3.559 \(\int \frac{e^x (1-\sin (x))}{1+\cos (x)} \, dx\)

Optimal. Leaf size=42 \[ -\frac{e^x \sin (x)}{\cos (x)+1}+(2-2 i) e^{(1+i) x} \, _2F_1\left (1-i,2;2-i;-e^{i x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.173968, antiderivative size = 44, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -4 i e^x \, _2F_1\left (-i,1;1-i;-e^{i x}\right )+2 i e^x+\frac{e^x \sin (x)}{\cos (x)+1} \]

Antiderivative was successfully verified.

[In]  Int[(E^x*(1 - Sin[x]))/(1 + Cos[x]),x]

[Out]

(2*I)*E^x - (4*I)*E^x*Hypergeometric2F1[-I, 1, 1 - I, -E^(I*x)] + (E^x*Sin[x])/(
1 + Cos[x])

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Rubi in Sympy [A]  time = 13.0771, size = 36, normalized size = 0.86 \[ - 4 i e^{x}{{}_{2}F_{1}\left (\begin{matrix} 1, - i \\ 1 - i \end{matrix}\middle |{- e^{i x}} \right )} + 2 i e^{x} + \frac{e^{x} \sin{\left (x \right )}}{\cos{\left (x \right )} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*I*exp(x)*hyper((1, -I), (1 - I,), -exp(I*x)) + 2*I*exp(x) + exp(x)*sin(x)/(co
s(x) + 1)

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Mathematica [B]  time = 0.117179, size = 87, normalized size = 2.07 \[ -\frac{2 e^x \cos \left (\frac{x}{2}\right ) \left (2 i \, _2F_1\left (-i,1;1-i;-e^{i x}\right ) \cos \left (\frac{x}{2}\right )-(1+i) e^{i x} \, _2F_1\left (1,1-i;2-i;-e^{i x}\right ) \cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )}{\cos (x)+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(E^x*(1 - Sin[x]))/(1 + Cos[x]),x]

[Out]

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

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Maple [F]  time = 0.102, size = 0, normalized size = 0. \[ \int{\frac{{{\rm e}^{x}} \left ( 1-\sin \left ( x \right ) \right ) }{1+\cos \left ( x \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x)*(1-sin(x))/(1+cos(x)),x)

[Out]

int(exp(x)*(1-sin(x))/(1+cos(x)),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(sin(x) - 1)*e^x/(cos(x) + 1),x, algorithm="maxima")

[Out]

-2*(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. \[{\rm integral}\left (-\frac{e^{x} \sin \left (x\right ) - e^{x}}{\cos \left (x\right ) + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(e^x*sin(x) - e^x)/(cos(x) + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(x)*(1-sin(x))/(1+cos(x)),x)

[Out]

-Integral(-exp(x)/(cos(x) + 1), x) - Integral(exp(x)*sin(x)/(cos(x) + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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