Optimal. Leaf size=17 \[ \frac{\log \left (e^{2 a+2 b x}+1\right )}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0173843, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2282, 12, 260} \[ \frac{\log \left (e^{2 a+2 b x}+1\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 12
Rule 260
Rubi steps
\begin{align*} \int e^{a+b x} \text{sech}(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{2 x}{1+x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\log \left (1+e^{2 a+2 b x}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0124252, size = 17, normalized size = 1. \[ \frac{\log \left (e^{2 a+2 b x}+1\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 19, normalized size = 1.1 \begin{align*} x+{\frac{\ln \left ( \cosh \left ( bx+a \right ) \right ) }{b}}+{\frac{a}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.59275, size = 22, normalized size = 1.29 \begin{align*} \frac{\log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.78251, size = 76, normalized size = 4.47 \begin{align*} \frac{\log \left (\frac{2 \, \cosh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b} \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*} e^{a} \int e^{b x} \operatorname{sech}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.16051, size = 22, normalized size = 1.29 \begin{align*} \frac{\log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]