\(\int e^{a+b x} \text {sech}^3(c+d x) \, dx\) [890]

Optimal result

Integrand size = 16, antiderivative size = 103 \[ \int e^{a+b x} \text {sech}^3(c+d x) \, dx=-\frac {(b-d) e^{a+c+b x+d x} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 d},\frac {1}{2} \left (3+\frac {b}{d}\right ),-e^{2 (c+d x)}\right )}{d^2}+\frac {b e^{a+b x} \text {sech}(c+d x)}{2 d^2}+\frac {e^{a+b x} \text {sech}(c+d x) \tanh (c+d x)}{2 d} \]

[Out]

-(b-d)*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))/d^2+1/2*b*exp(b*x+a)*sech(d*
x+c)/d^2+1/2*exp(b*x+a)*sech(d*x+c)*tanh(d*x+c)/d

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5598, 5600} \[ \int e^{a+b x} \text {sech}^3(c+d x) \, dx=-\frac {(b-d) e^{a+b x+c+d x} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 d},\frac {1}{2} \left (\frac {b}{d}+3\right ),-e^{2 (c+d x)}\right )}{d^2}+\frac {b e^{a+b x} \text {sech}(c+d x)}{2 d^2}+\frac {e^{a+b x} \tanh (c+d x) \text {sech}(c+d x)}{2 d} \]

[In]

Int[E^(a + b*x)*Sech[c + d*x]^3,x]

[Out]

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

Rule 5598

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sech[(d_.) + (e_.)*(x_)]^(n_), x_Symbol] :> Simp[b*c*Log[F]*F^(c*(a + b*
x))*(Sech[d + e*x]^(n - 2)/(e^2*(n - 1)*(n - 2))), x] + (Dist[(e^2*(n - 2)^2 - b^2*c^2*Log[F]^2)/(e^2*(n - 1)*
(n - 2)), Int[F^(c*(a + b*x))*Sech[d + e*x]^(n - 2), x], x] + Simp[F^(c*(a + b*x))*Sech[d + e*x]^(n - 1)*(Sinh
[d + e*x]/(e*(n - 1))), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*(n - 2)^2 - b^2*c^2*Log[F]^2, 0] && GtQ
[n, 1] && NeQ[n, 2]

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 {b e^{a+b x} \text {sech}(c+d x)}{2 d^2}+\frac {e^{a+b x} \text {sech}(c+d x) \tanh (c+d x)}{2 d}+\frac {1}{2} \left (1-\frac {b^2}{d^2}\right ) \int e^{a+b x} \text {sech}(c+d x) \, dx \\ & = -\frac {(b-d) 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 )}{d^2}+\frac {b e^{a+b x} \text {sech}(c+d x)}{2 d^2}+\frac {e^{a+b x} \text {sech}(c+d x) \tanh (c+d x)}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.78 \[ \int e^{a+b x} \text {sech}^3(c+d x) \, dx=\frac {e^{a+b x} \left (-2 (b-d) e^{c+d x} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 d},\frac {1}{2} \left (3+\frac {b}{d}\right ),-e^{2 (c+d x)}\right )+\text {sech}(c+d x) (b+d \tanh (c+d x))\right )}{2 d^2} \]

[In]

Integrate[E^(a + b*x)*Sech[c + d*x]^3,x]

[Out]

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

Maple [F]

\[\int {\mathrm e}^{b x +a} \operatorname {sech}\left (d x +c \right )^{3}d x\]

[In]

int(exp(b*x+a)*sech(d*x+c)^3,x)

[Out]

int(exp(b*x+a)*sech(d*x+c)^3,x)

Fricas [F]

\[ \int e^{a+b x} \text {sech}^3(c+d x) \, dx=\int { e^{\left (b x + a\right )} \operatorname {sech}\left (d x + c\right )^{3} \,d x } \]

[In]

integrate(exp(b*x+a)*sech(d*x+c)^3,x, algorithm="fricas")

[Out]

integral(e^(b*x + a)*sech(d*x + c)^3, x)

Sympy [F]

\[ \int e^{a+b x} \text {sech}^3(c+d x) \, dx=e^{a} \int e^{b x} \operatorname {sech}^{3}{\left (c + d x \right )}\, dx \]

[In]

integrate(exp(b*x+a)*sech(d*x+c)**3,x)

[Out]

exp(a)*Integral(exp(b*x)*sech(c + d*x)**3, x)

Maxima [F]

\[ \int e^{a+b x} \text {sech}^3(c+d x) \, dx=\int { e^{\left (b x + a\right )} \operatorname {sech}\left (d x + c\right )^{3} \,d x } \]

[In]

integrate(exp(b*x+a)*sech(d*x+c)^3,x, algorithm="maxima")

[Out]

-8*(b^2*e^c - d^2*e^c)*integrate(1/8*e^(b*x + d*x + a)/(d^2*e^(2*d*x + 2*c) + d^2), x) + ((b*e^(3*c) + d*e^(3*
c))*e^(b*x + 3*d*x + a) + (b*e^c - d*e^c)*e^(b*x + d*x + a))/(d^2*e^(4*d*x + 4*c) + 2*d^2*e^(2*d*x + 2*c) + d^
2)

Giac [F]

\[ \int e^{a+b x} \text {sech}^3(c+d x) \, dx=\int { e^{\left (b x + a\right )} \operatorname {sech}\left (d x + c\right )^{3} \,d x } \]

[In]

integrate(exp(b*x+a)*sech(d*x+c)^3,x, algorithm="giac")

[Out]

integrate(e^(b*x + a)*sech(d*x + c)^3, x)

Mupad [F(-1)]

Timed out. \[ \int e^{a+b x} \text {sech}^3(c+d x) \, dx=\int \frac {{\mathrm {e}}^{a+b\,x}}{{\mathrm {cosh}\left (c+d\,x\right )}^3} \,d x \]

[In]

int(exp(a + b*x)/cosh(c + d*x)^3,x)

[Out]

int(exp(a + b*x)/cosh(c + d*x)^3, x)