\(\int e^{a+b x} \text {sech}^2(c+d x) \, dx\) [889]

Optimal result

Integrand size = 16, antiderivative size = 56 \[ \int e^{a+b x} \text {sech}^2(c+d x) \, dx=\frac {4 e^{a+b x+2 (c+d x)} \operatorname {Hypergeometric2F1}\left (2,1+\frac {b}{2 d},2+\frac {b}{2 d},-e^{2 (c+d x)}\right )}{b+2 d} \]

[Out]

4*exp(b*x+2*d*x+a+2*c)*hypergeom([2, 1+1/2*b/d],[2+1/2*b/d],-exp(2*d*x+2*c))/(b+2*d)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {5600} \[ \int e^{a+b x} \text {sech}^2(c+d x) \, dx=\frac {4 e^{a+b x+2 (c+d x)} \operatorname {Hypergeometric2F1}\left (2,\frac {b}{2 d}+1,\frac {b}{2 d}+2,-e^{2 (c+d x)}\right )}{b+2 d} \]

[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 5600

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))/(e*n + b*c*Log[F]))*Hypergeometric2F1[n, n/2 + b*c*(Log[F]/(2*e)), 1 + n/2 + b*c*(Log[F]/(2*e))
, -E^(2*(d + e*x))], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = \frac {4 e^{a+b x+2 (c+d x)} \operatorname {Hypergeometric2F1}\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] (verified)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int e^{a+b x} \text {sech}^2(c+d x) \, dx=\frac {4 e^{a+b x+2 (c+d x)} \operatorname {Hypergeometric2F1}\left (2,1+\frac {b}{2 d},2+\frac {b}{2 d},-e^{2 (c+d x)}\right )}{b+2 d} \]

[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)

Maple [F]

\[\int {\mathrm e}^{b x +a} \operatorname {sech}\left (d x +c \right )^{2}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int e^{a+b x} \text {sech}^2(c+d x) \, dx=\int { e^{\left (b x + a\right )} \operatorname {sech}\left (d x + c\right )^{2} \,d x } \]

[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)

Sympy [F]

\[ \int e^{a+b x} \text {sech}^2(c+d x) \, dx=e^{a} \int e^{b x} \operatorname {sech}^{2}{\left (c + d x \right )}\, dx \]

[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)

Maxima [F]

\[ \int e^{a+b x} \text {sech}^2(c+d x) \, dx=\int { e^{\left (b x + a\right )} \operatorname {sech}\left (d x + c\right )^{2} \,d x } \]

[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)

Giac [F]

\[ \int e^{a+b x} \text {sech}^2(c+d x) \, dx=\int { e^{\left (b x + a\right )} \operatorname {sech}\left (d x + c\right )^{2} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int e^{a+b x} \text {sech}^2(c+d x) \, dx=\int \frac {{\mathrm {e}}^{a+b\,x}}{{\mathrm {cosh}\left (c+d\,x\right )}^2} \,d x \]

[In]

int(exp(a + b*x)/cosh(c + d*x)^2,x)

[Out]

int(exp(a + b*x)/cosh(c + d*x)^2, x)