Integrand size = 7, antiderivative size = 56 \[ \int \log \left (a \cot ^n(x)\right ) \, dx=-2 n x \text {arctanh}\left (e^{2 i x}\right )+x \log \left (a \cot ^n(x)\right )+\frac {1}{2} i n \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{2} i n \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \]
-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))
Time = 0.01 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.45 \[ \int \log \left (a \cot ^n(x)\right ) \, dx=-\frac {1}{2} i \log \left (a \cot ^n(x)\right ) \log (-i (i-\tan (x)))+\frac {1}{2} i \log \left (a \cot ^n(x)\right ) \log (-i (i+\tan (x)))+\frac {1}{2} i n \operatorname {PolyLog}(2,-i \tan (x))-\frac {1}{2} i n \operatorname {PolyLog}(2,i \tan (x)) \]
(-1/2*I)*Log[a*Cot[x]^n]*Log[(-I)*(I - Tan[x])] + (I/2)*Log[a*Cot[x]^n]*Lo g[(-I)*(I + Tan[x])] + (I/2)*n*PolyLog[2, (-I)*Tan[x]] - (I/2)*n*PolyLog[2 , I*Tan[x]]
Time = 0.32 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.143, Rules used = {3028, 25, 27, 4919, 3042, 4671, 2715, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \log \left (a \cot ^n(x)\right ) \, dx\) |
\(\Big \downarrow \) 3028 |
\(\displaystyle x \log \left (a \cot ^n(x)\right )-\int -n x \csc (x) \sec (x)dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int n x \csc (x) \sec (x)dx+x \log \left (a \cot ^n(x)\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle n \int x \csc (x) \sec (x)dx+x \log \left (a \cot ^n(x)\right )\) |
\(\Big \downarrow \) 4919 |
\(\displaystyle 2 n \int x \csc (2 x)dx+x \log \left (a \cot ^n(x)\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 n \int x \csc (2 x)dx+x \log \left (a \cot ^n(x)\right )\) |
\(\Big \downarrow \) 4671 |
\(\displaystyle x \log \left (a \cot ^n(x)\right )+2 n \left (-\frac {1}{2} \int \log \left (1-e^{2 i x}\right )dx+\frac {1}{2} \int \log \left (1+e^{2 i x}\right )dx-x \text {arctanh}\left (e^{2 i x}\right )\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle x \log \left (a \cot ^n(x)\right )+2 n \left (\frac {1}{4} i \int e^{-2 i x} \log \left (1-e^{2 i x}\right )de^{2 i x}-\frac {1}{4} i \int e^{-2 i x} \log \left (1+e^{2 i x}\right )de^{2 i x}-x \text {arctanh}\left (e^{2 i x}\right )\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle x \log \left (a \cot ^n(x)\right )+2 n \left (-x \text {arctanh}\left (e^{2 i x}\right )+\frac {1}{4} i \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{4} i \operatorname {PolyLog}\left (2,e^{2 i x}\right )\right )\) |
x*Log[a*Cot[x]^n] + 2*n*(-(x*ArcTanh[E^((2*I)*x)]) + (I/4)*PolyLog[2, -E^( (2*I)*x)] - (I/4)*PolyLog[2, E^((2*I)*x)])
3.2.72.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[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]
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]
Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/u), x], x] /; InverseFunctionFreeQ[u, x]
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] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Simp[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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 6.92 (sec) , antiderivative size = 2197, normalized size of antiderivative = 39.23
