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]

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

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Rubi [A]  time = 0.07, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [B]  time = 0.09, 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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fricas [B]  time = 0.69, size = 71, normalized size = 1.87 \[ \frac {1}{12} \, \sqrt {3} \log \left (\frac {4 \, \cos \relax (x)^{2} - 4 \, \sqrt {3} \cos \relax (x) + 3}{4 \, \cos \relax (x)^{2} - 3}\right ) + \cos \relax (x) - \frac {1}{12} \, \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + \frac {1}{12} \, \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + \frac {1}{12} \, \log \left (-2 \, \cos \relax (x) + 1\right ) - \frac {1}{12} \, \log \left (-2 \, \cos \relax (x) - 1\right ) \]

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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giac [B]  time = 0.17, size = 70, normalized size = 1.84 \[ \frac {1}{12} \, \sqrt {3} \log \left (\frac {{\left | -4 \, \sqrt {3} + 8 \, \cos \relax (x) \right |}}{{\left | 4 \, \sqrt {3} + 8 \, \cos \relax (x) \right |}}\right ) + \cos \relax (x) - \frac {1}{12} \, \log \left (\cos \relax (x) + 1\right ) + \frac {1}{12} \, \log \left (-\cos \relax (x) + 1\right ) - \frac {1}{12} \, \log \left ({\left | 2 \, \cos \relax (x) + 1 \right |}\right ) + \frac {1}{12} \, \log \left ({\left | 2 \, \cos \relax (x) - 1 \right |}\right ) \]

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))

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maple [A]  time = 0.56, size = 49, normalized size = 1.29 \[ \frac {\ln \left (2 \cos \relax (x )-1\right )}{12}-\frac {\ln \left (1+2 \cos \relax (x )\right )}{12}-\frac {\arctanh \left (\frac {2 \cos \relax (x ) \sqrt {3}}{3}\right ) \sqrt {3}}{6}+\frac {\ln \left (-1+\cos \relax (x )\right )}{12}-\frac {\ln \left (1+\cos \relax (x )\right )}{12}+\cos \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \cos \relax (x) + \int \frac {{\left (\sin \left (3 \, x\right ) - \sin \relax (x)\right )} \cos \left (4 \, x\right ) - {\left (\cos \left (3 \, x\right ) - \cos \relax (x)\right )} \sin \left (4 \, x\right ) - {\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (3 \, x\right ) + \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) - \cos \relax (x) \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) \sin \relax (x) - \sin \relax (x)}{2 \, {\left (2 \, {\left (\cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 2 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) - 1\right )}}\,{d x} - \frac {1}{24} \, \log \left (2 \, {\left (\cos \relax (x) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + \cos \relax (x)^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \left (2 \, x\right ) \sin \relax (x) + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + \frac {1}{24} \, \log \left (-2 \, {\left (\cos \relax (x) - 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + \cos \relax (x)^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \sin \left (2 \, x\right ) \sin \relax (x) + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) - \frac {1}{12} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + \frac {1}{12} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) \]

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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mupad [B]  time = 2.44, size = 86, normalized size = 2.26 \[ \frac {\mathrm {atanh}\left (\frac {1073741824}{10761687\,\left (\frac {427973089951744\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{14348907}-\frac {47552804159488}{4782969}\right )}+\frac {797161}{797162}\right )}{6}+\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{6}-\frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {303181204553728\,\sqrt {3}}{4782969\,\left (\frac {7314051205955584\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{4782969}-\frac {525125250187264}{4782969}\right )}-\frac {4222769432625152\,\sqrt {3}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{4782969\,\left (\frac {7314051205955584\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{4782969}-\frac {525125250187264}{4782969}\right )}\right )}{6}+\frac {2}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atanh(1073741824/(10761687*((427973089951744*tan(x/2)^2)/14348907 - 47552804159488/4782969)) + 797161/797162)/
6 + log(tan(x/2))/6 - (3^(1/2)*atanh((303181204553728*3^(1/2))/(4782969*((7314051205955584*tan(x/2)^2)/4782969
- 525125250187264/4782969)) - (4222769432625152*3^(1/2)*tan(x/2)^2)/(4782969*((7314051205955584*tan(x/2)^2)/4
782969 - 525125250187264/4782969))))/6 + 2/(tan(x/2)^2 + 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos {\relax (x )} \cot {\left (6 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(x)*cot(6*x), x)

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