3.110 \(\int \cos (x) \cot (2 x) \, dx\)

Optimal. Leaf size=10 \[ \cos (x)-\frac{1}{2} \tanh ^{-1}(\cos (x)) \]

[Out]

-ArcTanh[Cos[x]]/2 + Cos[x]

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Rubi [A]  time = 0.0203756, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {12, 388, 206} \[ \cos (x)-\frac{1}{2} \tanh ^{-1}(\cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Cot[2*x],x]

[Out]

-ArcTanh[Cos[x]]/2 + 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 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 206

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

Rubi steps

\begin{align*} \int \cos (x) \cot (2 x) \, dx &=-\operatorname{Subst}\left (\int \frac{-1+2 x^2}{2 \left (1-x^2\right )} \, dx,x,\cos (x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{-1+2 x^2}{1-x^2} \, dx,x,\cos (x)\right )\right )\\ &=\cos (x)-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (x)\right )\\ &=-\frac{1}{2} \tanh ^{-1}(\cos (x))+\cos (x)\\ \end{align*}

Mathematica [B]  time = 0.0149474, size = 25, normalized size = 2.5 \[ \cos (x)+\frac{1}{2} \log \left (\sin \left (\frac{x}{2}\right )\right )-\frac{1}{2} \log \left (\cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Cot[2*x],x]

[Out]

Cos[x] - Log[Cos[x/2]]/2 + Log[Sin[x/2]]/2

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Maple [A]  time = 0.025, size = 14, normalized size = 1.4 \begin{align*} \cos \left ( x \right ) +{\frac{\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*cot(2*x),x)

[Out]

cos(x)+1/2*ln(csc(x)-cot(x))

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Maxima [B]  time = 0.991714, size = 50, normalized size = 5. \begin{align*} \cos \left (x\right ) - \frac{1}{4} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \frac{1}{4} \, \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(2*x),x, algorithm="maxima")

[Out]

cos(x) - 1/4*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) + 1/4*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1)

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Fricas [B]  time = 2.51684, size = 88, normalized size = 8.8 \begin{align*} \cos \left (x\right ) - \frac{1}{4} \, \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + \frac{1}{4} \, \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(2*x),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.48852, size = 19, normalized size = 1.9 \begin{align*} \frac{\log{\left (\cos{\left (x \right )} - 1 \right )}}{4} - \frac{\log{\left (\cos{\left (x \right )} + 1 \right )}}{4} + \cos{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cot(2*x),x)

[Out]

log(cos(x) - 1)/4 - log(cos(x) + 1)/4 + cos(x)

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Giac [B]  time = 1.13916, size = 26, normalized size = 2.6 \begin{align*} \cos \left (x\right ) - \frac{1}{4} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac{1}{4} \, \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(2*x),x, algorithm="giac")

[Out]

cos(x) - 1/4*log(cos(x) + 1) + 1/4*log(-cos(x) + 1)