3.112 \(\int \cos (x) \cot (4 x) \, dx\)

Optimal. Leaf size=28 \[ \cos (x)-\frac{1}{4} \tanh ^{-1}(\cos (x))-\frac{\tanh ^{-1}\left (\sqrt{2} \cos (x)\right )}{2 \sqrt{2}} \]

[Out]

-ArcTanh[Cos[x]]/4 - ArcTanh[Sqrt[2]*Cos[x]]/(2*Sqrt[2]) + Cos[x]

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Rubi [A]  time = 0.0485803, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {1676, 1166, 207} \[ \cos (x)-\frac{1}{4} \tanh ^{-1}(\cos (x))-\frac{\tanh ^{-1}\left (\sqrt{2} \cos (x)\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Cot[4*x],x]

[Out]

-ArcTanh[Cos[x]]/4 - ArcTanh[Sqrt[2]*Cos[x]]/(2*Sqrt[2]) + Cos[x]

Rule 1676

Int[(Pq_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^2 + c*x^4), x], x
] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

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 (4 x) \, dx &=-\operatorname{Subst}\left (\int \frac{-1+8 x^2-8 x^4}{4-12 x^2+8 x^4} \, dx,x,\cos (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (-1+\frac{3-4 x^2}{4-12 x^2+8 x^4}\right ) \, dx,x,\cos (x)\right )\\ &=\cos (x)-\operatorname{Subst}\left (\int \frac{3-4 x^2}{4-12 x^2+8 x^4} \, dx,x,\cos (x)\right )\\ &=\cos (x)+2 \operatorname{Subst}\left (\int \frac{1}{-8+8 x^2} \, dx,x,\cos (x)\right )+2 \operatorname{Subst}\left (\int \frac{1}{-4+8 x^2} \, dx,x,\cos (x)\right )\\ &=-\frac{1}{4} \tanh ^{-1}(\cos (x))-\frac{\tanh ^{-1}\left (\sqrt{2} \cos (x)\right )}{2 \sqrt{2}}+\cos (x)\\ \end{align*}

Mathematica [C]  time = 0.0662992, size = 73, normalized size = 2.61 \[ \frac{1}{4} \left (4 \cos (x)+\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right )+(-1-i) (-1)^{3/4} \tanh ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right )-1}{\sqrt{2}}\right )-(1-i) \sqrt [4]{-1} \tanh ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right )+1}{\sqrt{2}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Cot[4*x],x]

[Out]

((-1 - I)*(-1)^(3/4)*ArcTanh[(-1 + Tan[x/2])/Sqrt[2]] - (1 - I)*(-1)^(1/4)*ArcTanh[(1 + Tan[x/2])/Sqrt[2]] + 4
*Cos[x] - Log[Cos[x/2]] + Log[Sin[x/2]])/4

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Maple [A]  time = 0.134, size = 30, normalized size = 1.1 \begin{align*} -{\frac{{\it Artanh} \left ( \cos \left ( x \right ) \sqrt{2} \right ) \sqrt{2}}{4}}-{\frac{\ln \left ( 1+\cos \left ( x \right ) \right ) }{8}}+{\frac{\ln \left ( -1+\cos \left ( x \right ) \right ) }{8}}+\cos \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*cot(4*x),x)

[Out]

-1/4*arctanh(cos(x)*2^(1/2))*2^(1/2)-1/8*ln(1+cos(x))+1/8*ln(-1+cos(x))+cos(x)

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Maxima [B]  time = 1.54073, size = 223, normalized size = 7.96 \begin{align*} -\frac{1}{16} \, \sqrt{2} \log \left (2 \, \sqrt{2} \sin \left (2 \, x\right ) \sin \left (x\right ) + 2 \,{\left (\sqrt{2} \cos \left (x\right ) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 2 \, \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \left (x\right )^{2} + 2 \, \sqrt{2} \cos \left (x\right ) + 1\right ) + \frac{1}{16} \, \sqrt{2} \log \left (-2 \, \sqrt{2} \sin \left (2 \, x\right ) \sin \left (x\right ) - 2 \,{\left (\sqrt{2} \cos \left (x\right ) - 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 2 \, \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \left (x\right )^{2} - 2 \, \sqrt{2} \cos \left (x\right ) + 1\right ) + \cos \left (x\right ) - \frac{1}{8} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \frac{1}{8} \, \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(4*x),x, algorithm="maxima")

[Out]

-1/16*sqrt(2)*log(2*sqrt(2)*sin(2*x)*sin(x) + 2*(sqrt(2)*cos(x) + 1)*cos(2*x) + cos(2*x)^2 + 2*cos(x)^2 + sin(
2*x)^2 + 2*sin(x)^2 + 2*sqrt(2)*cos(x) + 1) + 1/16*sqrt(2)*log(-2*sqrt(2)*sin(2*x)*sin(x) - 2*(sqrt(2)*cos(x)
- 1)*cos(2*x) + cos(2*x)^2 + 2*cos(x)^2 + sin(2*x)^2 + 2*sin(x)^2 - 2*sqrt(2)*cos(x) + 1) + cos(x) - 1/8*log(c
os(x)^2 + sin(x)^2 + 2*cos(x) + 1) + 1/8*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1)

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Fricas [B]  time = 2.57622, size = 185, normalized size = 6.61 \begin{align*} \frac{1}{8} \, \sqrt{2} \log \left (\frac{2 \, \cos \left (x\right )^{2} - 2 \, \sqrt{2} \cos \left (x\right ) + 1}{2 \, \cos \left (x\right )^{2} - 1}\right ) + \cos \left (x\right ) - \frac{1}{8} \, \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + \frac{1}{8} \, \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cot(4*x),x, algorithm="fricas")

[Out]

1/8*sqrt(2)*log((2*cos(x)^2 - 2*sqrt(2)*cos(x) + 1)/(2*cos(x)^2 - 1)) + cos(x) - 1/8*log(1/2*cos(x) + 1/2) + 1
/8*log(-1/2*cos(x) + 1/2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos{\left (x \right )} \cot{\left (4 x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cot(4*x),x)

[Out]

Integral(cos(x)*cot(4*x), x)

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Giac [B]  time = 1.13957, size = 68, normalized size = 2.43 \begin{align*} \frac{1}{8} \, \sqrt{2} \log \left (\frac{{\left | -2 \, \sqrt{2} + 4 \, \cos \left (x\right ) \right |}}{{\left | 2 \, \sqrt{2} + 4 \, \cos \left (x\right ) \right |}}\right ) + \cos \left (x\right ) - \frac{1}{8} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac{1}{8} \, \log \left (-\cos \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cot(4*x),x, algorithm="giac")

[Out]

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