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

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2648} \[ -\frac {\sin (x)}{1-\cos (x)} \]

Antiderivative was successfully verified.

[In]

Int[(1 - Cos[x])^(-1),x]

[Out]

-(Sin[x]/(1 - Cos[x]))

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin {align*} \int \frac {1}{1-\cos (x)} \, dx &=-\frac {\sin (x)}{1-\cos (x)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, 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]

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{1-\cos (x)} \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[(1 - Cos[x])^(-1),x]

[Out]

Could not integrate

________________________________________________________________________________________

fricas [A]  time = 1.05, size = 10, normalized size = 0.83 \[ -\frac {\cos \relax (x) + 1}{\sin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(cos(x) + 1)/sin(x)

________________________________________________________________________________________

giac [A]  time = 1.08, size = 8, normalized size = 0.67 \[ -\frac {1}{\tan \left (\frac {1}{2} \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/tan(1/2*x)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 9, normalized size = 0.75

method	result	size
default	\(-\frac {1}{\tan \left (\frac {x}{2}\right )}\)	\(9\)
norman	\(-\frac {1}{\tan \left (\frac {x}{2}\right )}\)	\(9\)
risch	\(-\frac {2 i}{{\mathrm e}^{i x}-1}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-cos(x)),x,method=_RETURNVERBOSE)

[Out]

-1/tan(1/2*x)

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 10, normalized size = 0.83 \[ -\frac {\cos \relax (x) + 1}{\sin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(cos(x) + 1)/sin(x)

________________________________________________________________________________________

mupad [B]  time = 0.00, size = 6, normalized size = 0.50 \[ -\mathrm {cot}\left (\frac {x}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(cos(x) - 1),x)

[Out]

-cot(x/2)

________________________________________________________________________________________

sympy [A]  time = 0.34, 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)

________________________________________________________________________________________