Integrand size = 5, antiderivative size = 38 \[ \int \log (a \text {sech}(x)) \, dx=-\frac {x^2}{2}+x \log \left (1+e^{2 x}\right )+x \log (a \text {sech}(x))+\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 x}\right ) \]
Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \log (a \text {sech}(x)) \, dx=\frac {x^2}{2}+x \log \left (1+e^{-2 x}\right )+x \log (a \text {sech}(x))-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{-2 x}\right ) \]
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.39, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.600, Rules used = {3028, 25, 3042, 26, 4201, 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.
\(\displaystyle \int \log (a \text {sech}(x)) \, dx\) |
\(\Big \downarrow \) 3028 |
\(\displaystyle x \log (a \text {sech}(x))-\int -x \tanh (x)dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int x \tanh (x)dx+x \log (a \text {sech}(x))\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle x \log (a \text {sech}(x))+\int -i x \tan (i x)dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle x \log (a \text {sech}(x))-i \int x \tan (i x)dx\) |
\(\Big \downarrow \) 4201 |
\(\displaystyle x \log (a \text {sech}(x))-i \left (2 i \int \frac {e^{2 x} x}{1+e^{2 x}}dx-\frac {i x^2}{2}\right )\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle x \log (a \text {sech}(x))-i \left (2 i \left (\frac {1}{2} x \log \left (e^{2 x}+1\right )-\frac {1}{2} \int \log \left (1+e^{2 x}\right )dx\right )-\frac {i x^2}{2}\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle x \log (a \text {sech}(x))-i \left (2 i \left (\frac {1}{2} x \log \left (e^{2 x}+1\right )-\frac {1}{4} \int e^{-2 x} \log \left (1+e^{2 x}\right )de^{2 x}\right )-\frac {i x^2}{2}\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle x \log (a \text {sech}(x))-i \left (2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 x}\right )+\frac {1}{2} x \log \left (e^{2 x}+1\right )\right )-\frac {i x^2}{2}\right )\) |
x*Log[a*Sech[x]] - I*((-1/2*I)*x^2 + (2*I)*((x*Log[1 + E^(2*x)])/2 + PolyL og[2, -E^(2*x)]/4))
3.3.15.3.1 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_.) + (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)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.59 (sec) , antiderivative size = 314, normalized size of antiderivative = 8.26
method | result | size |
risch | \(x \ln \left ({\mathrm e}^{x}\right )+\frac {i \pi \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{1+{\mathrm e}^{2 x}}\right )^{2} \operatorname {csgn}\left (i a \right ) x}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{1+{\mathrm e}^{2 x}}\right ) \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{1+{\mathrm e}^{2 x}}\right )^{2} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{1+{\mathrm e}^{2 x}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{1+{\mathrm e}^{2 x}}\right )^{2} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{x}\right ) \operatorname {csgn}\left (\frac {i}{1+{\mathrm e}^{2 x}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{1+{\mathrm e}^{2 x}}\right ) x}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{1+{\mathrm e}^{2 x}}\right ) \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{1+{\mathrm e}^{2 x}}\right ) \operatorname {csgn}\left (i a \right ) x}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{1+{\mathrm e}^{2 x}}\right )^{3} x}{2}+\ln \left (a \right ) x +x \ln \left (2\right )-\frac {x^{2}}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{1+{\mathrm e}^{2 x}}\right )^{3} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{1+{\mathrm e}^{2 x}}\right )^{2} x}{2}-\ln \left ({\mathrm e}^{x}\right ) \ln \left (1+{\mathrm e}^{2 x}\right )+\ln \left ({\mathrm e}^{x}\right ) \ln \left (1+i {\mathrm e}^{x}\right )+\ln \left ({\mathrm e}^{x}\right ) \ln \left (1-i {\mathrm e}^{x}\right )+\operatorname {dilog}\left (1+i {\mathrm e}^{x}\right )+\operatorname {dilog}\left (1-i {\mathrm e}^{x}\right )\) | \(314\) |
x*ln(exp(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*exp(x)/(1+exp(2*x)))*csgn(I*a/(1+exp(2*x))*exp(x))^2*x+1/2*I*Pi*cs gn(I/(1+exp(2*x)))*csgn(I*exp(x)/(1+exp(2*x)))^2*x-1/2*I*Pi*csgn(I*exp(x)) *csgn(I/(1+exp(2*x)))*csgn(I*exp(x)/(1+exp(2*x)))*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*e xp(x)/(1+exp(2*x)))^3*x+ln(a)*x+x*ln(2)-1/2*x^2-1/2*I*Pi*csgn(I*a/(1+exp(2 *x))*exp(x))^3*x+1/2*I*Pi*csgn(I*exp(x))*csgn(I*exp(x)/(1+exp(2*x)))^2*x-l n(exp(x))*ln(1+exp(2*x))+ln(exp(x))*ln(1+I*exp(x))+ln(exp(x))*ln(1-I*exp(x ))+dilog(1+I*exp(x))+dilog(1-I*exp(x))
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.21 \[ \int \log (a \text {sech}(x)) \, dx=-\frac {1}{2} \, x^{2} + x \log \left (\frac {2 \, {\left (a \cosh \left (x\right ) + a \sinh \left (x\right )\right )}}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}\right ) + x \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) + x \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) + {\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) + {\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\right ) \]
-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(I*cosh(x) + I*sinh(x) + 1) + x*log(-I*cosh(x) - I*sinh(x) + 1) + dilog(I*cosh(x) + I*sinh(x)) + dilog(-I*cosh(x) - I*sinh( x))
\[ \int \log (a \text {sech}(x)) \, dx=\int \log {\left (a \operatorname {sech}{\left (x \right )} \right )}\, dx \]
Time = 0.31 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.82 \[ \int \log (a \text {sech}(x)) \, dx=-\frac {1}{2} \, x^{2} + x \log \left (a \operatorname {sech}\left (x\right )\right ) + x \log \left (e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{2} \, {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) \]
\[ \int \log (a \text {sech}(x)) \, dx=\int { \log \left (a \operatorname {sech}\left (x\right )\right ) \,d x } \]
Timed out. \[ \int \log (a \text {sech}(x)) \, dx=-\int \ln \left (\mathrm {cosh}\left (x\right )\right )-\ln \left (a\right ) \,d x \]