Optimal. Leaf size=19 \[ \frac{2 \sinh (x)}{3}-\frac{\cosh (x)}{3 (\tanh (x)+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0300766, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3502, 2637} \[ \frac{2 \sinh (x)}{3}-\frac{\cosh (x)}{3 (\tanh (x)+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3502
Rule 2637
Rubi steps
\begin{align*} \int \frac{\cosh (x)}{1+\tanh (x)} \, dx &=-\frac{\cosh (x)}{3 (1+\tanh (x))}+\frac{2}{3} \int \cosh (x) \, dx\\ &=\frac{2 \sinh (x)}{3}-\frac{\cosh (x)}{3 (1+\tanh (x))}\\ \end{align*}
Mathematica [A] time = 0.0268828, size = 23, normalized size = 1.21 \[ \frac{1}{12} (9 \sinh (x)+\sinh (3 x)-3 \cosh (x)-\cosh (3 x)) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.027, size = 40, normalized size = 2.1 \begin{align*} -{\frac{2}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}+ \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}-{\frac{3}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.00966, size = 23, normalized size = 1.21 \begin{align*} -\frac{1}{2} \, e^{\left (-x\right )} - \frac{1}{12} \, e^{\left (-3 \, x\right )} + \frac{1}{4} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.21826, size = 99, normalized size = 5.21 \begin{align*} \frac{\cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 3}{6 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.435763, size = 48, normalized size = 2.53 \begin{align*} \frac{2 \sinh{\left (x \right )} \tanh{\left (x \right )}}{3 \tanh{\left (x \right )} + 3} + \frac{\sinh{\left (x \right )}}{3 \tanh{\left (x \right )} + 3} + \frac{\cosh{\left (x \right )} \tanh{\left (x \right )}}{3 \tanh{\left (x \right )} + 3} - \frac{\cosh{\left (x \right )}}{3 \tanh{\left (x \right )} + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.25906, size = 26, normalized size = 1.37 \begin{align*} -\frac{1}{12} \,{\left (6 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-3 \, x\right )} + \frac{1}{4} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]