\(\int \frac {\cot (6 x) \csc (6 x)}{(5-11 \csc ^2(6 x))^2} \, dx\) [738]

Optimal result

Integrand size = 21, antiderivative size = 43 \[ \int \frac {\cot (6 x) \csc (6 x)}{\left (5-11 \csc ^2(6 x)\right )^2} \, dx=-\frac {\text {arctanh}\left (\sqrt {\frac {5}{11}} \sin (6 x)\right )}{60 \sqrt {55}}+\frac {\sin (6 x)}{60 \left (11-5 \sin ^2(6 x)\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4423, 294, 212} \[ \int \frac {\cot (6 x) \csc (6 x)}{\left (5-11 \csc ^2(6 x)\right )^2} \, dx=\frac {\sin (6 x)}{60 \left (11-5 \sin ^2(6 x)\right )}-\frac {\text {arctanh}\left (\sqrt {\frac {5}{11}} \sin (6 x)\right )}{60 \sqrt {55}} \]

[In]

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

[Out]

-1/60*ArcTanh[Sqrt[5/11]*Sin[6*x]]/Sqrt[55] + Sin[6*x]/(60*(11 - 5*Sin[6*x]^2))

Rule 212

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

Rule 294

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 4423

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])

Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \text {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} \text {Subst}\left (\int \frac {1}{11-5 x^2} \, dx,x,\sin (6 x)\right ) \\ & = -\frac {\text {arctanh}\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] (verified)

Time = 0.37 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.95 \[ \int \frac {\cot (6 x) \csc (6 x)}{\left (5-11 \csc ^2(6 x)\right )^2} \, dx=-\frac {\text {arctanh}\left (\sqrt {\frac {5}{11}} \sin (6 x)\right )}{60 \sqrt {55}}+\frac {\sin (6 x)}{30 (17+5 \cos (12 x))} \]

[In]

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

[Out]

-1/60*ArcTanh[Sqrt[5/11]*Sin[6*x]]/Sqrt[55] + Sin[6*x]/(30*(17 + 5*Cos[12*x]))

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(\frac {\csc \left (6 x \right )}{660 \csc \left (6 x \right )^{2}-300}-\frac {\sqrt {55}\, \operatorname {arctanh}\left (\frac {\csc \left (6 x \right ) \sqrt {55}}{5}\right )}{3300}\)	\(35\)
default	\(\frac {\csc \left (6 x \right )}{660 \csc \left (6 x \right )^{2}-300}-\frac {\sqrt {55}\, \operatorname {arctanh}\left (\frac {\csc \left (6 x \right ) \sqrt {55}}{5}\right )}{3300}\)	\(35\)
risch	\(-\frac {i \left ({\mathrm e}^{18 i x}-{\mathrm e}^{6 i x}\right )}{30 \left (5 \,{\mathrm e}^{24 i x}+34 \,{\mathrm e}^{12 i x}+5\right )}-\frac {\sqrt {55}\, \ln \left ({\mathrm e}^{12 i x}+\frac {2 i \sqrt {55}\, {\mathrm e}^{6 i x}}{5}-1\right )}{6600}+\frac {\sqrt {55}\, \ln \left ({\mathrm e}^{12 i x}-\frac {2 i \sqrt {55}\, {\mathrm e}^{6 i x}}{5}-1\right )}{6600}\)	\(84\)

[In]

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

[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))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (34) = 68\).

Time = 0.25 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.70 \[ \int \frac {\cot (6 x) \csc (6 x)}{\left (5-11 \csc ^2(6 x)\right )^2} \, dx=\frac {{\left (5 \, \sqrt {55} \cos \left (6 \, x\right )^{2} + 6 \, \sqrt {55}\right )} \log \left (-\frac {5 \, \cos \left (6 \, x\right )^{2} + 2 \, \sqrt {55} \sin \left (6 \, x\right ) - 16}{5 \, \cos \left (6 \, x\right )^{2} + 6}\right ) + 110 \, \sin \left (6 \, x\right )}{6600 \, {\left (5 \, \cos \left (6 \, x\right )^{2} + 6\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (34) = 68\).

Time = 0.52 (sec) , antiderivative size = 151, normalized size of antiderivative = 3.51 \[ \int \frac {\cot (6 x) \csc (6 x)}{\left (5-11 \csc ^2(6 x)\right )^2} \, dx=\frac {11 \sqrt {55} \log {\left (\csc {\left (6 x \right )} - \frac {\sqrt {55}}{11} \right )} \csc ^{2}{\left (6 x \right )}}{72600 \csc ^{2}{\left (6 x \right )} - 33000} - \frac {5 \sqrt {55} \log {\left (\csc {\left (6 x \right )} - \frac {\sqrt {55}}{11} \right )}}{72600 \csc ^{2}{\left (6 x \right )} - 33000} - \frac {11 \sqrt {55} \log {\left (\csc {\left (6 x \right )} + \frac {\sqrt {55}}{11} \right )} \csc ^{2}{\left (6 x \right )}}{72600 \csc ^{2}{\left (6 x \right )} - 33000} + \frac {5 \sqrt {55} \log {\left (\csc {\left (6 x \right )} + \frac {\sqrt {55}}{11} \right )}}{72600 \csc ^{2}{\left (6 x \right )} - 33000} + \frac {110 \csc {\left (6 x \right )}}{72600 \csc ^{2}{\left (6 x \right )} - 33000} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.14 \[ \int \frac {\cot (6 x) \csc (6 x)}{\left (5-11 \csc ^2(6 x)\right )^2} \, dx=\frac {1}{6600} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sin \left (6 \, x\right )}{\sqrt {55} + 5 \, \sin \left (6 \, x\right )}\right ) - \frac {\sin \left (6 \, x\right )}{60 \, {\left (5 \, \sin \left (6 \, x\right )^{2} - 11\right )}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.12 \[ \int \frac {\cot (6 x) \csc (6 x)}{\left (5-11 \csc ^2(6 x)\right )^2} \, dx=\frac {1}{6600} \, \sqrt {55} \log \left (\frac {\sqrt {55} - 5 \, \sin \left (6 \, x\right )}{\sqrt {55} + 5 \, \sin \left (6 \, x\right )}\right ) - \frac {\sin \left (6 \, x\right )}{60 \, {\left (5 \, \sin \left (6 \, x\right )^{2} - 11\right )}} \]

[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)

Mupad [B] (verification not implemented)

Time = 27.44 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.33 \[ \int \frac {\cot (6 x) \csc (6 x)}{\left (5-11 \csc ^2(6 x)\right )^2} \, dx=-\frac {55\,\sin \left (6\,x\right )-11\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sin \left (6\,x\right )}{11}\right )+5\,\sqrt {55}\,{\sin \left (6\,x\right )}^2\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sin \left (6\,x\right )}{11}\right )}{16500\,{\sin \left (6\,x\right )}^2-36300} \]

[In]

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

[Out]

-(55*sin(6*x) - 11*55^(1/2)*atanh((55^(1/2)*sin(6*x))/11) + 5*55^(1/2)*sin(6*x)^2*atanh((55^(1/2)*sin(6*x))/11
))/(16500*sin(6*x)^2 - 36300)