∫ log(a cotn(x)) dx [172]

3.2.72 \(\int \log (a \cot ^n(x)) \, dx\) [172]

Optimal. Leaf size=56 \[ -2 n x \tanh ^{-1}\left (e^{2 i x}\right )+x \log \left (a \cot ^n(x)\right )+\frac {1}{2} i n \text {Li}_2\left (-e^{2 i x}\right )-\frac {1}{2} i n \text {Li}_2\left (e^{2 i x}\right ) \]

[Out]

-2*n*x*arctanh(exp(2*I*x))+x*ln(a*cot(x)^n)+1/2*I*n*polylog(2,-exp(2*I*x))-1/2*I*n*polylog(2,exp(2*I*x))

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Rubi [A]
time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, 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 = {2628, 12, 4504, 4268, 2317, 2438} \begin {gather*} \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 ) \end {gather*}

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 12

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

Rule 2317

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 2438

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]

Rule 2628

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

eeQ[u, x]

Rule 4268

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 4504

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]

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) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i x}\right )-\frac {1}{2} (i n) \text {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*}

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Mathematica [A]
time = 0.01, size = 81, normalized size = 1.45 \begin {gather*} -\frac {1}{2} i \log \left (a \cot ^n(x)\right ) \log (-i (i-\tan (x)))+\frac {1}{2} i \log \left (a \cot ^n(x)\right ) \log (-i (i+\tan (x)))+\frac {1}{2} i n \text {Li}_2(-i \tan (x))-\frac {1}{2} i n \text {Li}_2(i \tan (x)) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-1/2*I)*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*PolyL

og[2, (-I)*Tan[x]] - (I/2)*n*PolyLog[2, I*Tan[x]]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.62, size = 2357, normalized size = 42.09


method	result	size

risch	\(\text {Expression too large to display}\)	\(2357\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(ln(a*cot(x)^n),x,method=_RETURNVERBOSE)

[Out]

x*ln(exp(-1/2*n*(I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))^3*Pi-I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*csgn(I

/(exp(2*I*x)-1))*Pi-I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*csgn(I*(1+exp(2*I*x)))*Pi+I*csgn(I/(exp(2*I*x)-1

)*(1+exp(2*I*x)))*csgn(I/(exp(2*I*x)-1))*csgn(I*(1+exp(2*I*x)))*Pi-I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))*csg

n(1/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*Pi-I*csgn(1/(exp(2*I*x)-1)*(1+exp(2*I*x)))^3*Pi+I*csgn(I/(exp(2*I*x)-1)*(

1+exp(2*I*x)))*csgn(1/(exp(2*I*x)-1)*(1+exp(2*I*x)))*Pi+I*csgn(1/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*Pi-I*Pi+2*ln

(exp(2*I*x)-1)-2*ln(1+exp(2*I*x)))))+1/2*I*x*Pi*csgn(I*a)*csgn(I*a*exp(-1/2*n*(I*csgn(I/(exp(2*I*x)-1)*(1+exp(

2*I*x)))^3*Pi-I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*csgn(I/(exp(2*I*x)-1))*Pi-I*csgn(I/(exp(2*I*x)-1)*(1+e

xp(2*I*x)))^2*csgn(I*(1+exp(2*I*x)))*Pi+I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))*csgn(I/(exp(2*I*x)-1))*csgn(I*

(1+exp(2*I*x)))*Pi-I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))*csgn(1/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*Pi-I*csgn(1

/(exp(2*I*x)-1)*(1+exp(2*I*x)))^3*Pi+I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))*csgn(1/(exp(2*I*x)-1)*(1+exp(2*I*

x)))*Pi+I*csgn(1/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*Pi-I*Pi+2*ln(exp(2*I*x)-1)-2*ln(1+exp(2*I*x)))))^2+1/2*I*x*P

i*csgn(I*exp(-1/2*n*(I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))^3*Pi-I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*cs

gn(I/(exp(2*I*x)-1))*Pi-I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*csgn(I*(1+exp(2*I*x)))*Pi+I*csgn(I/(exp(2*I*

x)-1)*(1+exp(2*I*x)))*csgn(I/(exp(2*I*x)-1))*csgn(I*(1+exp(2*I*x)))*Pi-I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))

*csgn(1/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*Pi-I*csgn(1/(exp(2*I*x)-1)*(1+exp(2*I*x)))^3*Pi+I*csgn(I/(exp(2*I*x)-

1)*(1+exp(2*I*x)))*csgn(1/(exp(2*I*x)-1)*(1+exp(2*I*x)))*Pi+I*csgn(1/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*Pi-I*Pi+

