3.151 \(\int (\cos (x) \cot (x))^{3/2} \, dx\)

Optimal. Leaf size=31 \[ \frac {2}{3} \cos (x) \sqrt {\cos (x) \cot (x)}-\frac {8}{3} \sec (x) \sqrt {\cos (x) \cot (x)} \]

[Out]

2/3*cos(x)*(cos(x)*cot(x))^(1/2)-8/3*sec(x)*(cos(x)*cot(x))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4400, 2598, 2589} \[ \frac {2}{3} \cos (x) \sqrt {\cos (x) \cot (x)}-\frac {8}{3} \sec (x) \sqrt {\cos (x) \cot (x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Cos[x]*Sqrt[Cos[x]*Cot[x]])/3 - (8*Sqrt[Cos[x]*Cot[x]]*Sec[x])/3

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]

Rule 2598

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] + Dist[(a^2*(m + n - 1))/m, Int[(a*Sin[e + f*x])^(m - 2)*(b*Ta
n[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && (GtQ[m, 1] || (EqQ[m, 1] && EqQ[n, 1/2])) && IntegersQ[2
*m, 2*n]

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])

Rubi steps

\begin {align*} \int (\cos (x) \cot (x))^{3/2} \, dx &=\frac {\sqrt {\cos (x) \cot (x)} \int \cos ^{\frac {3}{2}}(x) \cot ^{\frac {3}{2}}(x) \, dx}{\sqrt {\cos (x)} \sqrt {\cot (x)}}\\ &=\frac {2}{3} \cos (x) \sqrt {\cos (x) \cot (x)}+\frac {\left (4 \sqrt {\cos (x) \cot (x)}\right ) \int \frac {\cot ^{\frac {3}{2}}(x)}{\sqrt {\cos (x)}} \, dx}{3 \sqrt {\cos (x)} \sqrt {\cot (x)}}\\ &=\frac {2}{3} \cos (x) \sqrt {\cos (x) \cot (x)}-\frac {8}{3} \sqrt {\cos (x) \cot (x)} \sec (x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 21, normalized size = 0.68 \[ \frac {2}{3} \left (\cos ^2(x)-4\right ) \sec (x) \sqrt {\cos (x) \cot (x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-4 + Cos[x]^2)*Sqrt[Cos[x]*Cot[x]]*Sec[x])/3

________________________________________________________________________________________

fricas [A]  time = 0.73, size = 23, normalized size = 0.74 \[ \frac {2 \, {\left (\cos \relax (x)^{2} - 4\right )} \sqrt {\frac {\cos \relax (x)^{2}}{\sin \relax (x)}}}{3 \, \cos \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(cos(x)^2 - 4)*sqrt(cos(x)^2/sin(x))/cos(x)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 19, normalized size = 0.61 \[ -\frac {2}{3} \, {\left (\sin \relax (x)^{\frac {3}{2}} + \frac {3}{\sqrt {\sin \relax (x)}}\right )} \mathrm {sgn}\left (\cos \relax (x)\right ) \mathrm {sgn}\left (\sin \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(sin(x)^(3/2) + 3/sqrt(sin(x)))*sgn(cos(x))*sgn(sin(x))

________________________________________________________________________________________

maple [A]  time = 0.25, size = 26, normalized size = 0.84 \[ \frac {2 \left (\cos ^{2}\relax (x )-4\right ) \left (\frac {\cos ^{2}\relax (x )}{\sin \relax (x )}\right )^{\frac {3}{2}} \sin \relax (x )}{3 \cos \relax (x )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(cos(x)^2-4)*(cos(x)^2/sin(x))^(3/2)*sin(x)/cos(x)^3

