3.633 \(\int \frac {\cot (x) \csc (x)}{\sqrt {\sin (2 x)}} \, dx\)

Optimal. Leaf size=16 \[ -\frac {2 \cos (x) \cot (x)}{3 \sqrt {\sin (2 x)}} \]

[Out]

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

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Rubi [A]  time = 0.09, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4390, 30} \[ -\frac {2 \cos (x) \cot (x)}{3 \sqrt {\sin (2 x)}} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[x]*Csc[x])/Sqrt[Sin[2*x]],x]

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 4390

Int[(u_)*((c_.)*sin[v_])^(m_), x_Symbol] :> With[{w = FunctionOfTrig[(u*Sin[v/2]^(2*m))/(c*Tan[v/2])^m, x]}, D
ist[((c*Sin[v])^m*(c*Tan[v/2])^m)/Sin[v/2]^(2*m), Int[(u*Sin[v/2]^(2*m))/(c*Tan[v/2])^m, x], x] /;  !FalseQ[w]
&& FunctionOfQ[NonfreeFactors[Tan[w], x], (u*Sin[v/2]^(2*m))/(c*Tan[v/2])^m, x]] /; FreeQ[c, x] && LinearQ[v,
x] && IntegerQ[m + 1/2] &&  !SumQ[u] && InverseFunctionFreeQ[u, x]

Rubi steps

\begin {align*} \int \frac {\cot (x) \csc (x)}{\sqrt {\sin (2 x)}} \, dx &=\frac {\sin (x) \int \frac {\csc ^2(x)}{\sqrt {\tan (x)}} \, dx}{\sqrt {\sin (2 x)} \sqrt {\tan (x)}}\\ &=\frac {\sin (x) \operatorname {Subst}\left (\int \frac {1}{x^{5/2}} \, dx,x,\tan (x)\right )}{\sqrt {\sin (2 x)} \sqrt {\tan (x)}}\\ &=-\frac {2 \cos (x) \cot (x)}{3 \sqrt {\sin (2 x)}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 16, normalized size = 1.00 \[ -\frac {1}{3} \sqrt {\sin (2 x)} \cot (x) \csc (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[x]*Csc[x])/Sqrt[Sin[2*x]],x]

[Out]

-1/3*(Cot[x]*Csc[x]*Sqrt[Sin[2*x]])

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fricas [B]  time = 1.87, size = 29, normalized size = 1.81 \[ \frac {\sqrt {2} \sqrt {\cos \relax (x) \sin \relax (x)} \cos \relax (x) + \cos \relax (x)^{2} - 1}{3 \, {\left (\cos \relax (x)^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(sqrt(2)*sqrt(cos(x)*sin(x))*cos(x) + cos(x)^2 - 1)/(cos(x)^2 - 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot \relax (x) \csc \relax (x)}{\sqrt {\sin \left (2 \, x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cot(x)*csc(x)/sqrt(sin(2*x)), x)

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maple [C]  time = 0.21, size = 119, normalized size = 7.44 \[ \frac {\sqrt {-\frac {\tan \left (\frac {x}{2}\right )}{\tan ^{2}\left (\frac {x}{2}\right )-1}}\, \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \left (4 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticF \left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \tan \left (\frac {x}{2}\right )+\tan ^{4}\left (\frac {x}{2}\right )-1\right )}{6 \tan \left (\frac {x}{2}\right ) \sqrt {\left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\, \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)*csc(x)/sin(2*x)^(1/2),x)

[Out]

1/6*(-tan(1/2*x)/(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)^2-1)/tan(1/2*x)*(4*(1+tan(1/2*x))^(1/2)*(-2*tan(1/2*x)+2)
^(1/2)*(-tan(1/2*x))^(1/2)*EllipticF((1+tan(1/2*x))^(1/2),1/2*2^(1/2))*tan(1/2*x)+tan(1/2*x)^4-1)/((tan(1/2*x)
^2-1)*tan(1/2*x))^(1/2)/(tan(1/2*x)^3-tan(1/2*x))^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot \relax (x) \csc \relax (x)}{\sqrt {\sin \left (2 \, x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)*csc(x)/sin(2*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(cot(x)*csc(x)/sqrt(sin(2*x)), x)

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mupad [B]  time = 3.10, size = 14, normalized size = 0.88 \[ -\frac {\sqrt {\sin \left (2\,x\right )}\,\cos \relax (x)}{3\,{\sin \relax (x)}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/(sin(2*x)^(1/2)*sin(x)),x)

[Out]

-(sin(2*x)^(1/2)*cos(x))/(3*sin(x)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cot(x)*csc(x)/sqrt(sin(2*x)), x)

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