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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]