Optimal. Leaf size=51 \[ -\frac {b x}{a^2-b^2}+\frac {b^2 \log (a \cosh (x)+b \sinh (x))}{a \left (a^2-b^2\right )}+\frac {\log (\sinh (x))}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3571, 3530, 3475} \[ -\frac {b x}{a^2-b^2}+\frac {b^2 \log (a \cosh (x)+b \sinh (x))}{a \left (a^2-b^2\right )}+\frac {\log (\sinh (x))}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3475
Rule 3530
Rule 3571
Rubi steps
\begin {align*} \int \frac {\coth (x)}{a+b \tanh (x)} \, dx &=-\frac {b x}{a^2-b^2}+\frac {\int \coth (x) \, dx}{a}+\frac {\left (i b^2\right ) \int \frac {-i b-i a \tanh (x)}{a+b \tanh (x)} \, dx}{a \left (a^2-b^2\right )}\\ &=-\frac {b x}{a^2-b^2}+\frac {\log (\sinh (x))}{a}+\frac {b^2 \log (a \cosh (x)+b \sinh (x))}{a \left (a^2-b^2\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 46, normalized size = 0.90 \[ \frac {\left (a^2-b^2\right ) \log (\sinh (x))+b (b \log (a \cosh (x)+b \sinh (x))-a x)}{a^3-a b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.54, size = 73, normalized size = 1.43 \[ \frac {b^{2} \log \left (\frac {2 \, {\left (a \cosh \relax (x) + b \sinh \relax (x)\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) - {\left (a^{2} + a b\right )} x + {\left (a^{2} - b^{2}\right )} \log \left (\frac {2 \, \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right )}{a^{3} - a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.11, size = 58, normalized size = 1.14 \[ \frac {b^{2} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{3} - a b^{2}} - \frac {x}{a - b} + \frac {\log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.13, size = 78, normalized size = 1.53 \[ \frac {b^{2} \ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \tanh \left (\frac {x}{2}\right ) b +a \right )}{\left (a +b \right ) \left (a -b \right ) a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a +b}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a -b}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 65, normalized size = 1.27 \[ \frac {b^{2} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{3} - a b^{2}} + \frac {x}{a + b} + \frac {\log \left (e^{\left (-x\right )} + 1\right )}{a} + \frac {\log \left (e^{\left (-x\right )} - 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.41, size = 58, normalized size = 1.14 \[ \frac {\ln \left ({\mathrm {e}}^{2\,x}-1\right )}{a}-\frac {x}{a-b}-\frac {b^2\,\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}{a\,b^2-a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth {\relax (x )}}{a + b \tanh {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________