Integrand size = 5, antiderivative size = 38 \[ \int \log (a \text {csch}(x)) \, dx=-\frac {x^2}{2}+x \log \left (1-e^{2 x}\right )+x \log (a \text {csch}(x))+\frac {\operatorname {PolyLog}\left (2,e^{2 x}\right )}{2} \]
Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \log (a \text {csch}(x)) \, dx=\frac {x^2}{2}+x \log \left (1-e^{-2 x}\right )+x \log (a \text {csch}(x))-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{-2 x}\right ) \]
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.39, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.800, Rules used = {3028, 25, 3042, 26, 4199, 25, 2620, 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.
\(\displaystyle \int \log (a \text {csch}(x)) \, dx\) |
\(\Big \downarrow \) 3028 |
\(\displaystyle x \log (a \text {csch}(x))-\int -x \coth (x)dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int x \coth (x)dx+x \log (a \text {csch}(x))\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle x \log (a \text {csch}(x))+\int -i x \tan \left (i x+\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle x \log (a \text {csch}(x))-i \int x \tan \left (i x+\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 4199 |
\(\displaystyle x \log (a \text {csch}(x))-i \left (2 i \int -\frac {e^{2 x} x}{1-e^{2 x}}dx-\frac {i x^2}{2}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle x \log (a \text {csch}(x))-i \left (-2 i \int \frac {e^{2 x} x}{1-e^{2 x}}dx-\frac {i x^2}{2}\right )\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle x \log (a \text {csch}(x))-i \left (-2 i \left (\frac {1}{2} \int \log \left (1-e^{2 x}\right )dx-\frac {1}{2} x \log \left (1-e^{2 x}\right )\right )-\frac {i x^2}{2}\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle x \log (a \text {csch}(x))-i \left (-2 i \left (\frac {1}{4} \int e^{-2 x} \log \left (1-e^{2 x}\right )de^{2 x}-\frac {1}{2} x \log \left (1-e^{2 x}\right )\right )-\frac {i x^2}{2}\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle x \log (a \text {csch}(x))-i \left (-2 i \left (-\frac {\operatorname {PolyLog}\left (2,e^{2 x}\right )}{4}-\frac {1}{2} x \log \left (1-e^{2 x}\right )\right )-\frac {i x^2}{2}\right )\) |
x*Log[a*Csch[x]] - I*((-1/2*I)*x^2 - (2*I)*(-1/2*(x*Log[1 - E^(2*x)]) - Po lyLog[2, E^(2*x)]/4))
3.3.18.3.1 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[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_ .)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp [2*I Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x ))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In tegerQ[4*k] && IGtQ[m, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.81 (sec) , antiderivative size = 293, normalized size of antiderivative = 7.71
method | result | size |
risch | \(x \ln \left ({\mathrm e}^{x}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{-1+{\mathrm e}^{2 x}}\right ) \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{-1+{\mathrm e}^{2 x}}\right )^{2} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{-1+{\mathrm e}^{2 x}}\right )^{2} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{-1+{\mathrm e}^{2 x}}\right ) \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{-1+{\mathrm e}^{2 x}}\right ) \operatorname {csgn}\left (i a \right ) x}{2}+\frac {i \pi \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{-1+{\mathrm e}^{2 x}}\right )^{2} \operatorname {csgn}\left (i a \right ) x}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{-1+{\mathrm e}^{2 x}}\right )^{3} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{x}\right ) \operatorname {csgn}\left (\frac {i}{-1+{\mathrm e}^{2 x}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{-1+{\mathrm e}^{2 x}}\right ) x}{2}+x \ln \left (2\right )+\ln \left (a \right ) x -\frac {x^{2}}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{-1+{\mathrm e}^{2 x}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{-1+{\mathrm e}^{2 x}}\right )^{2} x}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{-1+{\mathrm e}^{2 x}}\right )^{3} x}{2}-\ln \left ({\mathrm e}^{x}\right ) \ln \left (-1+{\mathrm e}^{2 x}\right )+\operatorname {dilog}\left (1+{\mathrm e}^{x}\right )+\ln \left ({\mathrm e}^{x}\right ) \ln \left (1+{\mathrm e}^{x}\right )-\operatorname {dilog}\left ({\mathrm e}^{x}\right )\) | \(293\) |
x*ln(exp(x))+1/2*I*Pi*csgn(I*exp(x)/(-1+exp(2*x)))*csgn(I*a/(-1+exp(2*x))* exp(x))^2*x+1/2*I*Pi*csgn(I*exp(x))*csgn(I*exp(x)/(-1+exp(2*x)))^2*x-1/2*I *Pi*csgn(I*exp(x)/(-1+exp(2*x)))*csgn(I*a/(-1+exp(2*x))*exp(x))*csgn(I*a)* x+1/2*I*Pi*csgn(I*a/(-1+exp(2*x))*exp(x))^2*csgn(I*a)*x-1/2*I*Pi*csgn(I*a/ (-1+exp(2*x))*exp(x))^3*x-1/2*I*Pi*csgn(I*exp(x))*csgn(I/(-1+exp(2*x)))*cs gn(I*exp(x)/(-1+exp(2*x)))*x+x*ln(2)+ln(a)*x-1/2*x^2+1/2*I*Pi*csgn(I/(-1+e xp(2*x)))*csgn(I*exp(x)/(-1+exp(2*x)))^2*x-1/2*I*Pi*csgn(I*exp(x)/(-1+exp( 2*x)))^3*x-ln(exp(x))*ln(-1+exp(2*x))+dilog(1+exp(x))+ln(exp(x))*ln(1+exp( x))-dilog(exp(x))
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (31) = 62\).
Time = 0.35 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.00 \[ \int \log (a \text {csch}(x)) \, dx=-\frac {1}{2} \, x^{2} + x \log \left (\frac {2 \, {\left (a \cosh \left (x\right ) + a \sinh \left (x\right )\right )}}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1}\right ) + x \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + x \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) \]
-1/2*x^2 + x*log(2*(a*cosh(x) + a*sinh(x))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)) + x*log(cosh(x) + sinh(x) + 1) + x*log(-cosh(x) - sinh(x ) + 1) + dilog(cosh(x) + sinh(x)) + dilog(-cosh(x) - sinh(x))
\[ \int \log (a \text {csch}(x)) \, dx=\int \log {\left (a \operatorname {csch}{\left (x \right )} \right )}\, dx \]
Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97 \[ \int \log (a \text {csch}(x)) \, dx=-\frac {1}{2} \, x^{2} + x \log \left (a \operatorname {csch}\left (x\right )\right ) + x \log \left (e^{x} + 1\right ) + x \log \left (-e^{x} + 1\right ) + {\rm Li}_2\left (-e^{x}\right ) + {\rm Li}_2\left (e^{x}\right ) \]
\[ \int \log (a \text {csch}(x)) \, dx=\int { \log \left (a \operatorname {csch}\left (x\right )\right ) \,d x } \]
Timed out. \[ \int \log (a \text {csch}(x)) \, dx=\int \ln \left (\frac {a}{\mathrm {sinh}\left (x\right )}\right ) \,d x \]