Integrand size = 7, antiderivative size = 37 \[ \int \log \left (a \coth ^2(x)\right ) \, dx=-4 x \text {arctanh}\left (e^{2 x}\right )+x \log \left (a \coth ^2(x)\right )-\operatorname {PolyLog}\left (2,-e^{2 x}\right )+\operatorname {PolyLog}\left (2,e^{2 x}\right ) \]
Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.27 \[ \int \log \left (a \coth ^2(x)\right ) \, dx=-\frac {1}{2} \log \left (a \coth ^2(x)\right ) \log (1-\tanh (x))+\frac {1}{2} \log \left (a \coth ^2(x)\right ) \log (1+\tanh (x))-\operatorname {PolyLog}(2,-\tanh (x))+\operatorname {PolyLog}(2,\tanh (x)) \]
-1/2*(Log[a*Coth[x]^2]*Log[1 - Tanh[x]]) + (Log[a*Coth[x]^2]*Log[1 + Tanh[ x]])/2 - PolyLog[2, -Tanh[x]] + PolyLog[2, Tanh[x]]
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.46, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.143, Rules used = {3028, 27, 5984, 3042, 26, 4670, 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.
| \(\displaystyle \int \log \left (a \coth ^2(x)\right ) \, dx\) |
| \(\Big \downarrow \) 3028 |
| \(\displaystyle x \log \left (a \coth ^2(x)\right )-\int -2 x \text {csch}(x) \text {sech}(x)dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int x \text {csch}(x) \text {sech}(x)dx+x \log \left (a \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 5984 |
| \(\displaystyle 4 \int x \text {csch}(2 x)dx+x \log \left (a \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle x \log \left (a \coth ^2(x)\right )+4 \int i x \csc (2 i x)dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle x \log \left (a \coth ^2(x)\right )+4 i \int x \csc (2 i x)dx\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle x \log \left (a \coth ^2(x)\right )+4 i \left (\frac {1}{2} i \int \log \left (1-e^{2 x}\right )dx-\frac {1}{2} i \int \log \left (1+e^{2 x}\right )dx+i x \text {arctanh}\left (e^{2 x}\right )\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle x \log \left (a \coth ^2(x)\right )+4 i \left (\frac {1}{4} i \int e^{-2 x} \log \left (1-e^{2 x}\right )de^{2 x}-\frac {1}{4} i \int e^{-2 x} \log \left (1+e^{2 x}\right )de^{2 x}+i x \text {arctanh}\left (e^{2 x}\right )\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle x \log \left (a \coth ^2(x)\right )+4 i \left (i x \text {arctanh}\left (e^{2 x}\right )+\frac {1}{4} i \operatorname {PolyLog}\left (2,-e^{2 x}\right )-\frac {1}{4} i \operatorname {PolyLog}\left (2,e^{2 x}\right )\right )\) |
x*Log[a*Coth[x]^2] + (4*I)*(I*x*ArcTanh[E^(2*x)] + (I/4)*PolyLog[2, -E^(2* x)] - (I/4)*PolyLog[2, E^(2*x)])
3.3.13.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n Int[(c + d*x)^m*Csch[2*a + 2*b*x ]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
Time = 1.14 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.27
| method | result | size |
| derivativedivides | \(-\frac {\ln \left (\coth \left (x \right )-1\right ) \ln \left (a \left (\coth ^{2}\left (x \right )\right )\right )}{2}+\operatorname {dilog}\left (\coth \left (x \right )\right )+\ln \left (\coth \left (x \right )-1\right ) \ln \left (\coth \left (x \right )\right )+\frac {\ln \left (\coth \left (x \right )+1\right ) \ln \left (a \left (\coth ^{2}\left (x \right )\right )\right )}{2}+\operatorname {dilog}\left (\coth \left (x \right )+1\right )\) | \(47\) |
| default | \(-\frac {\ln \left (\coth \left (x \right )-1\right ) \ln \left (a \left (\coth ^{2}\left (x \right )\right )\right )}{2}+\operatorname {dilog}\left (\coth \left (x \right )\right )+\ln \left (\coth \left (x \right )-1\right ) \ln \left (\coth \left (x \right )\right )+\frac {\ln \left (\coth \left (x \right )+1\right ) \ln \left (a \left (\coth ^{2}\left (x \right )\right )\right )}{2}+\operatorname {dilog}\left (\coth \left (x \right )+1\right )\) | \(47\) |
| risch | \(-2 x \ln \left (-1+{\mathrm e}^{2 x}\right )-\frac {i \pi {\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right )}^{2} \operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right ) x}{2}-\frac {i \pi {\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right )}^{3} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right ) {\operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 x}\right )^{2}}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right )}^{2} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 x}\right )^{2}}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right ) {\operatorname {csgn}\left (\frac {i a \left (1+{\mathrm e}^{2 x}\right )^{2}}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right )}^{2} x}{2}+\frac {i \pi {\operatorname {csgn}\left (\frac {i a \left (1+{\mathrm e}^{2 x}\right )^{2}}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right )}^{2} \operatorname {csgn}\left (i a \right ) x}{2}+\frac {i \pi {\operatorname {csgn}\left (i \left (-1+{\mathrm e}^{2 x}\right )^{2}\right )}^{3} x}{2}+\ln \left (a \right ) x +2 \operatorname {dilog}\left (1+{\mathrm e}^{x}\right )+2 x \ln \left (1+{\mathrm e}^{x}\right )-2 \operatorname {dilog}\left ({\mathrm e}^{x}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 x}\right )^{2}}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right )}^{2} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 x}\right )^{2}}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i a \left (1+{\mathrm e}^{2 x}\right )^{2}}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right ) \operatorname {csgn}\left (i a \right ) x}{2}-\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 x}\right )^{2}}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right )}^{3} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right ) \operatorname {csgn}\left (\frac {i}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 x}\right )^{2}}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right ) x}{2}-\frac {i \pi {\operatorname {csgn}\left (\frac {i a \left (1+{\mathrm e}^{2 x}\right )^{2}}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right )}^{3} x}{2}+\frac {i \pi {\operatorname {csgn}\left (i \left (-1+{\mathrm e}^{2 x}\right )\right )}^{2} \operatorname {csgn}\left (i \left (-1+{\mathrm e}^{2 x}\right )^{2}\right ) x}{2}+i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) {\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right )}^{2} x -i \pi \,\operatorname {csgn}\left (i \left (-1+{\mathrm e}^{2 x}\right )\right ) {\operatorname {csgn}\left (i \left (-1+{\mathrm e}^{2 x}\right )^{2}\right )}^{2} x +2 \ln \left ({\mathrm e}^{x}\right ) \ln \left (1+{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right ) \ln \left (1+i {\mathrm e}^{x}\right )-2 \ln \left ({\mathrm e}^{x}\right ) \ln \left (1-i {\mathrm e}^{x}\right )-2 \operatorname {dilog}\left (1+i {\mathrm e}^{x}\right )-2 \operatorname {dilog}\left (1-i {\mathrm e}^{x}\right )\) | \(562\) |
-1/2*ln(coth(x)-1)*ln(a*coth(x)^2)+dilog(coth(x))+ln(coth(x)-1)*ln(coth(x) )+1/2*ln(coth(x)+1)*ln(a*coth(x)^2)+dilog(coth(x)+1)
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 127, normalized size of antiderivative = 3.43 \[ \int \log \left (a \coth ^2(x)\right ) \, dx=x \log \left (\frac {a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + a}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 1}\right ) + 2 \, x \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - 2 \, x \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) - 2 \, x \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) + 2 \, x \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + 2 \, {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - 2 \, {\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) - 2 \, {\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\right ) + 2 \, {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) \]
x*log((a*cosh(x)^2 + a*sinh(x)^2 + a)/(cosh(x)^2 + sinh(x)^2 - 1)) + 2*x*l og(cosh(x) + sinh(x) + 1) - 2*x*log(I*cosh(x) + I*sinh(x) + 1) - 2*x*log(- I*cosh(x) - I*sinh(x) + 1) + 2*x*log(-cosh(x) - sinh(x) + 1) + 2*dilog(cos h(x) + sinh(x)) - 2*dilog(I*cosh(x) + I*sinh(x)) - 2*dilog(-I*cosh(x) - I* sinh(x)) + 2*dilog(-cosh(x) - sinh(x))
\[ \int \log \left (a \coth ^2(x)\right ) \, dx=\int \log {\left (a \coth ^{2}{\left (x \right )} \right )}\, dx \]
Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.59 \[ \int \log \left (a \coth ^2(x)\right ) \, dx=x \log \left (a \coth \left (x\right )^{2}\right ) - 2 \, x \log \left (e^{\left (2 \, x\right )} + 1\right ) + 2 \, x \log \left (e^{x} + 1\right ) + 2 \, x \log \left (-e^{x} + 1\right ) - {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) + 2 \, {\rm Li}_2\left (-e^{x}\right ) + 2 \, {\rm Li}_2\left (e^{x}\right ) \]
x*log(a*coth(x)^2) - 2*x*log(e^(2*x) + 1) + 2*x*log(e^x + 1) + 2*x*log(-e^ x + 1) - dilog(-e^(2*x)) + 2*dilog(-e^x) + 2*dilog(e^x)
\[ \int \log \left (a \coth ^2(x)\right ) \, dx=\int { \log \left (a \coth \left (x\right )^{2}\right ) \,d x } \]
Timed out. \[ \int \log \left (a \coth ^2(x)\right ) \, dx=\int \ln \left (a\,{\mathrm {coth}\left (x\right )}^2\right ) \,d x \]