3.127 \(\int \cos (x) \csc (6 x) \, dx\)

Optimal. Leaf size=36 \[ -\frac{1}{6} \tanh ^{-1}(\cos (x))-\frac{1}{6} \tanh ^{-1}(2 \cos (x))+\frac{\tanh ^{-1}\left (\frac{2 \cos (x)}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0414389, antiderivative size = 36, 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, 2057, 207} \[ -\frac{1}{6} \tanh ^{-1}(\cos (x))-\frac{1}{6} \tanh ^{-1}(2 \cos (x))+\frac{\tanh ^{-1}\left (\frac{2 \cos (x)}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

Rule 2057

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(u /. x -> x^2)^p, x], x
] /;  !SumQ[NonfreeFactors[u, x]]] /; PolyQ[P, x^2] && 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) \csc (6 x) \, dx &=-\operatorname{Subst}\left (\int \frac{1}{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}{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 (-\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 )\\ &=\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}}\\ \end{align*}

Mathematica [B]  time = 0.0727779, size = 83, normalized size = 2.31 \[ \frac{1}{12} \left (2 \log \left (\sin \left (\frac{x}{2}\right )\right )-2 \log \left (\cos \left (\frac{x}{2}\right )\right )+\log (1-2 \cos (x))-\log (2 \cos (x)+1)-2 \sqrt{3} \tanh ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right )-2}{\sqrt{3}}\right )+2 \sqrt{3} \tanh ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right )+2}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.059, size = 47, 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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*csc(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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\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)*csc(6*x),x, algorithm="maxima")

[Out]

-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 - cos(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) + co
s(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(cos(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 = 2.65906, size = 248, normalized size = 6.89 \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 ) - \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)*csc(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)) - 1/12*log(1/2*cos(x) + 1/2) + 1/12*lo
g(-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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos{\left (x \right )} \csc{\left (6 x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(x)*csc(6*x), x)

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Giac [B]  time = 1.178, size = 136, normalized size = 3.78 \begin{align*} \frac{1}{12} \, \sqrt{3} \log \left (\frac{{\left | -8 \, \sqrt{3} - \frac{2 \,{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - 14 \right |}}{{\left | 8 \, \sqrt{3} - \frac{2 \,{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - 14 \right |}}\right ) + \frac{1}{12} \, \log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) - \frac{1}{12} \, \log \left ({\left | -\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 3 \right |}\right ) + \frac{1}{12} \, \log \left ({\left | -\frac{3 \,{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(3)*log(abs(-8*sqrt(3) - 2*(cos(x) - 1)/(cos(x) + 1) - 14)/abs(8*sqrt(3) - 2*(cos(x) - 1)/(cos(x) + 1
) - 14)) + 1/12*log(-(cos(x) - 1)/(cos(x) + 1)) - 1/12*log(abs(-(cos(x) - 1)/(cos(x) + 1) - 3)) + 1/12*log(abs
(-3*(cos(x) - 1)/(cos(x) + 1) - 1))