Integrand size = 151, antiderivative size = 30 \[ \int \frac {e^{4+2 e^{x+2 x \left (i \pi +\log \left (-1+e^2\right )\right )+x \left (i \pi +\log \left (-1+e^2\right )\right )^2}+x+2 x \left (i \pi +\log \left (-1+e^2\right )\right )+x \left (i \pi +\log \left (-1+e^2\right )\right )^2} \left (2+4 \left (i \pi +\log \left (-1+e^2\right )\right )+2 \left (i \pi +\log \left (-1+e^2\right )\right )^2\right )}{-1+e^{4+2 e^{x+2 x \left (i \pi +\log \left (-1+e^2\right )\right )+x \left (i \pi +\log \left (-1+e^2\right )\right )^2}}} \, dx=\log \left (1-e^{4+2 e^{x \left (1+i \pi +\log \left (-1+e^2\right )\right )^2}}\right ) \]
Time = 0.23 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {e^{4+2 e^{x+2 x \left (i \pi +\log \left (-1+e^2\right )\right )+x \left (i \pi +\log \left (-1+e^2\right )\right )^2}+x+2 x \left (i \pi +\log \left (-1+e^2\right )\right )+x \left (i \pi +\log \left (-1+e^2\right )\right )^2} \left (2+4 \left (i \pi +\log \left (-1+e^2\right )\right )+2 \left (i \pi +\log \left (-1+e^2\right )\right )^2\right )}{-1+e^{4+2 e^{x+2 x \left (i \pi +\log \left (-1+e^2\right )\right )+x \left (i \pi +\log \left (-1+e^2\right )\right )^2}}} \, dx=\log \left (1-e^{4+2 e^{-x \left (\pi -i \left (1+\log \left (-1+e^2\right )\right )\right )^2}}\right ) \]
Integrate[(E^(4 + 2*E^(x + 2*x*(I*Pi + Log[-1 + E^2]) + x*(I*Pi + Log[-1 + E^2])^2) + x + 2*x*(I*Pi + Log[-1 + E^2]) + x*(I*Pi + Log[-1 + E^2])^2)*( 2 + 4*(I*Pi + Log[-1 + E^2]) + 2*(I*Pi + Log[-1 + E^2])^2))/(-1 + E^(4 + 2 *E^(x + 2*x*(I*Pi + Log[-1 + E^2]) + x*(I*Pi + Log[-1 + E^2])^2))),x]
Time = 0.75 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {27, 25, 2720, 2676, 16}
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.
| \(\displaystyle \int \frac {\left (2+4 \left (\log \left (e^2-1\right )+i \pi \right )+2 \left (\log \left (e^2-1\right )+i \pi \right )^2\right ) \exp \left (2 \exp \left (x+x \left (\log \left (e^2-1\right )+i \pi \right )^2+2 x \left (\log \left (e^2-1\right )+i \pi \right )\right )+x+x \left (\log \left (e^2-1\right )+i \pi \right )^2+2 x \left (\log \left (e^2-1\right )+i \pi \right )+4\right )}{-1+\exp \left (4+2 \exp \left (x+x \left (\log \left (e^2-1\right )+i \pi \right )^2+2 x \left (\log \left (e^2-1\right )+i \pi \right )\right )\right )} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \left (\pi -i \left (1+\log \left (e^2-1\right )\right )\right )^2 \int -\frac {\exp \left (\left (i \pi +\log \left (-1+e^2\right )\right )^2 x+2 \left (i \pi +\log \left (-1+e^2\right )\right ) x+x+2 e^{-x \left (\pi -i \left (1+\log \left (-1+e^2\right )\right )\right )^2}+4\right )}{1-\exp \left (4+2 e^{-x \left (\pi -i \left (1+\log \left (-1+e^2\right )\right )\right )^2}\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\pi -i \left (1+\log \left (e^2-1\right )\right )\right )^2 \int \frac {\exp \left (\left (i \pi +\log \left (-1+e^2\right )\right )^2 x+2 \left (i \pi +\log \left (-1+e^2\right )\right ) x+x+2 e^{-x \left (\pi -i \left (1+\log \left (-1+e^2\right )\right )\right )^2}+4\right )}{1-\exp \left (4+2 e^{-x \left (\pi -i \left (1+\log \left (-1+e^2\right )\right )\right )^2}\right )}dx\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -2 \int \frac {\exp \left (4+2 e^{-x \left (\pi -i \left (1+\log \left (-1+e^2\right )\right )\right )^2}\right )}{1-\exp \left (4+2 e^{-x \left (\pi -i \left (1+\log \left (-1+e^2\right )\right )\right )^2}\right )}de^{-x \left (\pi -i \left (1+\log \left (-1+e^2\right )\right )\right )^2}\) |
| \(\Big \downarrow \) 2676 |
| \(\displaystyle -\int \frac {1}{1-\exp \left (4+2 e^{-x \left (\pi -i \left (1+\log \left (-1+e^2\right )\right )\right )^2}\right )}d\exp \left (4+2 e^{-x \left (\pi -i \left (1+\log \left (-1+e^2\right )\right )\right )^2}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \log \left (1-\exp \left (4+2 e^{-x \left (\pi -i \left (1+\log \left (e^2-1\right )\right )\right )^2}\right )\right )\) |
Int[(E^(4 + 2*E^(x + 2*x*(I*Pi + Log[-1 + E^2]) + x*(I*Pi + Log[-1 + E^2]) ^2) + x + 2*x*(I*Pi + Log[-1 + E^2]) + x*(I*Pi + Log[-1 + E^2])^2)*(2 + 4* (I*Pi + Log[-1 + E^2]) + 2*(I*Pi + Log[-1 + E^2])^2))/(-1 + E^(4 + 2*E^(x + 2*x*(I*Pi + Log[-1 + E^2]) + x*(I*Pi + Log[-1 + E^2])^2))),x]
3.18.37.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)*((a_) + (b_.)*((F_)^((e_.)*(( c_.) + (d_.)*(x_))))^(n_.))^(p_.), x_Symbol] :> Simp[1/(d*e*n*Log[F]) Sub st[Int[(a + b*x)^p, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d , e, n, p}, x]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Leaf count of result is larger than twice the leaf count of optimal. \(61\) vs. \(2(24)=48\).
Time = 0.23 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.07
| method | result | size |
| derivativedivides | \(\ln \left ({\mathrm e}^{{\mathrm e}^{x \ln \left (1-{\mathrm e}^{2}\right )^{2}+2 x \ln \left (1-{\mathrm e}^{2}\right )+x}+2}-1\right )+\ln \left ({\mathrm e}^{{\mathrm e}^{x \ln \left (1-{\mathrm e}^{2}\right )^{2}+2 x \ln \left (1-{\mathrm e}^{2}\right )+x}+2}+1\right )\) | \(62\) |
| norman | \(\ln \left ({\mathrm e}^{{\mathrm e}^{x \ln \left (1-{\mathrm e}^{2}\right )^{2}+2 x \ln \left (1-{\mathrm e}^{2}\right )+x}+2}-1\right )+\ln \left ({\mathrm e}^{{\mathrm e}^{x \ln \left (1-{\mathrm e}^{2}\right )^{2}+2 x \ln \left (1-{\mathrm e}^{2}\right )+x}+2}+1\right )\) | \(62\) |
