Optimal. Leaf size=9 \[ \log (\sinh (x))-\tanh ^{-1}(\cosh (x)) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0079258, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3475, 3770} \[ \log (\sinh (x))-\tanh ^{-1}(\cosh (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3475
Rule 3770
Rubi steps
\begin{align*} \int (\coth (x)+\text{csch}(x)) \, dx &=\int \coth (x) \, dx+\int \text{csch}(x) \, dx\\ &=-\tanh ^{-1}(\cosh (x))+\log (\sinh (x))\\ \end{align*}
Mathematica [A] time = 0.003432, size = 11, normalized size = 1.22 \[ \log (\sinh (x))+\log \left (\tanh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.001, size = 10, normalized size = 1.1 \begin{align*} \ln \left ( \sinh \left ( x \right ) \right ) +\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.04251, size = 12, normalized size = 1.33 \begin{align*} \log \left (\sinh \left (x\right )\right ) + \log \left (\tanh \left (\frac{1}{2} \, x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.05436, size = 47, normalized size = 5.22 \begin{align*} -x + 2 \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\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 \left (\coth{\left (x \right )} + \operatorname{csch}{\left (x \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.13858, size = 34, normalized size = 3.78 \begin{align*} -x - \log \left (e^{x} + 1\right ) + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]