### 3.254 $$\int \frac{1}{1-\cos (x)} \, dx$$

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

[Out]

-(Sin[x]/(1 - Cos[x]))

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Rubi [A]  time = 0.008591, 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, 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 = 0.0081957, 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 = 0.006, size = 9, normalized size = 0.8 \begin{align*} - \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1} \end{align*}

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 = 0.920649, size = 14, normalized size = 1.17 \begin{align*} -\frac{\cos \left (x\right ) + 1}{\sin \left (x\right )} \end{align*}

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)

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Fricas [A]  time = 2.08231, size = 30, normalized size = 2.5 \begin{align*} -\frac{\cos \left (x\right ) + 1}{\sin \left (x\right )} \end{align*}

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)

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Sympy [A]  time = 0.348639, size = 7, normalized size = 0.58 \begin{align*} - \frac{1}{\tan{\left (\frac{x}{2} \right )}} \end{align*}

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 [A]  time = 1.06795, size = 11, normalized size = 0.92 \begin{align*} -\frac{1}{\tan \left (\frac{1}{2} \, x\right )} \end{align*}

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)