\(\int \log (a \cot (x)) \, dx\) [170]

Optimal result

Integrand size = 5, antiderivative size = 51 \[ \int \log (a \cot (x)) \, dx=-2 x \text {arctanh}\left (e^{2 i x}\right )+x \log (a \cot (x))+\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {2628, 4504, 4268, 2317, 2438} \[ \int \log (a \cot (x)) \, dx=x \log (a \cot (x))-2 x \text {arctanh}\left (e^{2 i x}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \]

[In]

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

[Out]

-2*x*ArcTanh[E^((2*I)*x)] + x*Log[a*Cot[x]] + (I/2)*PolyLog[2, -E^((2*I)*x)] - (I/2)*PolyLog[2, E^((2*I)*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*} \text {integral}& = x \log (a \cot (x))+\int x \csc (x) \sec (x) \, dx \\ & = x \log (a \cot (x))+2 \int x \csc (2 x) \, dx \\ & = -2 x \tanh ^{-1}\left (e^{2 i x}\right )+x \log (a \cot (x))-\int \log \left (1-e^{2 i x}\right ) \, dx+\int \log \left (1+e^{2 i x}\right ) \, dx \\ & = -2 x \tanh ^{-1}\left (e^{2 i x}\right )+x \log (a \cot (x))+\frac {1}{2} i \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i x}\right )-\frac {1}{2} i \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i x}\right ) \\ & = -2 x \tanh ^{-1}\left (e^{2 i x}\right )+x \log (a \cot (x))+\frac {1}{2} i \text {Li}_2\left (-e^{2 i x}\right )-\frac {1}{2} i \text {Li}_2\left (e^{2 i x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.47 \[ \int \log (a \cot (x)) \, dx=-\frac {1}{2} i \log (a \cot (x)) \log (-i (i-\tan (x)))+\frac {1}{2} i \log (a \cot (x)) \log (-i (i+\tan (x)))+\frac {1}{2} i \operatorname {PolyLog}(2,-i \tan (x))-\frac {1}{2} i \operatorname {PolyLog}(2,i \tan (x)) \]

[In]

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

[Out]

(-1/2*I)*Log[a*Cot[x]]*Log[(-I)*(I - Tan[x])] + (I/2)*Log[a*Cot[x]]*Log[(-I)*(I + Tan[x])] + (I/2)*PolyLog[2,
(-I)*Tan[x]] - (I/2)*PolyLog[2, I*Tan[x]]

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (39 ) = 78\).

