3.214 \(\int \frac {\csc (x)}{-\cot (x)+\csc (x)} \, dx\)

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

[Out]

-sin(x)/(1-cos(x))

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3166, 2648} \[ -\frac {\sin (x)}{1-\cos (x)} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]/(-Cot[x] + Csc[x]),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]

Rule 3166

Int[csc[(d_.) + (e_.)*(x_)]^(n_.)*((a_.) + csc[(d_.) + (e_.)*(x_)]*(b_.) + cot[(d_.) + (e_.)*(x_)]*(c_.))^(m_)
, x_Symbol] :> Int[1/(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + n, 0]
&& IntegerQ[n]

Rubi steps

\begin {align*} \int \frac {\csc (x)}{-\cot (x)+\csc (x)} \, dx &=\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[Csc[x]/(-Cot[x] + Csc[x]),x]

[Out]

-Cot[x/2]

________________________________________________________________________________________

fricas [A]  time = 0.91, 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(csc(x)/(-cot(x)+csc(x)),x, algorithm="fricas")

[Out]

-(cos(x) + 1)/sin(x)

________________________________________________________________________________________

giac [A]  time = 0.20, 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(csc(x)/(-cot(x)+csc(x)),x, algorithm="giac")

[Out]

-1/tan(1/2*x)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 9, normalized size = 0.75 \[ -\frac {1}{\tan \left (\frac {x}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)/(-cot(x)+csc(x)),x)

[Out]

-1/tan(1/2*x)

________________________________________________________________________________________

maxima [A]  time = 0.34, 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(csc(x)/(-cot(x)+csc(x)),x, algorithm="maxima")

[Out]

-(cos(x) + 1)/sin(x)

________________________________________________________________________________________

mupad [B]  time = 0.53, 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/(sin(x)*(cot(x) - 1/sin(x))),x)

[Out]

-cot(x/2)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\csc {\relax (x )}}{\cot {\relax (x )} - \csc {\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)/(-cot(x)+csc(x)),x)

[Out]

-Integral(csc(x)/(cot(x) - csc(x)), x)

________________________________________________________________________________________