\(\int e^{a+b x} \text {sech}(c+d x) \, dx\) [888]

Optimal result

Integrand size = 14, antiderivative size = 52 \[ \int e^{a+b x} \text {sech}(c+d x) \, dx=\frac {2 e^{a+c+b x+d x} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 d},\frac {1}{2} \left (3+\frac {b}{d}\right ),-e^{2 (c+d x)}\right )}{b+d} \]

[Out]

2*exp(b*x+d*x+a+c)*hypergeom([1, 1/2*(b+d)/d],[3/2+1/2*b/d],-exp(2*d*x+2*c))/(b+d)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {5600} \[ \int e^{a+b x} \text {sech}(c+d x) \, dx=\frac {2 e^{a+b x+c+d x} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 d},\frac {1}{2} \left (\frac {b}{d}+3\right ),-e^{2 (c+d x)}\right )}{b+d} \]

[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 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 {2 e^{a+c+b x+d x} \operatorname {Hypergeometric2F1}\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] (verified)

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.98 \[ \int e^{a+b x} \text {sech}(c+d x) \, dx=\frac {2 e^{a+c+(b+d) x} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 d},\frac {1}{2} \left (3+\frac {b}{d}\right ),-e^{2 (c+d x)}\right )}{b+d} \]

[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)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(exp(a + b*x)/cosh(c + d*x),x)

[Out]

int(exp(a + b*x)/cosh(c + d*x), x)