3.317 \(\int \sqrt {\csc (x)-\sin (x)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4397, 4400, 2589} \[ 2 \tan (x) \sqrt {\cos (x) \cot (x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Csc[x] - Sin[x]],x]

[Out]

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

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 4397

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

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 \sqrt {\csc (x)-\sin (x)} \, dx &=\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*}

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Mathematica [A]  time = 0.02, size = 13, normalized size = 1.00 \[ 2 \tan (x) \sqrt {\cos (x) \cot (x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Csc[x] - Sin[x]],x]

[Out]

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

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fricas [A]  time = 0.77, size = 19, normalized size = 1.46 \[ \frac {2 \, \sqrt {\frac {\cos \relax (x)^{2}}{\sin \relax (x)}} \sin \relax (x)}{\cos \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\csc \relax (x) - \sin \relax (x)}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(csc(x) - sin(x)), x)

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maple [A]  time = 0.23, size = 20, normalized size = 1.54 \[ \frac {2 \sin \relax (x ) \sqrt {\frac {\cos ^{2}\relax (x )}{\sin \relax (x )}}}{\cos \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.51, size = 188, normalized size = 14.46 \[ \frac {{\left ({\left (\cos \left (\frac {3}{2} \, x\right ) - \cos \left (\frac {1}{2} \, x\right ) + \sin \left (\frac {3}{2} \, x\right ) + \sin \left (\frac {1}{2} \, x\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right )\right ) - {\left (\cos \left (\frac {3}{2} \, x\right ) - \cos \left (\frac {1}{2} \, x\right ) - \sin \left (\frac {3}{2} \, x\right ) - \sin \left (\frac {1}{2} \, x\right )\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right )\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right )\right ) - {\left ({\left (\cos \left (\frac {3}{2} \, x\right ) - \cos \left (\frac {1}{2} \, x\right ) - \sin \left (\frac {3}{2} \, x\right ) - \sin \left (\frac {1}{2} \, x\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right )\right ) + {\left (\cos \left (\frac {3}{2} \, x\right ) - \cos \left (\frac {1}{2} \, x\right ) + \sin \left (\frac {3}{2} \, x\right ) + \sin \left (\frac {1}{2} \, x\right )\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right )\right )\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right )\right )}{{\left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right )}^{\frac {1}{4}} {\left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right )}^{\frac {1}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 2.48, size = 15, normalized size = 1.15 \[ \frac {2\,\left |\cos \relax (x)\right |}{\cos \relax (x)\,\sqrt {\frac {1}{\sin \relax (x)}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/sin(x) - sin(x))^(1/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- \sin {\relax (x )} + \csc {\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-sin(x) + csc(x)), x)

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