Optimal. Leaf size=19 \[ -\frac {x}{2}+\frac {1}{2 (\coth (x)+1)}+\log (\cosh (x)) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3551, 3479, 8, 3475} \[ -\frac {x}{2}+\frac {1}{2 (\coth (x)+1)}+\log (\cosh (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3475
Rule 3479
Rule 3551
Rubi steps
\begin {align*} \int \frac {\tanh (x)}{1+\coth (x)} \, dx &=-\int \frac {1}{1+\coth (x)} \, dx+\int \tanh (x) \, dx\\ &=\frac {1}{2 (1+\coth (x))}+\log (\cosh (x))-\frac {\int 1 \, dx}{2}\\ &=-\frac {x}{2}+\frac {1}{2 (1+\coth (x))}+\log (\cosh (x))\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 23, normalized size = 1.21 \[ \frac {1}{4} (-2 x+\sinh (2 x)-\cosh (2 x)+4 \log (\cosh (x))) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.41, size = 73, normalized size = 3.84 \[ -\frac {6 \, x \cosh \relax (x)^{2} + 12 \, x \cosh \relax (x) \sinh \relax (x) + 6 \, x \sinh \relax (x)^{2} - 4 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}\right )} \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 1}{4 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.13, size = 17, normalized size = 0.89 \[ -\frac {3}{2} \, x - \frac {1}{4} \, e^{\left (-2 \, x\right )} + \log \left (e^{\left (2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.10, size = 47, normalized size = 2.47 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{\tanh \left (\frac {x}{2}\right )+1}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}+\ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 17, normalized size = 0.89 \[ \frac {1}{2} \, x - \frac {1}{4} \, e^{\left (-2 \, x\right )} + \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.17, size = 17, normalized size = 0.89 \[ \ln \left ({\mathrm {e}}^{2\,x}+1\right )-\frac {3\,x}{2}-\frac {{\mathrm {e}}^{-2\,x}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh {\relax (x )}}{\coth {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________