Optimal. Leaf size=52 \[ \frac{2 e^{a+b x+c+d x} \, _2F_1\left (1,\frac{b+d}{2 d};\frac{1}{2} \left (\frac{b}{d}+3\right );-e^{2 (c+d x)}\right )}{b+d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0178902, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {5492} \[ \frac{2 e^{a+b x+c+d x} \, _2F_1\left (1,\frac{b+d}{2 d};\frac{1}{2} \left (\frac{b}{d}+3\right );-e^{2 (c+d x)}\right )}{b+d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5492
Rubi steps
\begin{align*} \int e^{a+b x} \text{sech}(c+d x) \, dx &=\frac{2 e^{a+c+b x+d x} \, _2F_1\left (1,\frac{b+d}{2 d};\frac{1}{2} \left (3+\frac{b}{d}\right );-e^{2 (c+d x)}\right )}{b+d}\\ \end{align*}
Mathematica [A] time = 0.0182925, size = 51, normalized size = 0.98 \[ \frac{2 e^{a+x (b+d)+c} \, _2F_1\left (1,\frac{b+d}{2 d};\frac{1}{2} \left (\frac{b}{d}+3\right );-e^{2 (c+d x)}\right )}{b+d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{bx+a}}{\rm sech} \left (dx+c\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left (b x + a\right )} \operatorname{sech}\left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (e^{\left (b x + a\right )} \operatorname{sech}\left (d x + c\right ), 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*} e^{a} \int e^{b x} \operatorname{sech}{\left (c + d 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 e^{\left (b x + a\right )} \operatorname{sech}\left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]