Optimal. Leaf size=14 \[ \frac{e^x \cos (x)}{1-\sin (x)} \]
[Out]
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Rubi [A] time = 0.0375474, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{e^x \cos (x)}{1-\sin (x)} \]
Antiderivative was successfully verified.
[In] Int[(E^x*(1 + Cos[x]))/(1 - Sin[x]),x]
[Out]
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Rubi in Sympy [A] time = 3.03878, size = 10, normalized size = 0.71 \[ \frac{e^{x} \cos{\left (x \right )}}{- \sin{\left (x \right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(exp(x)*(1+cos(x))/(1-sin(x)),x)
[Out]
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Mathematica [A] time = 0.0635755, size = 23, normalized size = 1.64 \[ -\frac{e^x \left (\tan \left (\frac{x}{2}\right )+1\right )}{\tan \left (\frac{x}{2}\right )-1} \]
Antiderivative was successfully verified.
[In] Integrate[(E^x*(1 + Cos[x]))/(1 - Sin[x]),x]
[Out]
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Maple [B] time = 0.106, size = 53, normalized size = 3.8 \[{1 \left ( -{{\rm e}^{x}}\tan \left ({\frac{x}{2}} \right ) -{{\rm e}^{x}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-{{\rm e}^{x}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}-{{\rm e}^{x}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-1} \left ( -1+\tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(exp(x)*(1+cos(x))/(1-sin(x)),x)
[Out]
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Maxima [A] time = 1.92892, size = 30, normalized size = 2.14 \[ \frac{2 \, \cos \left (x\right ) e^{x}}{\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(cos(x) + 1)*e^x/(sin(x) - 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.215362, size = 32, normalized size = 2.29 \[ \frac{{\left (\cos \left (x\right ) + 1\right )} e^{x} + e^{x} \sin \left (x\right )}{\cos \left (x\right ) - \sin \left (x\right ) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(cos(x) + 1)*e^x/(sin(x) - 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{e^{x}}{\sin{\left (x \right )} - 1}\, dx - \int \frac{e^{x} \cos{\left (x \right )}}{\sin{\left (x \right )} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(exp(x)*(1+cos(x))/(1-sin(x)),x)
[Out]
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GIAC/XCAS [A] time = 0.221522, size = 27, normalized size = 1.93 \[ -\frac{e^{x} \tan \left (\frac{1}{2} \, x\right ) + e^{x}}{\tan \left (\frac{1}{2} \, x\right ) - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(cos(x) + 1)*e^x/(sin(x) - 1),x, algorithm="giac")
[Out]