________________________________________________________________________________________

maxima [B]  time = 0.54, size = 314, normalized size = 10.13 \[ \frac {{\left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right )}^{\frac {1}{4}} {\left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right )}^{\frac {1}{4}} {\left ({\left ({\left (\cos \left (\frac {9}{2} \, x\right ) - 15 \, \cos \left (\frac {5}{2} \, x\right ) - \cos \left (\frac {3}{2} \, x\right ) + 15 \, \cos \left (\frac {1}{2} \, x\right ) - \sin \left (\frac {9}{2} \, x\right ) + 15 \, \sin \left (\frac {5}{2} \, x\right ) - \sin \left (\frac {3}{2} \, x\right ) - 15 \, \sin \left (\frac {1}{2} \, x\right )\right )} \cos \left (\frac {3}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right )\right ) + {\left (\cos \left (\frac {9}{2} \, x\right ) - 15 \, \cos \left (\frac {5}{2} \, x\right ) - \cos \left (\frac {3}{2} \, x\right ) + 15 \, \cos \left (\frac {1}{2} \, x\right ) + \sin \left (\frac {9}{2} \, x\right ) - 15 \, \sin \left (\frac {5}{2} \, x\right ) + \sin \left (\frac {3}{2} \, x\right ) + 15 \, \sin \left (\frac {1}{2} \, x\right )\right )} \sin \left (\frac {3}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right )\right )\right )} \cos \left (\frac {3}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right )\right ) + {\left ({\left (\cos \left (\frac {9}{2} \, x\right ) - 15 \, \cos \left (\frac {5}{2} \, x\right ) - \cos \left (\frac {3}{2} \, x\right ) + 15 \, \cos \left (\frac {1}{2} \, x\right ) + \sin \left (\frac {9}{2} \, x\right ) - 15 \, \sin \left (\frac {5}{2} \, x\right ) + \sin \left (\frac {3}{2} \, x\right ) + 15 \, \sin \left (\frac {1}{2} \, x\right )\right )} \cos \left (\frac {3}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right )\right ) - {\left (\cos \left (\frac {9}{2} \, x\right ) - 15 \, \cos \left (\frac {5}{2} \, x\right ) - \cos \left (\frac {3}{2} \, x\right ) + 15 \, \cos \left (\frac {1}{2} \, x\right ) - \sin \left (\frac {9}{2} \, x\right ) + 15 \, \sin \left (\frac {5}{2} \, x\right ) - \sin \left (\frac {3}{2} \, x\right ) - 15 \, \sin \left (\frac {1}{2} \, x\right )\right )} \sin \left (\frac {3}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right )\right )\right )} \sin \left (\frac {3}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right )\right )\right )}}{6 \, {\left (\cos \relax (x)^{4} + \sin \relax (x)^{4} + 2 \, {\left (\cos \relax (x)^{2} + 1\right )} \sin \relax (x)^{2} - 2 \, \cos \relax (x)^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1)^(1/4)*(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1)^(1/4)*(((cos(9/2*x) - 15*c
os(5/2*x) - cos(3/2*x) + 15*cos(1/2*x) - sin(9/2*x) + 15*sin(5/2*x) - sin(3/2*x) - 15*sin(1/2*x))*cos(3/2*arct
an2(sin(x), cos(x) - 1)) + (cos(9/2*x) - 15*cos(5/2*x) - cos(3/2*x) + 15*cos(1/2*x) + sin(9/2*x) - 15*sin(5/2*
x) + sin(3/2*x) + 15*sin(1/2*x))*sin(3/2*arctan2(sin(x), cos(x) - 1)))*cos(3/2*arctan2(sin(x), cos(x) + 1)) +
((cos(9/2*x) - 15*cos(5/2*x) - cos(3/2*x) + 15*cos(1/2*x) + sin(9/2*x) - 15*sin(5/2*x) + sin(3/2*x) + 15*sin(1
/2*x))*cos(3/2*arctan2(sin(x), cos(x) - 1)) - (cos(9/2*x) - 15*cos(5/2*x) - cos(3/2*x) + 15*cos(1/2*x) - sin(9
/2*x) + 15*sin(5/2*x) - sin(3/2*x) - 15*sin(1/2*x))*sin(3/2*arctan2(sin(x), cos(x) - 1)))*sin(3/2*arctan2(sin(
x), cos(x) + 1)))/(cos(x)^4 + sin(x)^4 + 2*(cos(x)^2 + 1)*sin(x)^2 - 2*cos(x)^2 + 1)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (\cos \relax (x)\,\mathrm {cot}\relax (x)\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((cos(x)*cot(x))^(3/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\cos {\relax (x )} \cot {\relax (x )}\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((cos(x)*cot(x))**(3/2), x)

________________________________________________________________________________________