3.37 \(\int \cot ^2(\frac{3 x}{4}) \, dx\)

Optimal. Leaf size=14 \[ -x-\frac{4}{3} \cot \left (\frac{3 x}{4}\right ) \]

[Out]

-x - (4*Cot[(3*x)/4])/3

________________________________________________________________________________________

Rubi [A]  time = 0.0062183, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3473, 8} \[ -x-\frac{4}{3} \cot \left (\frac{3 x}{4}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cot[(3*x)/4]^2,x]

[Out]

-x - (4*Cot[(3*x)/4])/3

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]

Rule 8

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

Rubi steps

\begin{align*} \int \cot ^2\left (\frac{3 x}{4}\right ) \, dx &=-\frac{4}{3} \cot \left (\frac{3 x}{4}\right )-\int 1 \, dx\\ &=-x-\frac{4}{3} \cot \left (\frac{3 x}{4}\right )\\ \end{align*}

Mathematica [C]  time = 0.013271, size = 28, normalized size = 2. \[ -\frac{4}{3} \cot \left (\frac{3 x}{4}\right ) \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-\tan ^2\left (\frac{3 x}{4}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[(3*x)/4]^2,x]

[Out]

(-4*Cot[(3*x)/4]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[(3*x)/4]^2])/3

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 14, normalized size = 1. \begin{align*} -{\frac{4}{3}\cot \left ({\frac{3\,x}{4}} \right ) }+{\frac{2\,\pi }{3}}-x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(3/4*x)^2,x)

[Out]

-4/3*cot(3/4*x)+2/3*Pi-x

________________________________________________________________________________________

Maxima [A]  time = 1.46041, size = 16, normalized size = 1.14 \begin{align*} -x - \frac{4}{3 \, \tan \left (\frac{3}{4} \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(3/4*x)^2,x, algorithm="maxima")

[Out]

-x - 4/3/tan(3/4*x)

________________________________________________________________________________________

Fricas [B]  time = 2.01595, size = 72, normalized size = 5.14 \begin{align*} -\frac{3 \, x \sin \left (\frac{3}{2} \, x\right ) + 4 \, \cos \left (\frac{3}{2} \, x\right ) + 4}{3 \, \sin \left (\frac{3}{2} \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(3/4*x)^2,x, algorithm="fricas")

[Out]

-1/3*(3*x*sin(3/2*x) + 4*cos(3/2*x) + 4)/sin(3/2*x)

________________________________________________________________________________________

Sympy [A]  time = 0.068754, size = 19, normalized size = 1.36 \begin{align*} - x - \frac{4 \cos{\left (\frac{3 x}{4} \right )}}{3 \sin{\left (\frac{3 x}{4} \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(3/4*x)**2,x)

[Out]

-x - 4*cos(3*x/4)/(3*sin(3*x/4))

________________________________________________________________________________________

Giac [A]  time = 1.0636, size = 24, normalized size = 1.71 \begin{align*} -x - \frac{2}{3 \, \tan \left (\frac{3}{8} \, x\right )} + \frac{2}{3} \, \tan \left (\frac{3}{8} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(3/4*x)^2,x, algorithm="giac")

[Out]

-x - 2/3/tan(3/8*x) + 2/3*tan(3/8*x)