \tanh \left (\frac {x}{2}\right )+a +b \right )}{2 a^{2} b}-\frac {C}{a \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {\left (-2 A a +b C \right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 a^{2}}-\frac {C \ln \left (1+\tanh \left (\frac {x}{2}\right )\right )}{2 b}\) | \(91\) |
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Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.77 \[ \int \frac {A+C \sinh (x)}{a+b \cosh (x)-b \sinh (x)} \, dx=-\frac {C a^{2} x - C a b \cosh \left (x\right ) - C a b \sinh \left (x\right ) - {\left (C a^{2} + 2 \, A a b - C b^{2}\right )} \log \left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b\right )}{2 \, a^{2} b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 852 vs. \(2 (66) = 132\).
Time = 2.28 (sec) , antiderivative size = 852, normalized size of antiderivative = 11.06 \[ \int \frac {A+C \sinh (x)}{a+b \cosh (x)-b \sinh (x)} \, dx=\text {Too large to display} \]
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Time = 0.20 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.84 \[ \int \frac {A+C \sinh (x)}{a+b \cosh (x)-b \sinh (x)} \, dx=A {\left (\frac {x}{a} + \frac {\log \left (b e^{\left (-x\right )} + a\right )}{a}\right )} - \frac {1}{2} \, C {\left (\frac {b x}{a^{2}} - \frac {e^{x}}{a} - \frac {{\left (a^{2} - b^{2}\right )} \log \left (b e^{\left (-x\right )} + a\right )}{a^{2} b}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.64 \[ \int \frac {A+C \sinh (x)}{a+b \cosh (x)-b \sinh (x)} \, dx=-\frac {C x}{2 \, b} + \frac {C e^{x}}{2 \, a} + \frac {{\left (C a^{2} + 2 \, A a b - C b^{2}\right )} \log \left ({\left | a e^{x} + b \right |}\right )}{2 \, a^{2} b} \]
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Time = 0.13 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.62 \[ \int \frac {A+C \sinh (x)}{a+b \cosh (x)-b \sinh (x)} \, dx=\frac {C\,{\mathrm {e}}^x}{2\,a}-\frac {C\,x}{2\,b}+\frac {\ln \left (b+a\,{\mathrm {e}}^x\right )\,\left (C\,a^2+2\,A\,a\,b-C\,b^2\right )}{2\,a^2\,b} \]
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