Time = 0.94 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.69

method	result	size
derivativedivides	\(-a \left (-\frac {i \ln \left (a \cot \left (x \right )\right ) \left (\ln \left (\frac {i \cot \left (x \right ) a +a}{a}\right )-\ln \left (-\frac {i \cot \left (x \right ) a -a}{a}\right )\right )}{2 a}-\frac {i \left (\operatorname {dilog}\left (\frac {i \cot \left (x \right ) a +a}{a}\right )-\operatorname {dilog}\left (-\frac {i \cot \left (x \right ) a -a}{a}\right )\right )}{2 a}\right )\)	\(86\)
default	\(-a \left (-\frac {i \ln \left (a \cot \left (x \right )\right ) \left (\ln \left (\frac {i \cot \left (x \right ) a +a}{a}\right )-\ln \left (-\frac {i \cot \left (x \right ) a -a}{a}\right )\right )}{2 a}-\frac {i \left (\operatorname {dilog}\left (\frac {i \cot \left (x \right ) a +a}{a}\right )-\operatorname {dilog}\left (-\frac {i \cot \left (x \right ) a -a}{a}\right )\right )}{2 a}\right )\)	\(86\)
risch	\(x \ln \left (1+{\mathrm e}^{2 i x}\right )-\frac {i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 i x}\right )\right ) \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i x}-1}\right ) \operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 i x}\right )}{{\mathrm e}^{2 i x}-1}\right ) x}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 i x}\right )}{{\mathrm e}^{2 i x}-1}\right ) \operatorname {csgn}\left (\frac {i a \left (1+{\mathrm e}^{2 i x}\right )}{{\mathrm e}^{2 i x}-1}\right ) \operatorname {csgn}\left (i a \right ) x}{2}+\frac {i \pi {\operatorname {csgn}\left (\frac {i a \left (1+{\mathrm e}^{2 i x}\right )}{{\mathrm e}^{2 i x}-1}\right )}^{2} \operatorname {csgn}\left (i a \right ) x}{2}-\frac {i \pi {\operatorname {csgn}\left (\frac {i a \left (1+{\mathrm e}^{2 i x}\right )}{{\mathrm e}^{2 i x}-1}\right )}^{3} x}{2}+\frac {i \pi x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 i x}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 i x}\right )}{{\mathrm e}^{2 i x}-1}\right )}^{2} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i a \left (1+{\mathrm e}^{2 i x}\right )}{{\mathrm e}^{2 i x}-1}\right ) \operatorname {csgn}\left (\frac {a \left (1+{\mathrm e}^{2 i x}\right )}{{\mathrm e}^{2 i x}-1}\right ) x}{2}+\ln \left (a \right ) x +\frac {i \pi {\operatorname {csgn}\left (\frac {a \left (1+{\mathrm e}^{2 i x}\right )}{{\mathrm e}^{2 i x}-1}\right )}^{3} x}{2}-\frac {i \pi {\operatorname {csgn}\left (\frac {a \left (1+{\mathrm e}^{2 i x}\right )}{{\mathrm e}^{2 i x}-1}\right )}^{2} x}{2}-i \ln \left ({\mathrm e}^{i x}\right ) \ln \left ({\mathrm e}^{i x}+1\right )-x \ln \left (1+i {\mathrm e}^{i x}\right )-x \ln \left (1-i {\mathrm e}^{i x}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i a \left (1+{\mathrm e}^{2 i x}\right )}{{\mathrm e}^{2 i x}-1}\right ) {\operatorname {csgn}\left (\frac {a \left (1+{\mathrm e}^{2 i x}\right )}{{\mathrm e}^{2 i x}-1}\right )}^{2} x}{2}+i \ln \left ({\mathrm e}^{i x}\right ) \ln \left ({\mathrm e}^{2 i x}-1\right )-i \operatorname {dilog}\left ({\mathrm e}^{i x}+1\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 i x}\right )}{{\mathrm e}^{2 i x}-1}\right ) {\operatorname {csgn}\left (\frac {i a \left (1+{\mathrm e}^{2 i x}\right )}{{\mathrm e}^{2 i x}-1}\right )}^{2} x}{2}+i \operatorname {dilog}\left ({\mathrm e}^{i x}\right )+i \operatorname {dilog}\left (1+i {\mathrm e}^{i x}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i x}-1}\right ) {\operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 i x}\right )}{{\mathrm e}^{2 i x}-1}\right )}^{2} x}{2}+i \operatorname {dilog}\left (1-i {\mathrm e}^{i x}\right )-\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 i x}\right )}{{\mathrm e}^{2 i x}-1}\right )}^{3} x}{2}\)	\(589\)

[In]

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

[Out]

-a*(-1/2*I*ln(a*cot(x))*(ln((I*cot(x)*a+a)/a)-ln(-(I*cot(x)*a-a)/a))/a-1/2*I*(dilog((I*cot(x)*a+a)/a)-dilog(-(
I*cot(x)*a-a)/a))/a)

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (32) = 64\).

Time = 0.34 (sec) , antiderivative size = 147, normalized size of antiderivative = 2.88 \[ \int \log (a \cot (x)) \, dx=x \log \left (\frac {a \cos \left (2 \, x\right ) + a}{\sin \left (2 \, x\right )}\right ) - \frac {1}{2} \, x \log \left (\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right ) + 1\right ) - \frac {1}{2} \, x \log \left (\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right ) + 1\right ) + \frac {1}{2} \, x \log \left (-\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right ) + 1\right ) + \frac {1}{2} \, x \log \left (-\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right ) + 1\right ) - \frac {1}{4} i \, {\rm Li}_2\left (\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right )\right ) + \frac {1}{4} i \, {\rm Li}_2\left (\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right )\right ) - \frac {1}{4} i \, {\rm Li}_2\left (-\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right )\right ) + \frac {1}{4} i \, {\rm Li}_2\left (-\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right )\right ) \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \log (a \cot (x)) \, dx=\int \log {\left (a \cot {\left (x \right )} \right )}\, dx \]

[In]

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

[Out]

Integral(log(a*cot(x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.84 \[ \int \log (a \cot (x)) \, dx=-\frac {1}{4} \, \pi \log \left (\tan \left (x\right )^{2} + 1\right ) + x \log \left (\frac {a}{\tan \left (x\right )}\right ) + x \log \left (\tan \left (x\right )\right ) - \frac {1}{2} i \, {\rm Li}_2\left (i \, \tan \left (x\right ) + 1\right ) + \frac {1}{2} i \, {\rm Li}_2\left (-i \, \tan \left (x\right ) + 1\right ) \]

[In]

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

[Out]

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

Giac [F]

\[ \int \log (a \cot (x)) \, dx=\int { \log \left (a \cot \left (x\right )\right ) \,d x } \]

[In]

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

[Out]

integrate(log(a*cot(x)), x)

Mupad [F(-1)]

Timed out. \[ \int \log (a \cot (x)) \, dx=\int \ln \left (a\,\mathrm {cot}\left (x\right )\right ) \,d x \]

[In]

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

[Out]

int(log(a*cot(x)), x)