Integrand size = 9, antiderivative size = 31 \[ \int (\cos (x) \cot (x))^{3/2} \, dx=\frac {2}{3} \cos (x) \sqrt {\cos (x) \cot (x)}-\frac {8}{3} \sqrt {\cos (x) \cot (x)} \sec (x) \]
Time = 0.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.68 \[ \int (\cos (x) \cot (x))^{3/2} \, dx=\frac {2}{3} \left (-4+\cos ^2(x)\right ) \sqrt {\cos (x) \cot (x)} \sec (x) \]
Time = 0.33 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.77, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 4900, 3042, 3078, 3042, 3069}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (\cos (x) \cot (x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (\cos (x) \cot (x))^{3/2}dx\) |
\(\Big \downarrow \) 4900 |
\(\displaystyle \frac {\sqrt {\cos (x) \cot (x)} \int \cos ^{\frac {3}{2}}(x) \cot ^{\frac {3}{2}}(x)dx}{\sqrt {\cos (x)} \sqrt {\cot (x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\cos (x) \cot (x)} \int \sin \left (x+\frac {\pi }{2}\right )^{3/2} \left (-\tan \left (x+\frac {\pi }{2}\right )\right )^{3/2}dx}{\sqrt {\cos (x)} \sqrt {\cot (x)}}\) |
\(\Big \downarrow \) 3078 |
\(\displaystyle \frac {\sqrt {\cos (x) \cot (x)} \left (\frac {4}{3} \int \frac {\cot ^{\frac {3}{2}}(x)}{\sqrt {\cos (x)}}dx+\frac {2}{3} \cos ^{\frac {3}{2}}(x) \sqrt {\cot (x)}\right )}{\sqrt {\cos (x)} \sqrt {\cot (x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\cos (x) \cot (x)} \left (\frac {4}{3} \int \frac {\left (-\tan \left (x+\frac {\pi }{2}\right )\right )^{3/2}}{\sqrt {\sin \left (x+\frac {\pi }{2}\right )}}dx+\frac {2}{3} \cos ^{\frac {3}{2}}(x) \sqrt {\cot (x)}\right )}{\sqrt {\cos (x)} \sqrt {\cot (x)}}\) |
\(\Big \downarrow \) 3069 |
\(\displaystyle \frac {\left (\frac {2}{3} \cos ^{\frac {3}{2}}(x) \sqrt {\cot (x)}-\frac {8 \sqrt {\cot (x)}}{3 \sqrt {\cos (x)}}\right ) \sqrt {\cos (x) \cot (x)}}{\sqrt {\cos (x)} \sqrt {\cot (x)}}\) |
(((-8*Sqrt[Cot[x]])/(3*Sqrt[Cos[x]]) + (2*Cos[x]^(3/2)*Sqrt[Cot[x]])/3)*Sq rt[Cos[x]*Cot[x]])/(Sqrt[Cos[x]]*Sqrt[Cot[x]])
3.2.51.3.1 Defintions of rubi rules used
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]
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] + Simp[a^2*((m + n - 1)/m) Int[(a*Sin[e + f*x])^(m - 2)*(b*Tan[ 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]
Int[(u_.)*((v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> With[{uu = ActivateTri g[u], vv = ActivateTrig[v], ww = ActivateTrig[w]}, Simp[(vv^m*ww^n)^FracPar t[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])
Time = 1.66 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55
method | result | size |
default | \(\frac {2 \sqrt {\cot \left (x \right ) \cos \left (x \right )}\, \left (\cos \left (x \right )-4 \sec \left (x \right )\right )}{3}\) | \(17\) |
Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int (\cos (x) \cot (x))^{3/2} \, dx=\frac {2 \, {\left (\cos \left (x\right )^{2} - 4\right )} \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}}}{3 \, \cos \left (x\right )} \]
\[ \int (\cos (x) \cot (x))^{3/2} \, dx=\int \left (\cos {\left (x \right )} \cot {\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (23) = 46\).
Time = 0.36 (sec) , antiderivative size = 314, normalized size of antiderivative = 10.13 \[ \int (\cos (x) \cot (x))^{3/2} \, dx=\frac {{\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}} {\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 \left (x\right ), \cos \left (x\right ) - 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 \left (x\right ), \cos \left (x\right ) - 1\right )\right )\right )} \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 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 \left (x\right ), \cos \left (x\right ) - 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 \left (x\right ), \cos \left (x\right ) - 1\right )\right )\right )} \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right )\right )\right )}}{6 \, {\left (\cos \left (x\right )^{4} + \sin \left (x\right )^{4} + 2 \, {\left (\cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )^{2} - 2 \, \cos \left (x\right )^{2} + 1\right )}} \]
1/6*(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1)^(1/4)*(cos(x)^2 + sin(x)^2 - 2*co s(x) + 1)^(1/4)*(((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*arctan 2(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)) + ((co s(9/2*x) - 15*cos(5/2*x) - cos(3/2*x) + 15*cos(1/2*x) + sin(9/2*x) - 15*si n(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)
Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.61 \[ \int (\cos (x) \cot (x))^{3/2} \, dx=-\frac {2}{3} \, {\left (\sin \left (x\right )^{\frac {3}{2}} + \frac {3}{\sqrt {\sin \left (x\right )}}\right )} \mathrm {sgn}\left (\cos \left (x\right )\right ) \mathrm {sgn}\left (\sin \left (x\right )\right ) \]
Timed out. \[ \int (\cos (x) \cot (x))^{3/2} \, dx=\int {\left (\cos \left (x\right )\,\mathrm {cot}\left (x\right )\right )}^{3/2} \,d x \]