\(\int \frac {e^x}{1+\cos (x)} \, dx\) [552]

Optimal result

Integrand size = 10, antiderivative size = 28 \[ \int \frac {e^x}{1+\cos (x)} \, dx=(1-i) e^{(1+i) x} \operatorname {Hypergeometric2F1}\left (1-i,2,2-i,-e^{i x}\right ) \]

[Out]

(1-I)*exp((1+I)*x)*hypergeom([2, 1-I],[2-I],-exp(I*x))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4542, 4536} \[ \int \frac {e^x}{1+\cos (x)} \, dx=(1-i) e^{(1+i) x} \operatorname {Hypergeometric2F1}\left (1-i,2,2-i,-e^{i x}\right ) \]

[In]

Int[E^x/(1 + Cos[x]),x]

[Out]

(1 - I)*E^((1 + I)*x)*Hypergeometric2F1[1 - I, 2, 2 - I, -E^(I*x)]

Rule 4536

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sec[(d_.) + (e_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n*E^(I*n*(d + e*x))*(
F^(c*(a + b*x))/(I*e*n + b*c*Log[F]))*Hypergeometric2F1[n, n/2 - I*b*c*(Log[F]/(2*e)), 1 + n/2 - I*b*c*(Log[F]
/(2*e)), -E^(2*I*(d + e*x))], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]

Rule 4542

Int[(Cos[(d_.) + (e_.)*(x_)]*(g_.) + (f_))^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Dist[2^n*f^n,
Int[F^(c*(a + b*x))*Cos[d/2 + e*(x/2)]^(2*n), x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f - g, 0] &
& ILtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int e^x \sec ^2\left (\frac {x}{2}\right ) \, dx \\ & = (1-i) e^{(1+i) x} \operatorname {Hypergeometric2F1}\left (1-i,2,2-i,-e^{i x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {e^x}{1+\cos (x)} \, dx=(1-i) e^{(1+i) x} \operatorname {Hypergeometric2F1}\left (1-i,2,2-i,-e^{i x}\right ) \]

[In]

Integrate[E^x/(1 + Cos[x]),x]

[Out]

(1 - I)*E^((1 + I)*x)*Hypergeometric2F1[1 - I, 2, 2 - I, -E^(I*x)]

Maple [F]

\[\int \frac {{\mathrm e}^{x}}{\cos \left (x \right )+1}d x\]

[In]

int(exp(x)/(cos(x)+1),x)

[Out]

int(exp(x)/(cos(x)+1),x)

Fricas [F]

\[ \int \frac {e^x}{1+\cos (x)} \, dx=\int { \frac {e^{x}}{\cos \left (x\right ) + 1} \,d x } \]

[In]

integrate(exp(x)/(1+cos(x)),x, algorithm="fricas")

[Out]

integral(e^x/(cos(x) + 1), x)

Sympy [F]

\[ \int \frac {e^x}{1+\cos (x)} \, dx=\int \frac {e^{x}}{\cos {\left (x \right )} + 1}\, dx \]

[In]

integrate(exp(x)/(1+cos(x)),x)

[Out]

Integral(exp(x)/(cos(x) + 1), x)

Maxima [F]

\[ \int \frac {e^x}{1+\cos (x)} \, dx=\int { \frac {e^{x}}{\cos \left (x\right ) + 1} \,d x } \]

[In]

integrate(exp(x)/(1+cos(x)),x, algorithm="maxima")

[Out]

-2*((cos(x)^2 + sin(x)^2 + 2*cos(x) + 1)*integrate(e^x*sin(x)/(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1), x) - e^x*s
in(x))/(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1)

Giac [F]

\[ \int \frac {e^x}{1+\cos (x)} \, dx=\int { \frac {e^{x}}{\cos \left (x\right ) + 1} \,d x } \]

[In]

integrate(exp(x)/(1+cos(x)),x, algorithm="giac")

[Out]

integrate(e^x/(cos(x) + 1), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^x}{1+\cos (x)} \, dx=\int \frac {{\mathrm {e}}^x}{\cos \left (x\right )+1} \,d x \]

[In]

int(exp(x)/(cos(x) + 1),x)

[Out]

int(exp(x)/(cos(x) + 1), x)