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*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)

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Rubi [A]  time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, 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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fricas [F]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (e^{\left (b x + a\right )} \operatorname {sech}\left (d x + c\right ), x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left (b x + a\right )} \operatorname {sech}\left (d x + c\right )\,{d x} \]

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)

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maple [F]  time = 0.26, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{b x +a} \mathrm {sech}\left (d x +c \right )\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left (b x + a\right )} \operatorname {sech}\left (d x + c\right )\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {e}}^{a+b\,x}}{\mathrm {cosh}\left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(a + b*x)/cosh(c + d*x),x)

[Out]

int(exp(a + b*x)/cosh(c + d*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ e^{a} \int e^{b x} \operatorname {sech}{\left (c + d x \right )}\, dx \]

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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