x*ln((exp(2*I*x)-1)^(-n)*(1+exp(2*I*x))^n*exp(-1/2*I*Pi*n*(-csgn(I*(1+exp( 2*I*x)))*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2+csgn(I*(1+exp(2*I*x)))*cs gn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))*csgn(I/(exp(2*I*x)-1))+csgn(I/(exp(2*I *x)-1)*(1+exp(2*I*x)))^3-csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*csgn(I/(e xp(2*I*x)-1))-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-csgn(1/(exp(2*I*x)-1)*(1+exp(2*I*x)))^3+csgn(1/(exp(2*I* x)-1)*(1+exp(2*I*x)))*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))+csgn(1/(exp(2* I*x)-1)*(1+exp(2*I*x)))^2-1)))+1/2*I*Pi*csgn(I*(exp(2*I*x)-1)^(-n)*(1+exp( 2*I*x))^n*exp(-1/2*I*Pi*n*(-csgn(I*(1+exp(2*I*x)))*csgn(I/(exp(2*I*x)-1)*( 1+exp(2*I*x)))^2+csgn(I*(1+exp(2*I*x)))*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x )))*csgn(I/(exp(2*I*x)-1))+csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))^3-csgn(I/ (exp(2*I*x)-1)*(1+exp(2*I*x)))^2*csgn(I/(exp(2*I*x)-1))-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-csgn(1/(exp(2* I*x)-1)*(1+exp(2*I*x)))^3+csgn(1/(exp(2*I*x)-1)*(1+exp(2*I*x)))*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)))^2-1)))*c sgn(I*a*(exp(2*I*x)-1)^(-n)*(1+exp(2*I*x))^n*exp(-1/2*I*Pi*n*(-csgn(I*(1+e xp(2*I*x)))*csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2+csgn(I*(1+exp(2*I*x))) *csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))*csgn(I/(exp(2*I*x)-1))+csgn(I/(exp( 2*I*x)-1)*(1+exp(2*I*x)))^3-csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))^2*csgn(I /(exp(2*I*x)-1))-csgn(I/(exp(2*I*x)-1)*(1+exp(2*I*x)))*csgn(1/(exp(2*I*...
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 = 0.35 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.82 \[ \int \log \left (a \cot ^n(x)\right ) \, dx=n x \log \left (\frac {\cos \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}\right ) - \frac {1}{2} \, n x \log \left (\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right ) + 1\right ) - \frac {1}{2} \, n x \log \left (\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right ) + 1\right ) + \frac {1}{2} \, n x \log \left (-\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right ) + 1\right ) + \frac {1}{2} \, n x \log \left (-\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right ) + 1\right ) - \frac {1}{4} i \, n {\rm Li}_2\left (\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right )\right ) + \frac {1}{4} i \, n {\rm Li}_2\left (\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right )\right ) - \frac {1}{4} i \, n {\rm Li}_2\left (-\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right )\right ) + \frac {1}{4} i \, n {\rm Li}_2\left (-\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right )\right ) + x \log \left (a\right ) \]
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(-co s(2*x) + I*sin(2*x)) + 1/4*I*n*dilog(-cos(2*x) - I*sin(2*x)) + x*log(a)
\[ \int \log \left (a \cot ^n(x)\right ) \, dx=\int \log {\left (a \cot ^{n}{\left (x \right )} \right )}\, dx \]
Time = 0.29 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88 \[ \int \log \left (a \cot ^n(x)\right ) \, dx=n x \log \left (\tan \left (x\right )\right ) - \frac {1}{4} \, {\left (\pi \log \left (\tan \left (x\right )^{2} + 1\right ) + 2 i \, {\rm Li}_2\left (i \, \tan \left (x\right ) + 1\right ) - 2 i \, {\rm Li}_2\left (-i \, \tan \left (x\right ) + 1\right )\right )} n + x \log \left (a \frac {1}{\tan \left (x\right )}^{n}\right ) \]
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)
\[ \int \log \left (a \cot ^n(x)\right ) \, dx=\int { \log \left (a \cot \left (x\right )^{n}\right ) \,d x } \]
Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.79 \[ \int \log \left (a \cot ^n(x)\right ) \, dx=x\,\ln \left (a\,{\mathrm {cot}\left (x\right )}^n\right )-\frac {n\,\mathrm {polylog}\left (2,{\mathrm {e}}^{x\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}+\frac {n\,\mathrm {polylog}\left (2,-{\mathrm {e}}^{x\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}-2\,n\,x\,\mathrm {atanh}\left ({\mathrm {e}}^{x\,2{}\mathrm {i}}\right ) \]