### 3.172 $$\int \log (a \cot ^n(x)) \, dx$$

Optimal. Leaf size=56 $\frac{1}{2} i n \text{PolyLog}\left (2,-e^{2 i x}\right )-\frac{1}{2} i n \text{PolyLog}\left (2,e^{2 i x}\right )+x \log \left (a \cot ^n(x)\right )-2 n x \tanh ^{-1}\left (e^{2 i x}\right )$

[Out]

-2*n*x*ArcTanh[E^((2*I)*x)] + x*Log[a*Cot[x]^n] + (I/2)*n*PolyLog[2, -E^((2*I)*x)] - (I/2)*n*PolyLog[2, E^((2*
I)*x)]

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Rubi [A]  time = 0.0486499, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.857, Rules used = {2548, 12, 4419, 4183, 2279, 2391} $\frac{1}{2} i n \text{PolyLog}\left (2,-e^{2 i x}\right )-\frac{1}{2} i n \text{PolyLog}\left (2,e^{2 i x}\right )+x \log \left (a \cot ^n(x)\right )-2 n x \tanh ^{-1}\left (e^{2 i x}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[Log[a*Cot[x]^n],x]

[Out]

-2*n*x*ArcTanh[E^((2*I)*x)] + x*Log[a*Cot[x]^n] + (I/2)*n*PolyLog[2, -E^((2*I)*x)] - (I/2)*n*PolyLog[2, E^((2*
I)*x)]

Rule 2548

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/u, x], x] /; InverseFunctionFr
eeQ[u, 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 4419

Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dist[
2^n, Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]

Rule 4183

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E^(I*(e + f*
x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[(d*m)/f, Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \log \left (a \cot ^n(x)\right ) \, dx &=x \log \left (a \cot ^n(x)\right )+\int n x \csc (x) \sec (x) \, dx\\ &=x \log \left (a \cot ^n(x)\right )+n \int x \csc (x) \sec (x) \, dx\\ &=x \log \left (a \cot ^n(x)\right )+(2 n) \int x \csc (2 x) \, dx\\ &=-2 n x \tanh ^{-1}\left (e^{2 i x}\right )+x \log \left (a \cot ^n(x)\right )-n \int \log \left (1-e^{2 i x}\right ) \, dx+n \int \log \left (1+e^{2 i x}\right ) \, dx\\ &=-2 n x \tanh ^{-1}\left (e^{2 i x}\right )+x \log \left (a \cot ^n(x)\right )+\frac{1}{2} (i n) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i x}\right )-\frac{1}{2} (i n) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i x}\right )\\ &=-2 n x \tanh ^{-1}\left (e^{2 i x}\right )+x \log \left (a \cot ^n(x)\right )+\frac{1}{2} i n \text{Li}_2\left (-e^{2 i x}\right )-\frac{1}{2} i n \text{Li}_2\left (e^{2 i x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0131143, size = 81, normalized size = 1.45 $\frac{1}{2} i n \text{PolyLog}(2,-i \tan (x))-\frac{1}{2} i n \text{PolyLog}(2,i \tan (x))-\frac{1}{2} i \log (-i (-\tan (x)+i)) \log \left (a \cot ^n(x)\right )+\frac{1}{2} i \log (-i (\tan (x)+i)) \log \left (a \cot ^n(x)\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Log[a*Cot[x]^n],x]

[Out]

(-I/2)*Log[a*Cot[x]^n]*Log[(-I)*(I - Tan[x])] + (I/2)*Log[a*Cot[x]^n]*Log[(-I)*(I + Tan[x])] + (I/2)*n*PolyLog
[2, (-I)*Tan[x]] - (I/2)*n*PolyLog[2, I*Tan[x]]

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Maple [C]  time = 2.596, size = 6531, normalized size = 116.6 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(ln(a*cot(x)^n),x)

[Out]

result too large to display

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Maxima [A]  time = 1.52, size = 66, normalized size = 1.18 \begin{align*} n x \log \left (\tan \left (x\right )\right ) - \frac{1}{4} \,{\left (\pi \log \left (\tan \left (x\right )^{2} + 1\right ) + 2 i \,{\rm Li}_2\left (i \, \tan \left (x\right ) + 1\right ) - 2 i \,{\rm Li}_2\left (-i \, \tan \left (x\right ) + 1\right )\right )} n + x \log \left (a \frac{1}{\tan \left (x\right )}^{n}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(a*cot(x)^n),x, algorithm="maxima")

[Out]

n*x*log(tan(x)) - 1/4*(pi*log(tan(x)^2 + 1) + 2*I*dilog(I*tan(x) + 1) - 2*I*dilog(-I*tan(x) + 1))*n + x*log(a*
(1/tan(x))^n)

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Fricas [B]  time = 2.14417, size = 498, normalized size = 8.89 \begin{align*} n x \log \left (\frac{\cos \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}\right ) - \frac{1}{2} \, n x \log \left (\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right ) + 1\right ) - \frac{1}{2} \, n x \log \left (\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right ) + 1\right ) + \frac{1}{2} \, n x \log \left (-\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right ) + 1\right ) + \frac{1}{2} \, n x \log \left (-\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right ) + 1\right ) - \frac{1}{4} i \, n{\rm Li}_2\left (\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right )\right ) + \frac{1}{4} i \, n{\rm Li}_2\left (\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right )\right ) - \frac{1}{4} i \, n{\rm Li}_2\left (-\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right )\right ) + \frac{1}{4} i \, n{\rm Li}_2\left (-\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right )\right ) + x \log \left (a\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(a*cot(x)^n),x, algorithm="fricas")

[Out]

n*x*log((cos(2*x) + 1)/sin(2*x)) - 1/2*n*x*log(cos(2*x) + I*sin(2*x) + 1) - 1/2*n*x*log(cos(2*x) - I*sin(2*x)
+ 1) + 1/2*n*x*log(-cos(2*x) + I*sin(2*x) + 1) + 1/2*n*x*log(-cos(2*x) - I*sin(2*x) + 1) - 1/4*I*n*dilog(cos(2
*x) + I*sin(2*x)) + 1/4*I*n*dilog(cos(2*x) - I*sin(2*x)) - 1/4*I*n*dilog(-cos(2*x) + I*sin(2*x)) + 1/4*I*n*dil
og(-cos(2*x) - I*sin(2*x)) + x*log(a)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(a*cot(x)**n),x)

[Out]

Integral(log(a*cot(x)**n), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(a*cot(x)^n),x, algorithm="giac")

[Out]

integrate(log(a*cot(x)^n), x)