Integrand size = 3, antiderivative size = 27 \[ \int \coth ^{-1}(\cosh (x)) \, dx=x \coth ^{-1}(\cosh (x))-2 x \text {arctanh}\left (e^x\right )-\operatorname {PolyLog}\left (2,-e^x\right )+\operatorname {PolyLog}\left (2,e^x\right ) \]
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \coth ^{-1}(\cosh (x)) \, dx=x \coth ^{-1}(\cosh (x))+x \left (\log \left (1-e^x\right )-\log \left (1+e^x\right )\right )-\operatorname {PolyLog}\left (2,-e^x\right )+\operatorname {PolyLog}\left (2,e^x\right ) \]
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.333, Rules used = {6826, 25, 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.
\(\displaystyle \int \coth ^{-1}(\cosh (x)) \, dx\) |
\(\Big \downarrow \) 6826 |
\(\displaystyle x \coth ^{-1}(\cosh (x))-\int -x \text {csch}(x)dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int x \text {csch}(x)dx+x \coth ^{-1}(\cosh (x))\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle x \coth ^{-1}(\cosh (x))+\int i x \csc (i x)dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle x \coth ^{-1}(\cosh (x))+i \int x \csc (i x)dx\) |
\(\Big \downarrow \) 4670 |
\(\displaystyle x \coth ^{-1}(\cosh (x))+i \left (i \int \log \left (1-e^x\right )dx-i \int \log \left (1+e^x\right )dx+2 i x \text {arctanh}\left (e^x\right )\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle x \coth ^{-1}(\cosh (x))+i \left (i \int e^{-x} \log \left (1-e^x\right )de^x-i \int e^{-x} \log \left (1+e^x\right )de^x+2 i x \text {arctanh}\left (e^x\right )\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle x \coth ^{-1}(\cosh (x))+i \left (2 i x \text {arctanh}\left (e^x\right )+i \operatorname {PolyLog}\left (2,-e^x\right )-i \operatorname {PolyLog}\left (2,e^x\right )\right )\) |
3.2.100.3.1 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[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[ArcCoth[u_], x_Symbol] :> Simp[x*ArcCoth[u], x] - Int[SimplifyIntegrand [x*(D[u, x]/(1 - u^2)), x], x] /; InverseFunctionFreeQ[u, x]
Time = 0.30 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33
method | result | size |
default | \(x \,\operatorname {arccoth}\left (\cosh \left (x \right )\right )+x \ln \left (1-{\mathrm e}^{x}\right )+\operatorname {polylog}\left (2, {\mathrm e}^{x}\right )-x \ln \left (1+{\mathrm e}^{x}\right )-\operatorname {polylog}\left (2, -{\mathrm e}^{x}\right )\) | \(36\) |
parts | \(x \,\operatorname {arccoth}\left (\cosh \left (x \right )\right )+x \ln \left (1-{\mathrm e}^{x}\right )+\operatorname {polylog}\left (2, {\mathrm e}^{x}\right )-x \ln \left (1+{\mathrm e}^{x}\right )-\operatorname {polylog}\left (2, -{\mathrm e}^{x}\right )\) | \(36\) |
risch | \(-\frac {i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{2} x}{4}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{2} x}{4}+\frac {i \pi {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right )}^{3} x}{4}-x \ln \left ({\mathrm e}^{x}-1\right )-\operatorname {dilog}\left ({\mathrm e}^{x}\right )-\operatorname {dilog}\left (1+{\mathrm e}^{x}\right )-\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{x}\right )^{2}\right )^{3} x}{4}+\frac {i \pi {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )\right )}^{2} \operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right ) x}{4}+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{x}\right )^{2}\right )^{2} x}{4}+\frac {i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{x}\right )^{2}\right )^{2} x}{4}-\frac {i \pi {\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )\right )}^{2} \operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )^{2}\right ) x}{4}-\frac {i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{x}\right )^{2}\right ) x}{4}-\frac {i \pi {\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )^{2}\right )}^{3} x}{4}+\frac {i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right ) x}{4}+\frac {i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )\right ) {\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )^{2}\right )}^{2} x}{2}+\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{3} x}{4}-\frac {i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )\right ) {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right )}^{2} x}{2}\) | \(392\) |
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (22) = 44\).
Time = 0.25 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.11 \[ \int \coth ^{-1}(\cosh (x)) \, dx=\frac {1}{2} \, x \log \left (\frac {\cosh \left (x\right ) + 1}{\cosh \left (x\right ) - 1}\right ) - x \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + x \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) \]
1/2*x*log((cosh(x) + 1)/(cosh(x) - 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))
\[ \int \coth ^{-1}(\cosh (x)) \, dx=\int \operatorname {acoth}{\left (\cosh {\left (x \right )} \right )}\, dx \]
Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \coth ^{-1}(\cosh (x)) \, dx=x \operatorname {arcoth}\left (\cosh \left (x\right )\right ) - x \log \left (e^{x} + 1\right ) + x \log \left (-e^{x} + 1\right ) - {\rm Li}_2\left (-e^{x}\right ) + {\rm Li}_2\left (e^{x}\right ) \]
\[ \int \coth ^{-1}(\cosh (x)) \, dx=\int { \operatorname {arcoth}\left (\cosh \left (x\right )\right ) \,d x } \]
Timed out. \[ \int \coth ^{-1}(\cosh (x)) \, dx=\int \mathrm {acoth}\left (\mathrm {cosh}\left (x\right )\right ) \,d x \]