\(\int (\csc (x)-\sin (x))^{3/2} \, dx\) [316]

Optimal result

Integrand size = 11, antiderivative size = 31 \[ \int (\csc (x)-\sin (x))^{3/2} \, dx=\frac {2}{3} \cos (x) \sqrt {\cos (x) \cot (x)}-\frac {8}{3} \sqrt {\cos (x) \cot (x)} \sec (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4482, 4485, 2678, 2669} \[ \int (\csc (x)-\sin (x))^{3/2} \, dx=\frac {2}{3} \cos (x) \sqrt {\cos (x) \cot (x)}-\frac {8}{3} \sec (x) \sqrt {\cos (x) \cot (x)} \]

[In]

Int[(Csc[x] - Sin[x])^(3/2),x]

[Out]

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

Rule 2669

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 2678

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*
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]

Rule 4482

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]

Rule 4485

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*} \text {integral}& = \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] (verified)

Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.68 \[ \int (\csc (x)-\sin (x))^{3/2} \, dx=\frac {2}{3} \left (-4+\cos ^2(x)\right ) \sqrt {\cos (x) \cot (x)} \sec (x) \]

[In]

Integrate[(Csc[x] - Sin[x])^(3/2),x]

[Out]

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

Maple [A] (verified)

Time = 1.78 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55

method	result	size
default	\(\frac {2 \sqrt {\cot \left (x \right ) \cos \left (x \right )}\, \left (\cos \left (x \right )-4 \sec \left (x \right )\right )}{3}\)	\(17\)

[In]

int((csc(x)-sin(x))^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/3*(cot(x)*cos(x))^(1/2)*(cos(x)-4*sec(x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int (\csc (x)-\sin (x))^{3/2} \, dx=\frac {2 \, {\left (\cos \left (x\right )^{2} - 4\right )} \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}}}{3 \, \cos \left (x\right )} \]

[In]

integrate((csc(x)-sin(x))^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int (\csc (x)-\sin (x))^{3/2} \, dx=\int \left (- \sin {\left (x \right )} + \csc {\left (x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

integrate((csc(x)-sin(x))**(3/2),x)

[Out]

Integral((-sin(x) + csc(x))**(3/2), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (23) = 46\).

Time = 0.39 (sec) , antiderivative size = 314, normalized size of antiderivative = 10.13 \[ \int (\csc (x)-\sin (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 )}} \]

[In]

integrate((csc(x)-sin(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)

Giac [F]

\[ \int (\csc (x)-\sin (x))^{3/2} \, dx=\int { {\left (\csc \left (x\right ) - \sin \left (x\right )\right )}^{\frac {3}{2}} \,d x } \]

[In]

integrate((csc(x)-sin(x))^(3/2),x, algorithm="giac")

[Out]

integrate((csc(x) - sin(x))^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int (\csc (x)-\sin (x))^{3/2} \, dx=\int {\left (\frac {1}{\sin \left (x\right )}-\sin \left (x\right )\right )}^{3/2} \,d x \]

[In]

int((1/sin(x) - sin(x))^(3/2),x)

[Out]

int((1/sin(x) - sin(x))^(3/2), x)