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 = 0.0524635, 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 \[ (1-i) e^{(1+i) x} \, _2F_1\left (1-i,2;2-i;-e^{i x}\right ) \]

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)]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{e^{x}}{\cos ^{2}{\left (\frac{x}{2} \right )}}\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(x)/(1+cos(x)),x)

[Out]

Integral(exp(x)/cos(x/2)**2, x)/2

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Mathematica [B]  time = 0.0709591, size = 89, normalized size = 3.18 \[ -\frac{(1+i) e^x \cos \left (\frac{x}{2}\right ) \left ((1+i) \, _2F_1\left (-i,1;1-i;-e^{i x}\right ) \cos \left (\frac{x}{2}\right )-e^{i x} \, _2F_1\left (1,1-i;2-i;-e^{i x}\right ) \cos \left (\frac{x}{2}\right )-(1-i) \sin \left (\frac{x}{2}\right )\right )}{\cos (x)+1} \]

Antiderivative was successfully verified.

[In]  Integrate[E^x/(1 + Cos[x]),x]

[Out]

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

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Maple [F]  time = 0.048, size = 0, normalized size = 0. \[ \int{\frac{{{\rm e}^{x}}}{1+\cos \left ( x \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x)/(1+cos(x)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^x/(cos(x) + 1),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*sin(x))/(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^x/(cos(x) + 1),x, algorithm="fricas")

[Out]

integral(e^x/(cos(x) + 1), x)

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

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e^{x}}{\cos \left (x\right ) + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^x/(cos(x) + 1),x, algorithm="giac")

[Out]

integrate(e^x/(cos(x) + 1), x)