Optimal. Leaf size=29 \[ \frac{3 x}{2}-\frac{3 \coth (x)}{2}-\log (\sinh (x))+\frac{\coth (x)}{2 (\tanh (x)+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0676766, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {3552, 3529, 3531, 3475} \[ \frac{3 x}{2}-\frac{3 \coth (x)}{2}-\log (\sinh (x))+\frac{\coth (x)}{2 (\tanh (x)+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3552
Rule 3529
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \frac{\coth ^2(x)}{1+\tanh (x)} \, dx &=\frac{\coth (x)}{2 (1+\tanh (x))}-\frac{1}{2} \int \coth ^2(x) (-3+2 \tanh (x)) \, dx\\ &=-\frac{3 \coth (x)}{2}+\frac{\coth (x)}{2 (1+\tanh (x))}-\frac{1}{2} i \int \coth (x) (-2 i+3 i \tanh (x)) \, dx\\ &=\frac{3 x}{2}-\frac{3 \coth (x)}{2}+\frac{\coth (x)}{2 (1+\tanh (x))}-\int \coth (x) \, dx\\ &=\frac{3 x}{2}-\frac{3 \coth (x)}{2}-\log (\sinh (x))+\frac{\coth (x)}{2 (1+\tanh (x))}\\ \end{align*}
Mathematica [A] time = 0.0433458, size = 27, normalized size = 0.93 \[ \frac{1}{4} (6 x+\sinh (2 x)-\cosh (2 x)-4 \coth (x)-4 \log (\sinh (x))) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.029, size = 59, normalized size = 2. \begin{align*} -{\frac{1}{2}\tanh \left ({\frac{x}{2}} \right ) }- \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}+ \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}+{\frac{5}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) -{\frac{1}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.21798, size = 51, normalized size = 1.76 \begin{align*} \frac{1}{2} \, x + \frac{2}{e^{\left (-2 \, x\right )} - 1} - \frac{1}{4} \, e^{\left (-2 \, x\right )} - \log \left (e^{\left (-x\right )} + 1\right ) - \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.25379, size = 632, normalized size = 21.79 \begin{align*} \frac{10 \, x \cosh \left (x\right )^{4} + 40 \, x \cosh \left (x\right ) \sinh \left (x\right )^{3} + 10 \, x \sinh \left (x\right )^{4} -{\left (10 \, x + 9\right )} \cosh \left (x\right )^{2} +{\left (60 \, x \cosh \left (x\right )^{2} - 10 \, x - 9\right )} \sinh \left (x\right )^{2} - 4 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} +{\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \,{\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \,{\left (20 \, x \cosh \left (x\right )^{3} -{\left (10 \, x + 9\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}{4 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} +{\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \,{\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth ^{2}{\left (x \right )}}{\tanh{\left (x \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.20825, size = 49, normalized size = 1.69 \begin{align*} \frac{5}{2} \, x - \frac{{\left (9 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )}}{4 \,{\left (e^{\left (2 \, x\right )} - 1\right )}} - \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]