3.94 \(\int \frac{1}{1-\cos (x)} \, dx\)

Optimal. Leaf size=12 \[ -\frac{\sin (x)}{1-\cos (x)} \]

[Out]

-(Sin[x]/(1 - Cos[x]))

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Rubi [A]  time = 0.015229, antiderivative size = 12, 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 \[ -\frac{\sin (x)}{1-\cos (x)} \]

Antiderivative was successfully verified.

[In]  Int[(1 - Cos[x])^(-1),x]

[Out]

-(Sin[x]/(1 - Cos[x]))

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Rubi in Sympy [A]  time = 0.510592, size = 8, normalized size = 0.67 \[ - \frac{\sin{\left (x \right )}}{- \cos{\left (x \right )} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-cos(x)),x)

[Out]

-sin(x)/(-cos(x) + 1)

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Mathematica [A]  time = 0.00803861, size = 8, normalized size = 0.67 \[ -\cot \left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - Cos[x])^(-1),x]

[Out]

-Cot[x/2]

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Maple [A]  time = 0.003, size = 9, normalized size = 0.8 \[ - \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-cos(x)),x)

[Out]

-1/tan(1/2*x)

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Maxima [A]  time = 1.32959, size = 14, normalized size = 1.17 \[ -\frac{\cos \left (x\right ) + 1}{\sin \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(cos(x) - 1),x, algorithm="maxima")

[Out]

-(cos(x) + 1)/sin(x)

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Fricas [A]  time = 0.22252, size = 14, normalized size = 1.17 \[ -\frac{\cos \left (x\right ) + 1}{\sin \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(cos(x) - 1),x, algorithm="fricas")

[Out]

-(cos(x) + 1)/sin(x)

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Sympy [A]  time = 0.656155, size = 7, normalized size = 0.58 \[ - \frac{1}{\tan{\left (\frac{x}{2} \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1-cos(x)),x)

[Out]

-1/tan(x/2)

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GIAC/XCAS [A]  time = 0.212029, size = 11, normalized size = 0.92 \[ -\frac{1}{\tan \left (\frac{1}{2} \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(cos(x) - 1),x, algorithm="giac")

[Out]

-1/tan(1/2*x)