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

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Rubi [A]  time = 0.0868706, antiderivative size = 16, normalized size of antiderivative = 1., 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 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]

Rule 30

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

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*}

Mathematica [A]  time = 0.0331401, size = 16, normalized size = 1. \[ -\frac{1}{3} \sqrt{\sin (2 x)} \cot (x) \csc (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.09, size = 119, normalized size = 7.4 \begin{align*}{\frac{1}{6}\sqrt{-{\tan \left ({\frac{x}{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-1 \right ) ^{-1}}} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-1 \right ) \left ( 4\,\sqrt{1+\tan \left ( x/2 \right ) }\sqrt{-2\,\tan \left ( x/2 \right ) +2}\sqrt{-\tan \left ( x/2 \right ) }{\it EllipticF} \left ( \sqrt{1+\tan \left ( x/2 \right ) },1/2\,\sqrt{2} \right ) \tan \left ( x/2 \right ) + \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{4}-1 \right ) \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{\tan \left ({\frac{x}{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-1 \right ) }}}{\frac{1}{\sqrt{ \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}-\tan \left ({\frac{x}{2}} \right ) }}}} \end{align*}

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)*
(tan(1/2*x)^2-1))^(1/2)/(tan(1/2*x)^3-tan(1/2*x))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (x\right ) \csc \left (x\right )}{\sqrt{\sin \left (2 \, x\right )}}\,{d x} \end{align*}

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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Fricas [B]  time = 2.51186, size = 97, normalized size = 6.06 \begin{align*} \frac{\sqrt{2} \sqrt{\cos \left (x\right ) \sin \left (x\right )} \cos \left (x\right ) + \cos \left (x\right )^{2} - 1}{3 \,{\left (\cos \left (x\right )^{2} - 1\right )}} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot{\left (x \right )} \csc{\left (x \right )}}{\sqrt{\sin{\left (2 x \right )}}}\, dx \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (x\right ) \csc \left (x\right )}{\sqrt{\sin \left (2 \, x\right )}}\,{d x} \end{align*}

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)