3.150 \(\int \sqrt{\cos (x) \cot (x)} \, dx\)

Optimal. Leaf size=13 \[ 2 \tan (x) \sqrt{\cos (x) \cot (x)} \]

[Out]

2*Sqrt[Cos[x]*Cot[x]]*Tan[x]

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Rubi [A]  time = 0.0381709, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4400, 2589} \[ 2 \tan (x) \sqrt{\cos (x) \cot (x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Cos[x]*Cot[x]],x]

[Out]

2*Sqrt[Cos[x]*Cot[x]]*Tan[x]

Rule 4400

Int[(u_.)*((v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> With[{uu = ActivateTrig[u], vv = ActivateTrig[v], ww = Ac
tivateTrig[w]}, Dist[(vv^m*ww^n)^FracPart[p]/(vv^(m*FracPart[p])*ww^(n*FracPart[p])), Int[uu*vv^(m*p)*ww^(n*p)
, x], x]] /; FreeQ[{m, n, p}, x] &&  !IntegerQ[p] && ( !InertTrigFreeQ[v] ||  !InertTrigFreeQ[w])

Rule 2589

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*(a*Sin[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*m), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n - 1, 0]

Rubi steps

\begin{align*} \int \sqrt{\cos (x) \cot (x)} \, dx &=\frac{\sqrt{\cos (x) \cot (x)} \int \sqrt{\cos (x)} \sqrt{\cot (x)} \, dx}{\sqrt{\cos (x)} \sqrt{\cot (x)}}\\ &=2 \sqrt{\cos (x) \cot (x)} \tan (x)\\ \end{align*}

Mathematica [A]  time = 0.0687166, size = 13, normalized size = 1. \[ 2 \tan (x) \sqrt{\cos (x) \cot (x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Cos[x]*Cot[x]],x]

[Out]

2*Sqrt[Cos[x]*Cot[x]]*Tan[x]

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Maple [A]  time = 0.132, size = 20, normalized size = 1.5 \begin{align*} 2\,{\frac{\sin \left ( x \right ) }{\cos \left ( x \right ) }\sqrt{{\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}}{\sin \left ( x \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(x)*cot(x))^(1/2),x)

[Out]

2*sin(x)*(cos(x)^2/sin(x))^(1/2)/cos(x)

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Maxima [B]  time = 1.79664, size = 254, normalized size = 19.54 \begin{align*} \frac{{\left ({\left (\cos \left (\frac{3}{2} \, x\right ) - \cos \left (\frac{1}{2} \, x\right ) + \sin \left (\frac{3}{2} \, x\right ) + \sin \left (\frac{1}{2} \, x\right )\right )} \cos \left (\frac{1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right ) -{\left (\cos \left (\frac{3}{2} \, x\right ) - \cos \left (\frac{1}{2} \, x\right ) - \sin \left (\frac{3}{2} \, x\right ) - \sin \left (\frac{1}{2} \, x\right )\right )} \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right )\right )} \cos \left (\frac{1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right )\right ) -{\left ({\left (\cos \left (\frac{3}{2} \, x\right ) - \cos \left (\frac{1}{2} \, x\right ) - \sin \left (\frac{3}{2} \, x\right ) - \sin \left (\frac{1}{2} \, x\right )\right )} \cos \left (\frac{1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right ) +{\left (\cos \left (\frac{3}{2} \, x\right ) - \cos \left (\frac{1}{2} \, x\right ) + \sin \left (\frac{3}{2} \, x\right ) + \sin \left (\frac{1}{2} \, x\right )\right )} \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right )\right )} \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right )\right )}{{\left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right )}^{\frac{1}{4}}{\left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right )}^{\frac{1}{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cos(x)*cot(x))^(1/2),x, algorithm="maxima")

[Out]

(((cos(3/2*x) - cos(1/2*x) + sin(3/2*x) + sin(1/2*x))*cos(1/2*arctan2(sin(x), cos(x) - 1)) - (cos(3/2*x) - cos
(1/2*x) - sin(3/2*x) - sin(1/2*x))*sin(1/2*arctan2(sin(x), cos(x) - 1)))*cos(1/2*arctan2(sin(x), cos(x) + 1))
- ((cos(3/2*x) - cos(1/2*x) - sin(3/2*x) - sin(1/2*x))*cos(1/2*arctan2(sin(x), cos(x) - 1)) + (cos(3/2*x) - co
s(1/2*x) + sin(3/2*x) + sin(1/2*x))*sin(1/2*arctan2(sin(x), cos(x) - 1)))*sin(1/2*arctan2(sin(x), cos(x) + 1))
)/((cos(x)^2 + sin(x)^2 + 2*cos(x) + 1)^(1/4)*(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1)^(1/4))

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Fricas [A]  time = 2.38723, size = 53, normalized size = 4.08 \begin{align*} \frac{2 \, \sqrt{\frac{\cos \left (x\right )^{2}}{\sin \left (x\right )}} \sin \left (x\right )}{\cos \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cos(x)*cot(x))^(1/2),x, algorithm="fricas")

[Out]

2*sqrt(cos(x)^2/sin(x))*sin(x)/cos(x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\cos{\left (x \right )} \cot{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cos(x)*cot(x))**(1/2),x)

[Out]

Integral(sqrt(cos(x)*cot(x)), x)

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Giac [A]  time = 1.14641, size = 16, normalized size = 1.23 \begin{align*} 2 \, \mathrm{sgn}\left (\cos \left (x\right )\right ) \mathrm{sgn}\left (\sin \left (x\right )\right ) \sqrt{\sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cos(x)*cot(x))^(1/2),x, algorithm="giac")

[Out]

2*sgn(cos(x))*sgn(sin(x))*sqrt(sin(x))