3.44 \(\int \cot ^2(x) \, dx\)

Optimal. Leaf size=8 \[ -x-\cot (x) \]

[Out]

-x-cot(x)

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Rubi [A]  time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3473, 8} \[ -x-\cot (x) \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^2,x]

[Out]

-x - Cot[x]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rubi steps

\begin {align*} \int \cot ^2(x) \, dx &=-\cot (x)-\int 1 \, dx\\ &=-x-\cot (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 1.00 \[ -x-\cot (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^2,x]

[Out]

-x - Cot[x]

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fricas [B]  time = 0.41, size = 20, normalized size = 2.50 \[ -\frac {x \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^2,x, algorithm="fricas")

[Out]

-(x*sin(2*x) + cos(2*x) + 1)/sin(2*x)

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giac [B]  time = 0.02, size = 18, normalized size = 2.25 \[ -x - \frac {1}{2 \, \tan \left (\frac {1}{2} \, x\right )} + \frac {1}{2} \, \tan \left (\frac {1}{2} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^2,x, algorithm="giac")

[Out]

-x - 1/2/tan(1/2*x) + 1/2*tan(1/2*x)

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maple [A]  time = 0.00, size = 12, normalized size = 1.50 \[ -x -\cot \relax (x )+\frac {\pi }{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^2,x)

[Out]

-cot(x)+1/2*Pi-x

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maxima [A]  time = 1.27, size = 10, normalized size = 1.25 \[ -x - \frac {1}{\tan \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^2,x, algorithm="maxima")

[Out]

-x - 1/tan(x)

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mupad [B]  time = 0.06, size = 8, normalized size = 1.00 \[ -x-\mathrm {cot}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^2,x)

[Out]

- x - cot(x)

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sympy [A]  time = 0.07, size = 8, normalized size = 1.00 \[ - x - \frac {\cos {\relax (x )}}{\sin {\relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)**2,x)

[Out]

-x - cos(x)/sin(x)

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