Optimal. Leaf size=81 \[ \frac {\left (-\left (a^2 (B-C)\right )+2 a A b-b^2 (B+C)\right ) \log (a-b \sinh (x)+b \cosh (x))}{2 a^2 b}+\frac {x (2 a A-b (B+C))}{2 a^2}+\frac {(B+C) (\sinh (x)+\cosh (x))}{2 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {3130} \[ \frac {\left (a^2 (-(B-C))+2 a A b-b^2 (B+C)\right ) \log (a-b \sinh (x)+b \cosh (x))}{2 a^2 b}+\frac {x (2 a A-b (B+C))}{2 a^2}+\frac {(B+C) (\sinh (x)+\cosh (x))}{2 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3130
Rubi steps
\begin {align*} \int \frac {A+B \cosh (x)+C \sinh (x)}{a+b \cosh (x)-b \sinh (x)} \, dx &=\frac {(2 a A-b (B+C)) x}{2 a^2}+\frac {\left (2 a A b-a^2 (B-C)-b^2 (B+C)\right ) \log (a+b \cosh (x)-b \sinh (x))}{2 a^2 b}+\frac {(B+C) (\cosh (x)+\sinh (x))}{2 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.30, size = 102, normalized size = 1.26 \[ \frac {x \left (a^2 (B-C)+2 a A b-b^2 (B+C)\right )-2 \left (a^2 (B-C)-2 a A b+b^2 (B+C)\right ) \log \left ((a-b) \sinh \left (\frac {x}{2}\right )+(a+b) \cosh \left (\frac {x}{2}\right )\right )+2 a b (B+C) \sinh (x)+2 a b (B+C) \cosh (x)}{4 a^2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 70, normalized size = 0.86 \[ \frac {{\left (B - C\right )} a^{2} x + {\left (B + C\right )} a b \cosh \relax (x) + {\left (B + C\right )} a b \sinh \relax (x) - {\left ({\left (B - C\right )} a^{2} - 2 \, A a b + {\left (B + C\right )} b^{2}\right )} \log \left (a \cosh \relax (x) + a \sinh \relax (x) + b\right )}{2 \, a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.13, size = 69, normalized size = 0.85 \[ \frac {{\left (B - C\right )} x}{2 \, b} + \frac {B e^{x} + C e^{x}}{2 \, a} - \frac {{\left (B a^{2} - C a^{2} - 2 \, A a b + B b^{2} + C b^{2}\right )} \log \left ({\left | a e^{x} + b \right |}\right )}{2 \, a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.21, size = 213, normalized size = 2.63 \[ -\frac {B}{a \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {C}{a \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right ) A}{a}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right ) B b}{2 a^{2}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right ) b C}{2 a^{2}}+\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 b}-\frac {C \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 b}+\frac {\ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b +a +b \right ) A}{a}-\frac {\ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b +a +b \right ) B}{2 b}-\frac {b \ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b +a +b \right ) B}{2 a^{2}}+\frac {\ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b +a +b \right ) C}{2 b}-\frac {b \ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b +a +b \right ) C}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 105, normalized size = 1.30 \[ A {\left (\frac {x}{a} + \frac {\log \left (b e^{\left (-x\right )} + a\right )}{a}\right )} - \frac {1}{2} \, B {\left (\frac {b x}{a^{2}} - \frac {e^{x}}{a} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (b e^{\left (-x\right )} + a\right )}{a^{2} b}\right )} - \frac {1}{2} \, C {\left (\frac {b x}{a^{2}} - \frac {e^{x}}{a} - \frac {{\left (a^{2} - b^{2}\right )} \log \left (b e^{\left (-x\right )} + a\right )}{a^{2} b}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.63, size = 64, normalized size = 0.79 \[ \frac {x\,\left (B-C\right )}{2\,b}+\frac {{\mathrm {e}}^x\,\left (B+C\right )}{2\,a}-\frac {\ln \left (b+a\,{\mathrm {e}}^x\right )\,\left (B\,a^2+B\,b^2-C\,a^2+C\,b^2-2\,A\,a\,b\right )}{2\,a^2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 5.97, size = 1420, normalized size = 17.53 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________