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