Optimal. Leaf size=14 \[ 2 \coth (x) \sqrt{-\sinh (x) \tanh (x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0568221, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4397, 4400, 2589} \[ 2 \coth (x) \sqrt{-\sinh (x) \tanh (x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4397
Rule 4400
Rule 2589
Rubi steps
\begin{align*} \int \sqrt{-\cosh (x)+\text{sech}(x)} \, dx &=\int \sqrt{-\sinh (x) \tanh (x)} \, dx\\ &=\frac{\sqrt{-\sinh (x) \tanh (x)} \int \sqrt{i \sinh (x)} \sqrt{i \tanh (x)} \, dx}{\sqrt{i \sinh (x)} \sqrt{i \tanh (x)}}\\ &=2 \coth (x) \sqrt{-\sinh (x) \tanh (x)}\\ \end{align*}
Mathematica [A] time = 0.0521067, size = 14, normalized size = 1. \[ 2 \coth (x) \sqrt{-\sinh (x) \tanh (x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.185, size = 43, normalized size = 3.1 \begin{align*}{\frac{\sqrt{2} \left ({{\rm e}^{2\,x}}+1 \right ) }{{{\rm e}^{2\,x}}-1}\sqrt{-{\frac{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}{{\rm e}^{-x}}}{{{\rm e}^{2\,x}}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.78975, size = 53, normalized size = 3.79 \begin{align*} -\frac{\sqrt{2} e^{\left (\frac{1}{2} \, x\right )}}{\sqrt{-e^{\left (-2 \, x\right )} - 1}} - \frac{\sqrt{2} e^{\left (-\frac{3}{2} \, x\right )}}{\sqrt{-e^{\left (-2 \, x\right )} - 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.82606, size = 208, normalized size = 14.86 \begin{align*} 2 \, \sqrt{\frac{1}{2}}{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \sqrt{-\frac{1}{\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} +{\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right ) + \cosh \left (x\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 \sqrt{- \cosh{\left (x \right )} + \operatorname{sech}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-\cosh \left (x\right ) + \operatorname{sech}\left (x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]