3.552 \(\int \frac{e^x}{1+\cos (x)} \, dx\)

Optimal. Leaf size=28 \[ (1-i) e^{(1+i) x} \, _2F_1\left (1-i,2;2-i;-e^{i x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0312807, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4457, 4451} \[ (1-i) e^{(1+i) x} \text{Hypergeometric2F1}\left (1-i,2,2-i,-e^{i x}\right ) \]

Antiderivative was successfully verified.

[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 4457

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]

Rule 4451

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))*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))])/(I*e*n + b*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]

Rubi steps

\begin{align*} \int \frac{e^x}{1+\cos (x)} \, dx &=\frac{1}{2} \int e^x \sec ^2\left (\frac{x}{2}\right ) \, dx\\ &=(1-i) e^{(1+i) x} \, _2F_1\left (1-i,2;2-i;-e^{i x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0097364, size = 28, normalized size = 1. \[ (1-i) e^{(1+i) x} \, _2F_1\left (1-i,2;2-i;-e^{i x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x/(1 + Cos[x]),x]

[Out]

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

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{x}}}{\cos \left ( x \right ) +1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \,{\left ({\left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right )} \int \frac{e^{x} \sin \left (x\right )}{\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1}\,{d x} - e^{x} \sin \left (x\right )\right )}}{\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{e^{x}}{\cos \left (x\right ) + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(e^x/(cos(x) + 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{x}}{\cos{\left (x \right )} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(x)/(cos(x) + 1), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{x}}{\cos \left (x\right ) + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^x/(cos(x) + 1), x)