3.738 \(\int \frac{\cot (6 x) \csc (6 x)}{(5-11 \csc ^2(6 x))^2} \, dx\)

Optimal. Leaf size=43 \[ \frac{\sin (6 x)}{60 \left (11-5 \sin ^2(6 x)\right )}-\frac{\tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sin (6 x)\right )}{60 \sqrt{55}} \]

[Out]

-ArcTanh[Sqrt[5/11]*Sin[6*x]]/(60*Sqrt[55]) + Sin[6*x]/(60*(11 - 5*Sin[6*x]^2))

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Rubi [A]  time = 0.0570532, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4338, 288, 206} \[ \frac{\sin (6 x)}{60 \left (11-5 \sin ^2(6 x)\right )}-\frac{\tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sin (6 x)\right )}{60 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[6*x]*Csc[6*x])/(5 - 11*Csc[6*x]^2)^2,x]

[Out]

-ArcTanh[Sqrt[5/11]*Sin[6*x]]/(60*Sqrt[55]) + Sin[6*x]/(60*(11 - 5*Sin[6*x]^2))

Rule 4338

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[1/(b
*c), Subst[Int[SubstFor[1/x, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a
+ b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cot] || EqQ[F, cot])

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\cot (6 x) \csc (6 x)}{\left (5-11 \csc ^2(6 x)\right )^2} \, dx &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{x^2}{\left (11-5 x^2\right )^2} \, dx,x,\sin (6 x)\right )\\ &=\frac{\sin (6 x)}{60 \left (11-5 \sin ^2(6 x)\right )}-\frac{1}{60} \operatorname{Subst}\left (\int \frac{1}{11-5 x^2} \, dx,x,\sin (6 x)\right )\\ &=-\frac{\tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sin (6 x)\right )}{60 \sqrt{55}}+\frac{\sin (6 x)}{60 \left (11-5 \sin ^2(6 x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.680499, size = 41, normalized size = 0.95 \[ \frac{\sin (6 x)}{30 (5 \cos (12 x)+17)}-\frac{\tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sin (6 x)\right )}{60 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[6*x]*Csc[6*x])/(5 - 11*Csc[6*x]^2)^2,x]

[Out]

-ArcTanh[Sqrt[5/11]*Sin[6*x]]/(60*Sqrt[55]) + Sin[6*x]/(30*(17 + 5*Cos[12*x]))

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Maple [A]  time = 0.033, size = 35, normalized size = 0.8 \begin{align*}{\frac{\csc \left ( 6\,x \right ) }{660\, \left ( \csc \left ( 6\,x \right ) \right ) ^{2}-300}}-{\frac{\sqrt{55}}{3300}{\it Artanh} \left ({\frac{\csc \left ( 6\,x \right ) \sqrt{55}}{5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(6*x)*csc(6*x)/(5-11*csc(6*x)^2)^2,x)

[Out]

1/60*csc(6*x)/(11*csc(6*x)^2-5)-1/3300*55^(1/2)*arctanh(1/5*csc(6*x)*55^(1/2))

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Maxima [A]  time = 1.47708, size = 66, normalized size = 1.53 \begin{align*} \frac{1}{6600} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sin \left (6 \, x\right )}{\sqrt{55} + 5 \, \sin \left (6 \, x\right )}\right ) - \frac{\sin \left (6 \, x\right )}{60 \,{\left (5 \, \sin \left (6 \, x\right )^{2} - 11\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(6*x)*csc(6*x)/(5-11*csc(6*x)^2)^2,x, algorithm="maxima")

[Out]

1/6600*sqrt(55)*log(-(sqrt(55) - 5*sin(6*x))/(sqrt(55) + 5*sin(6*x))) - 1/60*sin(6*x)/(5*sin(6*x)^2 - 11)

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Fricas [B]  time = 2.16254, size = 200, normalized size = 4.65 \begin{align*} \frac{{\left (5 \, \sqrt{55} \cos \left (6 \, x\right )^{2} + 6 \, \sqrt{55}\right )} \log \left (-\frac{5 \, \cos \left (6 \, x\right )^{2} + 2 \, \sqrt{55} \sin \left (6 \, x\right ) - 16}{5 \, \cos \left (6 \, x\right )^{2} + 6}\right ) + 110 \, \sin \left (6 \, x\right )}{6600 \,{\left (5 \, \cos \left (6 \, x\right )^{2} + 6\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(6*x)*csc(6*x)/(5-11*csc(6*x)^2)^2,x, algorithm="fricas")

[Out]

1/6600*((5*sqrt(55)*cos(6*x)^2 + 6*sqrt(55))*log(-(5*cos(6*x)^2 + 2*sqrt(55)*sin(6*x) - 16)/(5*cos(6*x)^2 + 6)
) + 110*sin(6*x))/(5*cos(6*x)^2 + 6)

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Sympy [B]  time = 2.60155, size = 151, normalized size = 3.51 \begin{align*} \frac{11 \sqrt{55} \log{\left (\csc{\left (6 x \right )} - \frac{\sqrt{55}}{11} \right )} \csc ^{2}{\left (6 x \right )}}{72600 \csc ^{2}{\left (6 x \right )} - 33000} - \frac{5 \sqrt{55} \log{\left (\csc{\left (6 x \right )} - \frac{\sqrt{55}}{11} \right )}}{72600 \csc ^{2}{\left (6 x \right )} - 33000} - \frac{11 \sqrt{55} \log{\left (\csc{\left (6 x \right )} + \frac{\sqrt{55}}{11} \right )} \csc ^{2}{\left (6 x \right )}}{72600 \csc ^{2}{\left (6 x \right )} - 33000} + \frac{5 \sqrt{55} \log{\left (\csc{\left (6 x \right )} + \frac{\sqrt{55}}{11} \right )}}{72600 \csc ^{2}{\left (6 x \right )} - 33000} + \frac{110 \csc{\left (6 x \right )}}{72600 \csc ^{2}{\left (6 x \right )} - 33000} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(6*x)*csc(6*x)/(5-11*csc(6*x)**2)**2,x)

[Out]

11*sqrt(55)*log(csc(6*x) - sqrt(55)/11)*csc(6*x)**2/(72600*csc(6*x)**2 - 33000) - 5*sqrt(55)*log(csc(6*x) - sq
rt(55)/11)/(72600*csc(6*x)**2 - 33000) - 11*sqrt(55)*log(csc(6*x) + sqrt(55)/11)*csc(6*x)**2/(72600*csc(6*x)**
2 - 33000) + 5*sqrt(55)*log(csc(6*x) + sqrt(55)/11)/(72600*csc(6*x)**2 - 33000) + 110*csc(6*x)/(72600*csc(6*x)
**2 - 33000)

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Giac [A]  time = 1.16217, size = 65, normalized size = 1.51 \begin{align*} \frac{1}{6600} \, \sqrt{55} \log \left (\frac{\sqrt{55} - 5 \, \sin \left (6 \, x\right )}{\sqrt{55} + 5 \, \sin \left (6 \, x\right )}\right ) - \frac{\sin \left (6 \, x\right )}{60 \,{\left (5 \, \sin \left (6 \, x\right )^{2} - 11\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(6*x)*csc(6*x)/(5-11*csc(6*x)^2)^2,x, algorithm="giac")

[Out]

1/6600*sqrt(55)*log((sqrt(55) - 5*sin(6*x))/(sqrt(55) + 5*sin(6*x))) - 1/60*sin(6*x)/(5*sin(6*x)^2 - 11)