2*ln(exp(2*I*x)-1)-2*ln(1+exp(2*I*x)))))*csgn(I*a*exp(-1/2*n*(I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))^3*Pi-I*c

sgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*csgn(I/(exp(2*I*x)-1))*Pi-I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*csg

n(I*(1+exp(2*I*x)))*Pi+I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))*csgn(I/(exp(2*I*x)-1))*csgn(I*(1+exp(2*I*x)))*P

i-I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))*csgn(1/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*Pi-I*csgn(1/(exp(2*I*x)-1)*(

1+exp(2*I*x)))^3*Pi+I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))*csgn(1/(exp(2*I*x)-1)*(1+exp(2*I*x)))*Pi+I*csgn(1/

(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*Pi-I*Pi+2*ln(exp(2*I*x)-1)-2*ln(1+exp(2*I*x)))))^2+x*ln(a)-n*x*ln(1+I*exp(I*x

))-n*x*ln(1-I*exp(I*x))+I*n*dilog(1+I*exp(I*x))+I*n*dilog(1-I*exp(I*x))+I*n*dilog(exp(I*x))-I*n*dilog(exp(I*x)

+1)+n*x*ln(exp(I*x)+1)-1/2*I*x*Pi*csgn(I*a*exp(-1/2*n*(I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))^3*Pi-I*csgn(I/(

exp(2*I*x)-1)*(1+exp(2*I*x)))^2*csgn(I/(exp(2*I*x)-1))*Pi-I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*csgn(I*(1+

exp(2*I*x)))*Pi+I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))*csgn(I/(exp(2*I*x)-1))*csgn(I*(1+exp(2*I*x)))*Pi-I*csg

n(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))*csgn(1/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*Pi-I*csgn(1/(exp(2*I*x)-1)*(1+exp(2

*I*x)))^3*Pi+I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))*csgn(1/(exp(2*I*x)-1)*(1+exp(2*I*x)))*Pi+I*csgn(1/(exp(2*

I*x)-1)*(1+exp(2*I*x)))^2*Pi-I*Pi+2*ln(exp(2*I*x)-1)-2*ln(1+exp(2*I*x)))))^3-1/2*I*x*Pi*csgn(I*a)*csgn(I*exp(-

1/2*n*(I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))^3*Pi-I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*csgn(I/(exp(2*I*

x)-1))*Pi-I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*csgn(I*(1+exp(2*I*x)))*Pi+I*csgn(I/(exp(2*I*x)-1)*(1+exp(2

*I*x)))*csgn(I/(exp(2*I*x)-1))*csgn(I*(1+exp(2*I*x)))*Pi-I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))*csgn(1/(exp(2

*I*x)-1)*(1+exp(2*I*x)))^2*Pi-I*csgn(1/(exp(2*I*x)-1)*(1+exp(2*I*x)))^3*Pi+I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*

x)))*csgn(1/(exp(2*I*x)-1)*(1+exp(2*I*x)))*Pi+I*csgn(1/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*Pi-I*Pi+2*ln(exp(2*I*x

)-1)-2*ln(1+exp(2*I*x)))))*csgn(I*a*exp(-1/2*n*(I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))^3*Pi-I*csgn(I/(exp(2*I

*x)-1)*(1+exp(2*I*x)))^2*csgn(I/(exp(2*I*x)-1))*Pi-I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*csgn(I*(1+exp(2*I

*x)))*Pi+I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))*csgn(I/(exp(2*I*x)-1))*csgn(I*(1+exp(2*I*x)))*Pi-I*csgn(I/(ex

p(2*I*x)-1)*(1+exp(2*I*x)))*csgn(1/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*Pi-I*csgn(1/(exp(2*I*x)-1)*(1+exp(2*I*x)))

^3*Pi+I*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))*csgn(1/(exp(2*I*x)-1)*(1+exp(2*I*x)))*Pi+I*csgn(1/(exp(2*I*x)-1)

*(1+exp(2*I*x)))^2*Pi-I*Pi+2*ln(exp(2*I*x)-1)-2*ln(1+exp(2*I*x)))))

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Maxima [A]
time = 0.52, size = 49, normalized size = 0.88 \begin {gather*} 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 {gather*}

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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (37) = 74\).
time = 0.43, size = 158, normalized size = 2.82 \begin {gather*} 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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \log {\left (a \cot ^{n}{\left (x \right )} \right )}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 0.07, size = 44, normalized size = 0.79 \begin {gather*} x\,\ln \left (a\,{\mathrm {cot}\left (x\right )}^n\right )-\frac {n\,\mathrm {polylog}\left (2,{\mathrm {e}}^{x\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}+\frac {n\,\mathrm {polylog}\left (2,-{\mathrm {e}}^{x\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}-2\,n\,x\,\mathrm {atanh}\left ({\mathrm {e}}^{x\,2{}\mathrm {i}}\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(a*cot(x)^n) - (n*polylog(2, exp(x*2i))*1i)/2 + (n*polylog(2, -exp(x*2i))*1i)/2 - 2*n*x*atanh(exp(x*2i))

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