Optimal. Leaf size=5 \[ \tan ^{-1}\left (\sinh \left (e^x\right )\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.009588, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2282, 3770} \[ \tan ^{-1}\left (\sinh \left (e^x\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 3770
Rubi steps
\begin{align*} \int e^x \text{sech}\left (e^x\right ) \, dx &=\operatorname{Subst}\left (\int \text{sech}(x) \, dx,x,e^x\right )\\ &=\tan ^{-1}\left (\sinh \left (e^x\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0049734, size = 5, normalized size = 1. \[ \tan ^{-1}\left (\sinh \left (e^x\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 5, normalized size = 1. \begin{align*} \arctan \left ( \sinh \left ({{\rm e}^{x}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.953006, size = 5, normalized size = 1. \begin{align*} \arctan \left (\sinh \left (e^{x}\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.92324, size = 82, normalized size = 16.4 \begin{align*} 2 \, \arctan \left (\cosh \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + \sinh \left (\cosh \left (x\right ) + \sinh \left (x\right )\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 e^{x} \operatorname{sech}{\left (e^{x} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.06723, size = 8, normalized size = 1.6 \begin{align*} 2 \, \arctan \left (e^{\left (e^{x}\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]