### 3.888 $$\int e^{a+b x} \text{sech}(c+d x) \, dx$$

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]

(2*E^(a + c + b*x + d*x)*Hypergeometric2F1[1, (b + d)/(2*d), (3 + b/d)/2, -E^(2*(c + d*x))])/(b + d)

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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]

Int[E^(a + b*x)*Sech[c + d*x],x]

[Out]

(2*E^(a + c + b*x + d*x)*Hypergeometric2F1[1, (b + d)/(2*d), (3 + b/d)/2, -E^(2*(c + d*x))])/(b + d)

Rule 5492

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sech[(d_.) + (e_.)*(x_)]^(n_.), x_Symbol] :> Simp[(2^n*E^(n*(d + e*x))*F
^(c*(a + b*x))*Hypergeometric2F1[n, n/2 + (b*c*Log[F])/(2*e), 1 + n/2 + (b*c*Log[F])/(2*e), -E^(2*(d + e*x))])
/(e*n + b*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]

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]

Integrate[E^(a + b*x)*Sech[c + d*x],x]

[Out]

(2*E^(a + c + (b + d)*x)*Hypergeometric2F1[1, (b + d)/(2*d), (3 + b/d)/2, -E^(2*(c + d*x))])/(b + d)

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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]

int(exp(b*x+a)*sech(d*x+c),x)

[Out]

int(exp(b*x+a)*sech(d*x+c),x)

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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]

integrate(exp(b*x+a)*sech(d*x+c),x, algorithm="maxima")

[Out]

integrate(e^(b*x + a)*sech(d*x + c), x)

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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]

integrate(exp(b*x+a)*sech(d*x+c),x, algorithm="fricas")

[Out]

integral(e^(b*x + a)*sech(d*x + c), x)

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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]

integrate(exp(b*x+a)*sech(d*x+c),x)

[Out]

exp(a)*Integral(exp(b*x)*sech(c + d*x), x)

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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]

integrate(exp(b*x+a)*sech(d*x+c),x, algorithm="giac")

[Out]

integrate(e^(b*x + a)*sech(d*x + c), x)