3.114 \(\int \cos (x) \cot (6 x) \, dx\)

Optimal. Leaf size=38 \[ \cos (x)-\frac{1}{6} \tanh ^{-1}(\cos (x))-\frac{1}{6} \tanh ^{-1}(2 \cos (x))-\frac{\tanh ^{-1}\left (\frac{2 \cos (x)}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

[Out]

-ArcTanh[Cos[x]]/6 - ArcTanh[2*Cos[x]]/6 - ArcTanh[(2*Cos[x])/Sqrt[3]]/(2*Sqrt[3]) + Cos[x]

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Rubi [A]  time = 0.0712837, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {12, 2073, 207} \[ \cos (x)-\frac{1}{6} \tanh ^{-1}(\cos (x))-\frac{1}{6} \tanh ^{-1}(2 \cos (x))-\frac{\tanh ^{-1}\left (\frac{2 \cos (x)}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Cot[6*x],x]

[Out]

-ArcTanh[Cos[x]]/6 - ArcTanh[2*Cos[x]]/6 - ArcTanh[(2*Cos[x])/Sqrt[3]]/(2*Sqrt[3]) + Cos[x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2073

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(PP /. x ->
x^2)^p*Q^q, x], x] /;  !SumQ[NonfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x^2] && PolyQ[Q, x] && ILtQ[p,
0]

Rule 207

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

Rubi steps

\begin{align*} \int \cos (x) \cot (6 x) \, dx &=-\operatorname{Subst}\left (\int \frac{-1+18 x^2-48 x^4+32 x^6}{2 \left (3-19 x^2+32 x^4-16 x^6\right )} \, dx,x,\cos (x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{-1+18 x^2-48 x^4+32 x^6}{3-19 x^2+32 x^4-16 x^6} \, dx,x,\cos (x)\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (-2-\frac{1}{3 \left (-1+x^2\right )}-\frac{2}{-3+4 x^2}-\frac{2}{3 \left (-1+4 x^2\right )}\right ) \, dx,x,\cos (x)\right )\right )\\ &=\cos (x)+\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\cos (x)\right )+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-1+4 x^2} \, dx,x,\cos (x)\right )+\operatorname{Subst}\left (\int \frac{1}{-3+4 x^2} \, dx,x,\cos (x)\right )\\ &=-\frac{1}{6} \tanh ^{-1}(\cos (x))-\frac{1}{6} \tanh ^{-1}(2 \cos (x))-\frac{\tanh ^{-1}\left (\frac{2 \cos (x)}{\sqrt{3}}\right )}{2 \sqrt{3}}+\cos (x)\\ \end{align*}

Mathematica [B]  time = 0.0860789, size = 87, normalized size = 2.29 \[ \frac{1}{12} \left (12 \cos (x)+2 \log \left (\sin \left (\frac{x}{2}\right )\right )-2 \log \left (\cos \left (\frac{x}{2}\right )\right )+\log (1-2 \cos (x))-\log (2 \cos (x)+1)+2 \sqrt{3} \tanh ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right )-2}{\sqrt{3}}\right )-2 \sqrt{3} \tanh ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right )+2}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Cot[6*x],x]

[Out]

(2*Sqrt[3]*ArcTanh[(-2 + Tan[x/2])/Sqrt[3]] - 2*Sqrt[3]*ArcTanh[(2 + Tan[x/2])/Sqrt[3]] + 12*Cos[x] - 2*Log[Co
s[x/2]] + Log[1 - 2*Cos[x]] - Log[1 + 2*Cos[x]] + 2*Log[Sin[x/2]])/12

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Maple [A]  time = 0.2, size = 49, normalized size = 1.3 \begin{align*} -{\frac{\ln \left ( 1+\cos \left ( x \right ) \right ) }{12}}+{\frac{\ln \left ( -1+\cos \left ( x \right ) \right ) }{12}}-{\frac{\ln \left ( 1+2\,\cos \left ( x \right ) \right ) }{12}}+{\frac{\ln \left ( 2\,\cos \left ( x \right ) -1 \right ) }{12}}-{\frac{\sqrt{3}}{6}{\it Artanh} \left ({\frac{2\,\cos \left ( x \right ) \sqrt{3}}{3}} \right ) }+\cos \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*cot(6*x),x)

[Out]

