Optimal. Leaf size=11 \[ \frac{\cos (x)}{1-\sin (x)} \]
[Out]
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Rubi [A] time = 0.015708, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\cos (x)}{1-\sin (x)} \]
Antiderivative was successfully verified.
[In] Int[(1 - Sin[x])^(-1),x]
[Out]
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Rubi in Sympy [A] time = 0.51639, size = 7, normalized size = 0.64 \[ \frac{\cos{\left (x \right )}}{- \sin{\left (x \right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1-sin(x)),x)
[Out]
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Mathematica [B] time = 0.0104177, size = 25, normalized size = 2.27 \[ \frac{2 \sin \left (\frac{x}{2}\right )}{\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(1 - Sin[x])^(-1),x]
[Out]
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Maple [A] time = 0.003, size = 11, normalized size = 1. \[ -2\, \left ( -1+\tan \left ( x/2 \right ) \right ) ^{-1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1-sin(x)),x)
[Out]
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Maxima [A] time = 1.35804, size = 20, normalized size = 1.82 \[ -\frac{2}{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(sin(x) - 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.211194, size = 23, normalized size = 2.09 \[ \frac{\cos \left (x\right ) + \sin \left (x\right ) + 1}{\cos \left (x\right ) - \sin \left (x\right ) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(sin(x) - 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.714072, size = 8, normalized size = 0.73 \[ - \frac{2}{\tan{\left (\frac{x}{2} \right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(1-sin(x)),x)
[Out]
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GIAC/XCAS [A] time = 0.215994, size = 14, normalized size = 1.27 \[ -\frac{2}{\tan \left (\frac{1}{2} \, x\right ) - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(sin(x) - 1),x, algorithm="giac")
[Out]