### 3.889 $$\int e^{a+b x} \text{sech}^2(c+d x) \, dx$$

Optimal. Leaf size=56 $\frac{4 e^{a+b x+2 (c+d x)} \, _2F_1\left (2,\frac{b}{2 d}+1;\frac{b}{2 d}+2;-e^{2 (c+d x)}\right )}{b+2 d}$

[Out]

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

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Rubi [A]  time = 0.026573, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.062, Rules used = {5492} $\frac{4 e^{a+b x+2 (c+d x)} \, _2F_1\left (2,\frac{b}{2 d}+1;\frac{b}{2 d}+2;-e^{2 (c+d x)}\right )}{b+2 d}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(a + b*x)*Sech[c + d*x]^2,x]

[Out]

(4*E^(a + b*x + 2*(c + d*x))*Hypergeometric2F1[2, 1 + b/(2*d), 2 + b/(2*d), -E^(2*(c + d*x))])/(b + 2*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}^2(c+d x) \, dx &=\frac{4 e^{a+b x+2 (c+d x)} \, _2F_1\left (2,1+\frac{b}{2 d};2+\frac{b}{2 d};-e^{2 (c+d x)}\right )}{b+2 d}\\ \end{align*}

Mathematica [A]  time = 0.0167047, size = 56, normalized size = 1. $\frac{4 e^{a+b x+2 (c+d x)} \, _2F_1\left (2,\frac{b}{2 d}+1;\frac{b}{2 d}+2;-e^{2 (c+d x)}\right )}{b+2 d}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^(a + b*x)*Sech[c + d*x]^2,x]

[Out]

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

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{bx+a}} \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(exp(b*x+a)*sech(d*x+c)^2,x)

[Out]

int(exp(b*x+a)*sech(d*x+c)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 4 \, b \int \frac{e^{\left (b x + a\right )}}{2 \,{\left (d e^{\left (2 \, d x + 2 \, c\right )} + d\right )}}\,{d x} - \frac{2 \, e^{\left (b x + a\right )}}{d e^{\left (2 \, d x + 2 \, c\right )} + d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)*sech(d*x+c)^2,x, algorithm="maxima")

[Out]

4*b*integrate(1/2*e^(b*x + a)/(d*e^(2*d*x + 2*c) + d), x) - 2*e^(b*x + a)/(d*e^(2*d*x + 2*c) + d)

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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 )^{2}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)*sech(d*x+c)^2,x, algorithm="fricas")

[Out]

integral(e^(b*x + a)*sech(d*x + c)^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} e^{a} \int e^{b x} \operatorname{sech}^{2}{\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)**2,x)

[Out]

exp(a)*Integral(exp(b*x)*sech(c + d*x)**2, 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 )^{2}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)*sech(d*x+c)^2,x, algorithm="giac")

[Out]

integrate(e^(b*x + a)*sech(d*x + c)^2, x)