Optimal. Leaf size=37 \[ -\frac {3 x}{2}-\tanh ^2(x)+\frac {3 \tanh (x)}{2}+2 \log (\cosh (x))+\frac {\tanh ^2(x)}{2 (\coth (x)+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3552, 3529, 3531, 3475} \[ -\frac {3 x}{2}-\tanh ^2(x)+\frac {3 \tanh (x)}{2}+2 \log (\cosh (x))+\frac {\tanh ^2(x)}{2 (\coth (x)+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3475
Rule 3529
Rule 3531
Rule 3552
Rubi steps
\begin {align*} \int \frac {\tanh ^3(x)}{1+\coth (x)} \, dx &=\frac {\tanh ^2(x)}{2 (1+\coth (x))}-\frac {1}{2} \int (-4+3 \coth (x)) \tanh ^3(x) \, dx\\ &=-\tanh ^2(x)+\frac {\tanh ^2(x)}{2 (1+\coth (x))}-\frac {1}{2} i \int (-3 i+4 i \coth (x)) \tanh ^2(x) \, dx\\ &=\frac {3 \tanh (x)}{2}-\tanh ^2(x)+\frac {\tanh ^2(x)}{2 (1+\coth (x))}+\frac {1}{2} \int (4-3 \coth (x)) \tanh (x) \, dx\\ &=-\frac {3 x}{2}+\frac {3 \tanh (x)}{2}-\tanh ^2(x)+\frac {\tanh ^2(x)}{2 (1+\coth (x))}+2 \int \tanh (x) \, dx\\ &=-\frac {3 x}{2}+2 \log (\cosh (x))+\frac {3 \tanh (x)}{2}-\tanh ^2(x)+\frac {\tanh ^2(x)}{2 (1+\coth (x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 33, normalized size = 0.89 \[ \frac {1}{4} \left (-6 x+\sinh (2 x)-\cosh (2 x)+4 \tanh (x)+2 \text {sech}^2(x)+8 \log (\cosh (x))\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.41, size = 354, normalized size = 9.57 \[ -\frac {14 \, x \cosh \relax (x)^{6} + 84 \, x \cosh \relax (x) \sinh \relax (x)^{5} + 14 \, x \sinh \relax (x)^{6} + {\left (28 \, x + 1\right )} \cosh \relax (x)^{4} + {\left (210 \, x \cosh \relax (x)^{2} + 28 \, x + 1\right )} \sinh \relax (x)^{4} + 4 \, {\left (70 \, x \cosh \relax (x)^{3} + {\left (28 \, x + 1\right )} \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 2 \, {\left (7 \, x + 5\right )} \cosh \relax (x)^{2} + 2 \, {\left (105 \, x \cosh \relax (x)^{4} + 3 \, {\left (28 \, x + 1\right )} \cosh \relax (x)^{2} + 7 \, x + 5\right )} \sinh \relax (x)^{2} - 8 \, {\left (\cosh \relax (x)^{6} + 6 \, \cosh \relax (x) \sinh \relax (x)^{5} + \sinh \relax (x)^{6} + {\left (15 \, \cosh \relax (x)^{2} + 2\right )} \sinh \relax (x)^{4} + 2 \, \cosh \relax (x)^{4} + 4 \, {\left (5 \, \cosh \relax (x)^{3} + 2 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + {\left (15 \, \cosh \relax (x)^{4} + 12 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + \cosh \relax (x)^{2} + 2 \, {\left (3 \, \cosh \relax (x)^{5} + 4 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x)\right )} \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 4 \, {\left (21 \, x \cosh \relax (x)^{5} + {\left (28 \, x + 1\right )} \cosh \relax (x)^{3} + {\left (7 \, x + 5\right )} \cosh \relax (x)\right )} \sinh \relax (x) + 1}{4 \, {\left (\cosh \relax (x)^{6} + 6 \, \cosh \relax (x) \sinh \relax (x)^{5} + \sinh \relax (x)^{6} + {\left (15 \, \cosh \relax (x)^{2} + 2\right )} \sinh \relax (x)^{4} + 2 \, \cosh \relax (x)^{4} + 4 \, {\left (5 \, \cosh \relax (x)^{3} + 2 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + {\left (15 \, \cosh \relax (x)^{4} + 12 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + \cosh \relax (x)^{2} + 2 \, {\left (3 \, \cosh \relax (x)^{5} + 4 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.13, size = 39, normalized size = 1.05 \[ -\frac {7}{2} \, x - \frac {{\left (e^{\left (4 \, x\right )} + 10 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )}}{4 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{2}} + 2 \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.11, size = 80, normalized size = 2.16 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{\tanh \left (\frac {x}{2}\right )+1}-\frac {7 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}+\frac {2 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )-2 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \tanh \left (\frac {x}{2}\right )}{\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+2 \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.41, size = 43, normalized size = 1.16 \[ \frac {1}{2} \, x + \frac {2 \, {\left (2 \, e^{\left (-2 \, x\right )} + 1\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} - \frac {1}{4} \, e^{\left (-2 \, x\right )} + 2 \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.21, size = 35, normalized size = 0.95 \[ 2\,\ln \left ({\mathrm {e}}^{2\,x}+1\right )-\frac {7\,x}{2}-\frac {{\mathrm {e}}^{-2\,x}}{4}-\frac {2}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh ^{3}{\relax (x )}}{\coth {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________