-1/12*ln(1+cos(x))+1/12*ln(-1+cos(x))-1/12*ln(1+2*cos(x))+1/12*ln(2*cos(x)-1)-1/6*arctanh(2/3*cos(x)*3^(1/2))*
3^(1/2)+cos(x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \cos \left (x\right ) + \int \frac{{\left (\sin \left (3 \, x\right ) - \sin \left (x\right )\right )} \cos \left (4 \, x\right ) -{\left (\cos \left (3 \, x\right ) - \cos \left (x\right )\right )} \sin \left (4 \, x\right ) -{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (3 \, x\right ) + \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) - \cos \left (x\right ) \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) \sin \left (x\right ) - \sin \left (x\right )}{2 \,{\left (2 \,{\left (\cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 2 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) - 1\right )}}\,{d x} - \frac{1}{24} \, \log \left (2 \,{\left (\cos \left (x\right ) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \frac{1}{24} \, \log \left (-2 \,{\left (\cos \left (x\right ) - 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) - \frac{1}{12} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \frac{1}{12} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cot(6*x),x, algorithm="maxima")

[Out]

cos(x) + integrate(1/2*((sin(3*x) - sin(x))*cos(4*x) - (cos(3*x) - cos(x))*sin(4*x) - (cos(2*x) - 1)*sin(3*x)
+ cos(3*x)*sin(2*x) - cos(x)*sin(2*x) + cos(2*x)*sin(x) - sin(x))/(2*(cos(2*x) - 1)*cos(4*x) - cos(4*x)^2 - co
s(2*x)^2 - sin(4*x)^2 + 2*sin(4*x)*sin(2*x) - sin(2*x)^2 + 2*cos(2*x) - 1), x) - 1/24*log(2*(cos(x) + 1)*cos(2
*x) + cos(2*x)^2 + cos(x)^2 + sin(2*x)^2 + 2*sin(2*x)*sin(x) + sin(x)^2 + 2*cos(x) + 1) + 1/24*log(-2*(cos(x)
- 1)*cos(2*x) + cos(2*x)^2 + cos(x)^2 + sin(2*x)^2 - 2*sin(2*x)*sin(x) + sin(x)^2 - 2*cos(x) + 1) - 1/12*log(c
os(x)^2 + sin(x)^2 + 2*cos(x) + 1) + 1/12*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1)

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Fricas [B]  time = 2.61589, size = 259, normalized size = 6.82 \begin{align*} \frac{1}{12} \, \sqrt{3} \log \left (\frac{4 \, \cos \left (x\right )^{2} - 4 \, \sqrt{3} \cos \left (x\right ) + 3}{4 \, \cos \left (x\right )^{2} - 3}\right ) + \cos \left (x\right ) - \frac{1}{12} \, \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + \frac{1}{12} \, \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + \frac{1}{12} \, \log \left (-2 \, \cos \left (x\right ) + 1\right ) - \frac{1}{12} \, \log \left (-2 \, \cos \left (x\right ) - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cot(6*x),x, algorithm="fricas")

[Out]

1/12*sqrt(3)*log((4*cos(x)^2 - 4*sqrt(3)*cos(x) + 3)/(4*cos(x)^2 - 3)) + cos(x) - 1/12*log(1/2*cos(x) + 1/2) +
1/12*log(-1/2*cos(x) + 1/2) + 1/12*log(-2*cos(x) + 1) - 1/12*log(-2*cos(x) - 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cot(6*x),x)

[Out]

Timed out

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Giac [B]  time = 1.14683, size = 95, normalized size = 2.5 \begin{align*} \frac{1}{12} \, \sqrt{3} \log \left (\frac{{\left | -4 \, \sqrt{3} + 8 \, \cos \left (x\right ) \right |}}{{\left | 4 \, \sqrt{3} + 8 \, \cos \left (x\right ) \right |}}\right ) + \cos \left (x\right ) - \frac{1}{12} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac{1}{12} \, \log \left (-\cos \left (x\right ) + 1\right ) - \frac{1}{12} \, \log \left ({\left | 2 \, \cos \left (x\right ) + 1 \right |}\right ) + \frac{1}{12} \, \log \left ({\left | 2 \, \cos \left (x\right ) - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cot(6*x),x, algorithm="giac")

[Out]

1/12*sqrt(3)*log(abs(-4*sqrt(3) + 8*cos(x))/abs(4*sqrt(3) + 8*cos(x))) + cos(x) - 1/12*log(cos(x) + 1) + 1/12*
log(-cos(x) + 1) - 1/12*log(abs(2*cos(x) + 1)) + 1/12*log(abs(2*cos(x) - 1))