\(\int \sqrt {\csc (x)-\sin (x)} \, dx\) [32]

Optimal result

Integrand size = 11, antiderivative size = 13 \[ \int \sqrt {\csc (x)-\sin (x)} \, dx=2 \sqrt {\cos (x) \cot (x)} \tan (x) \]

[Out]

2*(cos(x)*cot(x))^(1/2)*tan(x)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4482, 4485, 2669} \[ \int \sqrt {\csc (x)-\sin (x)} \, dx=2 \tan (x) \sqrt {\cos (x) \cot (x)} \]

[In]

Int[Sqrt[Csc[x] - Sin[x]],x]

[Out]

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

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 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 \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] (verified)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \sqrt {\csc (x)-\sin (x)} \, dx=2 \sqrt {\cos (x) \cot (x)} \tan (x) \]

[In]

Integrate[Sqrt[Csc[x] - Sin[x]],x]

[Out]

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

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92

method	result	size
default	\(2 \sqrt {\cos \left (x \right ) \cot \left (x \right )}\, \tan \left (x \right )\)	\(12\)
risch	\(-\frac {i \sqrt {2}\, \sqrt {\frac {i \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-i x}}{{\mathrm e}^{2 i x}-1}}\, \left ({\mathrm e}^{2 i x}-1\right )}{{\mathrm e}^{2 i x}+1}\)	\(51\)

[In]

int((csc(x)-sin(x))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2*(cos(x)*cot(x))^(1/2)*tan(x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.46 \[ \int \sqrt {\csc (x)-\sin (x)} \, dx=\frac {2 \, \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} \sin \left (x\right )}{\cos \left (x\right )} \]

[In]

integrate((csc(x)-sin(x))^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \sqrt {\csc (x)-\sin (x)} \, dx=\int \sqrt {- \sin {\left (x \right )} + \csc {\left (x \right )}}\, dx \]

[In]

integrate((csc(x)-sin(x))**(1/2),x)

[Out]

Integral(sqrt(-sin(x) + csc(x)), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (11) = 22\).

Time = 0.37 (sec) , antiderivative size = 188, normalized size of antiderivative = 14.46 \[ \int \sqrt {\csc (x)-\sin (x)} \, dx=\frac {{\left ({\left (\cos \left (\frac {3}{2} \, x\right ) - \cos \left (\frac {1}{2} \, x\right ) + \sin \left (\frac {3}{2} \, x\right ) + \sin \left (\frac {1}{2} \, x\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right ) - {\left (\cos \left (\frac {3}{2} \, x\right ) - \cos \left (\frac {1}{2} \, x\right ) - \sin \left (\frac {3}{2} \, x\right ) - \sin \left (\frac {1}{2} \, x\right )\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right )\right ) - {\left ({\left (\cos \left (\frac {3}{2} \, x\right ) - \cos \left (\frac {1}{2} \, x\right ) - \sin \left (\frac {3}{2} \, x\right ) - \sin \left (\frac {1}{2} \, x\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right ) + {\left (\cos \left (\frac {3}{2} \, x\right ) - \cos \left (\frac {1}{2} \, x\right ) + \sin \left (\frac {3}{2} \, x\right ) + \sin \left (\frac {1}{2} \, x\right )\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right )\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right )\right )}{{\left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right )}^{\frac {1}{4}} {\left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right )}^{\frac {1}{4}}} \]

[In]

integrate((csc(x)-sin(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))

Giac [F]

\[ \int \sqrt {\csc (x)-\sin (x)} \, dx=\int { \sqrt {\csc \left (x\right ) - \sin \left (x\right )} \,d x } \]

[In]

integrate((csc(x)-sin(x))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(csc(x) - sin(x)), x)

Mupad [B] (verification not implemented)

Time = 15.46 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \sqrt {\csc (x)-\sin (x)} \, dx=\frac {2\,\left |\cos \left (x\right )\right |}{\cos \left (x\right )\,\sqrt {\frac {1}{\sin \left (x\right )}}} \]

[In]

int((1/sin(x) - sin(x))^(1/2),x)

[Out]

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