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

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

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4392, 2735, 2648} \[ x-\frac {\sin (x)}{\cos (x)+1} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]/(Cot[x] + Csc[x]),x]

[Out]

x - 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 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 4392

Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b_.))^(p_)*(u_.), x_Symbol] :> Int[A
ctivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a*Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 10, normalized size = 0.83 \[ x-\tan \left (\frac {x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]/(Cot[x] + Csc[x]),x]

[Out]

x - Tan[x/2]

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fricas [A]  time = 0.59, size = 17, normalized size = 1.42 \[ \frac {x \cos \relax (x) + x - \sin \relax (x)}{\cos \relax (x) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(cot(x)+csc(x)),x, algorithm="fricas")

[Out]

(x*cos(x) + x - sin(x))/(cos(x) + 1)

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giac [A]  time = 0.22, size = 8, normalized size = 0.67 \[ x - \tan \left (\frac {1}{2} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(cot(x)+csc(x)),x, algorithm="giac")

[Out]

x - tan(1/2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-tan(1/2*x)+x

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maxima [A]  time = 0.43, size = 23, normalized size = 1.92 \[ -\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 2 \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(cot(x)+csc(x)),x, algorithm="maxima")

[Out]

-sin(x)/(cos(x) + 1) + 2*arctan(sin(x)/(cos(x) + 1))

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mupad [B]  time = 0.57, size = 8, normalized size = 0.67 \[ x-\mathrm {tan}\left (\frac {x}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/(cot(x) + 1/sin(x)),x)

[Out]

x - tan(x/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cot(x)/(cot(x) + csc(x)), x)

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