3.115 \(\int \cos (x) \cot (n x) \, dx\)

Optimal. Leaf size=92 \[ e^{-i x} \text{Hypergeometric2F1}\left (1,-\frac{1}{2 n},1-\frac{1}{2 n},e^{2 i n x}\right )-e^{i x} \text{Hypergeometric2F1}\left (1,\frac{1}{2 n},\frac{1}{2} \left (\frac{1}{n}+2\right ),e^{2 i n x}\right )-\frac{e^{-i x}}{2}+\frac{e^{i x}}{2} \]

[Out]

-1/(2*E^(I*x)) + E^(I*x)/2 + Hypergeometric2F1[1, -1/(2*n), 1 - 1/(2*n), E^((2*I)*n*x)]/E^(I*x) - E^(I*x)*Hype
rgeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, E^((2*I)*n*x)]

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Rubi [A]  time = 0.0908161, antiderivative size = 92, 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 = {4558, 2194, 2251} \[ e^{-i x} \, _2F_1\left (1,-\frac{1}{2 n};1-\frac{1}{2 n};e^{2 i n x}\right )-e^{i x} \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );e^{2 i n x}\right )-\frac{e^{-i x}}{2}+\frac{e^{i x}}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*E^(I*x)) + E^(I*x)/2 + Hypergeometric2F1[1, -1/(2*n), 1 - 1/(2*n), E^((2*I)*n*x)]/E^(I*x) - E^(I*x)*Hype
rgeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, E^((2*I)*n*x)]

Rule 4558

Int[Cos[(a_.) + (b_.)*(x_)]*Cot[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[I/(E^(I*(a + b*x))*2) + (I*E^(I*(a + b*x
)))/2 - I/(E^(I*(a + b*x))*(1 - E^(2*I*(c + d*x)))) - (I*E^(I*(a + b*x)))/(1 - E^(2*I*(c + d*x))), x] /; FreeQ
[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2251

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Simp
[(a^p*G^(h*(f + g*x))*Hypergeometric2F1[-p, (g*h*Log[G])/(d*e*Log[F]), (g*h*Log[G])/(d*e*Log[F]) + 1, Simplify
[-((b*F^(e*(c + d*x)))/a)]])/(g*h*Log[G]), x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && (ILtQ[p, 0] ||
GtQ[a, 0])

Rubi steps

\begin{align*} \int \cos (x) \cot (n x) \, dx &=\int \left (\frac{1}{2} i e^{-i x}+\frac{1}{2} i e^{i x}-\frac{i e^{-i x}}{1-e^{2 i n x}}-\frac{i e^{i x}}{1-e^{2 i n x}}\right ) \, dx\\ &=\frac{1}{2} i \int e^{-i x} \, dx+\frac{1}{2} i \int e^{i x} \, dx-i \int \frac{e^{-i x}}{1-e^{2 i n x}} \, dx-i \int \frac{e^{i x}}{1-e^{2 i n x}} \, dx\\ &=-\frac{1}{2} e^{-i x}+\frac{e^{i x}}{2}+e^{-i x} \, _2F_1\left (1,-\frac{1}{2 n};1-\frac{1}{2 n};e^{2 i n x}\right )-e^{i x} \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );e^{2 i n x}\right )\\ \end{align*}

Mathematica [A]  time = 0.190771, size = 179, normalized size = 1.95 \[ \frac{1}{2} e^{-2 i x} \left (-\frac{e^{i (2 n x+x)} \text{Hypergeometric2F1}\left (1,1-\frac{1}{2 n},2-\frac{1}{2 n},e^{2 i n x}\right )}{2 n-1}-\frac{e^{i (2 n+3) x} \text{Hypergeometric2F1}\left (1,\frac{1}{2 n}+1,\frac{1}{2 n}+2,e^{2 i n x}\right )}{2 n+1}+e^{i x} \text{Hypergeometric2F1}\left (1,-\frac{1}{2 n},1-\frac{1}{2 n},e^{2 i n x}\right )-e^{3 i x} \text{Hypergeometric2F1}\left (1,\frac{1}{2 n},\frac{1}{2 n}+1,e^{2 i n x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((E^(I*(x + 2*n*x))*Hypergeometric2F1[1, 1 - 1/(2*n), 2 - 1/(2*n), E^((2*I)*n*x)])/(-1 + 2*n)) - (E^(I*(3 +
2*n)*x)*Hypergeometric2F1[1, 1 + 1/(2*n), 2 + 1/(2*n), E^((2*I)*n*x)])/(1 + 2*n) + E^(I*x)*Hypergeometric2F1[1
, -1/(2*n), 1 - 1/(2*n), E^((2*I)*n*x)] - E^((3*I)*x)*Hypergeometric2F1[1, 1/(2*n), 1 + 1/(2*n), E^((2*I)*n*x)
])/(2*E^((2*I)*x))

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Maple [F]  time = 0.223, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( x \right ) \cot \left ( nx \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cos(x)*cot(n*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\cos \left (x\right ) \cot \left (n x\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cos(x)*cot(n*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(x)*cot(n*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cos(x)*cot(n*x), x)