Integrand size = 3, antiderivative size = 39 \[ \int \log (\tanh (x)) \, dx=2 x \text {arctanh}\left (e^{2 x}\right )+x \log (\tanh (x))+\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 x}\right )-\frac {\operatorname {PolyLog}\left (2,e^{2 x}\right )}{2} \]
Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.90 \[ \int \log (\tanh (x)) \, dx=\frac {1}{2} \log (\tanh (x)) \log (1+\tanh (x))+\frac {1}{2} \operatorname {PolyLog}(2,1-\tanh (x))+\frac {1}{2} \operatorname {PolyLog}(2,-\tanh (x)) \]
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.28, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.333, Rules used = {3028, 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.
\(\displaystyle \int \log (\tanh (x)) \, dx\) |
\(\Big \downarrow \) 3028 |
\(\displaystyle x \log (\tanh (x))-\int x \text {csch}(x) \text {sech}(x)dx\) |
\(\Big \downarrow \) 5984 |
\(\displaystyle x \log (\tanh (x))-2 \int x \text {csch}(2 x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle x \log (\tanh (x))-2 \int i x \csc (2 i x)dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle x \log (\tanh (x))-2 i \int x \csc (2 i x)dx\) |
\(\Big \downarrow \) 4670 |
\(\displaystyle x \log (\tanh (x))-2 i \left (\frac {1}{2} i \int \log \left (1-e^{2 x}\right )dx-\frac {1}{2} i \int \log \left (1+e^{2 x}\right )dx+i x \text {arctanh}\left (e^{2 x}\right )\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle x \log (\tanh (x))-2 i \left (\frac {1}{4} i \int e^{-2 x} \log \left (1-e^{2 x}\right )de^{2 x}-\frac {1}{4} i \int e^{-2 x} \log \left (1+e^{2 x}\right )de^{2 x}+i x \text {arctanh}\left (e^{2 x}\right )\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle x \log (\tanh (x))-2 i \left (i x \text {arctanh}\left (e^{2 x}\right )+\frac {1}{4} i \operatorname {PolyLog}\left (2,-e^{2 x}\right )-\frac {1}{4} i \operatorname {PolyLog}\left (2,e^{2 x}\right )\right )\) |
x*Log[Tanh[x]] - (2*I)*(I*x*ArcTanh[E^(2*x)] + (I/4)*PolyLog[2, -E^(2*x)] - (I/4)*PolyLog[2, E^(2*x)])
3.3.7.3.1 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[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 = 0.50 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.62
method | result | size |
derivativedivides | \(\frac {\operatorname {dilog}\left (\tanh \left (x \right )\right )}{2}+\frac {\operatorname {dilog}\left (\tanh \left (x \right )+1\right )}{2}+\frac {\ln \left (\tanh \left (x \right )\right ) \ln \left (\tanh \left (x \right )+1\right )}{2}\) | \(24\) |
default | \(\frac {\operatorname {dilog}\left (\tanh \left (x \right )\right )}{2}+\frac {\operatorname {dilog}\left (\tanh \left (x \right )+1\right )}{2}+\frac {\ln \left (\tanh \left (x \right )\right ) \ln \left (\tanh \left (x \right )+1\right )}{2}\) | \(24\) |
risch | \(x \ln \left (-1+{\mathrm e}^{2 x}\right )+\frac {i \pi \,\operatorname {csgn}\left (i \left (-1+{\mathrm e}^{2 x}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (-1+{\mathrm e}^{2 x}\right )}{1+{\mathrm e}^{2 x}}\right )}^{2} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{1+{\mathrm e}^{2 x}}\right ) {\operatorname {csgn}\left (\frac {i \left (-1+{\mathrm e}^{2 x}\right )}{1+{\mathrm e}^{2 x}}\right )}^{2} x}{2}-\operatorname {dilog}\left (1+{\mathrm e}^{x}\right )-x \ln \left (1+{\mathrm e}^{x}\right )+\operatorname {dilog}\left ({\mathrm e}^{x}\right )-\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (-1+{\mathrm e}^{2 x}\right )}{1+{\mathrm e}^{2 x}}\right )}^{3} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (-1+{\mathrm e}^{2 x}\right )\right ) \operatorname {csgn}\left (\frac {i}{1+{\mathrm e}^{2 x}}\right ) \operatorname {csgn}\left (\frac {i \left (-1+{\mathrm e}^{2 x}\right )}{1+{\mathrm e}^{2 x}}\right ) x}{2}-\ln \left ({\mathrm e}^{x}\right ) \ln \left (1+{\mathrm e}^{2 x}\right )+\ln \left ({\mathrm e}^{x}\right ) \ln \left (1+i {\mathrm e}^{x}\right )+\ln \left ({\mathrm e}^{x}\right ) \ln \left (1-i {\mathrm e}^{x}\right )+\operatorname {dilog}\left (1+i {\mathrm e}^{x}\right )+\operatorname {dilog}\left (1-i {\mathrm e}^{x}\right )\) | \(223\) |
Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.59 \[ \int \log (\tanh (x)) \, dx=x \log \left (\frac {\sinh \left (x\right )}{\cosh \left (x\right )}\right ) - x \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + x \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) + x \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) - x \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) - {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) + {\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\right ) - {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) \]
x*log(sinh(x)/cosh(x)) - x*log(cosh(x) + sinh(x) + 1) + x*log(I*cosh(x) + I*sinh(x) + 1) + x*log(-I*cosh(x) - I*sinh(x) + 1) - x*log(-cosh(x) - sinh (x) + 1) - dilog(cosh(x) + sinh(x)) + dilog(I*cosh(x) + I*sinh(x)) + dilog (-I*cosh(x) - I*sinh(x)) - dilog(-cosh(x) - sinh(x))
\[ \int \log (\tanh (x)) \, dx=\int \log {\left (\tanh {\left (x \right )} \right )}\, dx \]
Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.38 \[ \int \log (\tanh (x)) \, dx=x \log \left (e^{\left (2 \, x\right )} + 1\right ) - x \log \left (e^{x} + 1\right ) - x \log \left (-e^{x} + 1\right ) + x \log \left (\tanh \left (x\right )\right ) + \frac {1}{2} \, {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) - {\rm Li}_2\left (-e^{x}\right ) - {\rm Li}_2\left (e^{x}\right ) \]
x*log(e^(2*x) + 1) - x*log(e^x + 1) - x*log(-e^x + 1) + x*log(tanh(x)) + 1 /2*dilog(-e^(2*x)) - dilog(-e^x) - dilog(e^x)
\[ \int \log (\tanh (x)) \, dx=\int { \log \left (\tanh \left (x\right )\right ) \,d x } \]
Time = 1.65 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.51 \[ \int \log (\tanh (x)) \, dx=x\,\ln \left (\mathrm {tanh}\left (x\right )\right )-\frac {\mathrm {polylog}\left (2,\mathrm {tanh}\left (x\right )\right )}{2}+\frac {\mathrm {polylog}\left (2,-\mathrm {tanh}\left (x\right )\right )}{2} \]