| parallelrisch | \(\ln \left ({\mathrm e}^{{\mathrm e}^{x \left (\ln \left (1-{\mathrm e}^{2}\right )^{2}+2 \ln \left (1-{\mathrm e}^{2}\right )+1\right )}+2}-1\right )+\ln \left ({\mathrm e}^{{\mathrm e}^{x \left (\ln \left (1-{\mathrm e}^{2}\right )^{2}+2 \ln \left (1-{\mathrm e}^{2}\right )+1\right )}+2}+1\right )\) | \(106\) |
| default | \(\frac {\left (2 \ln \left (1-{\mathrm e}^{2}\right )^{2}+4 \ln \left (1-{\mathrm e}^{2}\right )+2\right ) \left (\frac {\ln \left ({\mathrm e}^{{\mathrm e}^{x \ln \left (1-{\mathrm e}^{2}\right )^{2}+2 x \ln \left (1-{\mathrm e}^{2}\right )+x}+2}-1\right )}{2}+\frac {\ln \left ({\mathrm e}^{{\mathrm e}^{x \ln \left (1-{\mathrm e}^{2}\right )^{2}+2 x \ln \left (1-{\mathrm e}^{2}\right )+x}+2}+1\right )}{2}\right )}{\ln \left (1-{\mathrm e}^{2}\right )^{2}+2 \ln \left (1-{\mathrm e}^{2}\right )+1}\) | \(111\) |
| risch | \(\frac {\left (2 \ln \left (1-{\mathrm e}^{2}\right )^{2}+4 \ln \left (1-{\mathrm e}^{2}\right )+2\right ) \left (1-{\mathrm e}^{2}\right )^{2 x} {\mathrm e}^{x \left (\ln \left (1-{\mathrm e}^{2}\right )^{2}+1\right )}}{\ln \left (1-{\mathrm e}^{2}\right )^{2}+2 \ln \left (1-{\mathrm e}^{2}\right )+1}-\frac {2 \left (1-{\mathrm e}^{2}\right )^{2 x} {\mathrm e}^{x \left (\ln \left (1-{\mathrm e}^{2}\right )^{2}+1\right )} \ln \left (1-{\mathrm e}^{2}\right )^{2}}{\ln \left (1-{\mathrm e}^{2}\right )^{2}+2 \ln \left (1-{\mathrm e}^{2}\right )+1}-\frac {4 \left (1-{\mathrm e}^{2}\right )^{2 x} {\mathrm e}^{x \left (\ln \left (1-{\mathrm e}^{2}\right )^{2}+1\right )} \ln \left (1-{\mathrm e}^{2}\right )}{\ln \left (1-{\mathrm e}^{2}\right )^{2}+2 \ln \left (1-{\mathrm e}^{2}\right )+1}-\frac {2 \left (1-{\mathrm e}^{2}\right )^{2 x} {\mathrm e}^{x \left (\ln \left (1-{\mathrm e}^{2}\right )^{2}+1\right )}}{\ln \left (1-{\mathrm e}^{2}\right )^{2}+2 \ln \left (1-{\mathrm e}^{2}\right )+1}-\frac {4 \ln \left (1-{\mathrm e}^{2}\right )^{2}}{\ln \left (1-{\mathrm e}^{2}\right )^{2}+2 \ln \left (1-{\mathrm e}^{2}\right )+1}-\frac {8 \ln \left (1-{\mathrm e}^{2}\right )}{\ln \left (1-{\mathrm e}^{2}\right )^{2}+2 \ln \left (1-{\mathrm e}^{2}\right )+1}-\frac {4}{\ln \left (1-{\mathrm e}^{2}\right )^{2}+2 \ln \left (1-{\mathrm e}^{2}\right )+1}+\frac {\ln \left ({\mathrm e}^{2 \left (1-{\mathrm e}^{2}\right )^{2 x} {\mathrm e}^{x \left (\ln \left (1-{\mathrm e}^{2}\right )^{2}+1\right )}+4}-1\right ) \ln \left (1-{\mathrm e}^{2}\right )^{2}}{\ln \left (1-{\mathrm e}^{2}\right )^{2}+2 \ln \left (1-{\mathrm e}^{2}\right )+1}+\frac {2 \ln \left ({\mathrm e}^{2 \left (1-{\mathrm e}^{2}\right )^{2 x} {\mathrm e}^{x \left (\ln \left (1-{\mathrm e}^{2}\right )^{2}+1\right )}+4}-1\right ) \ln \left (1-{\mathrm e}^{2}\right )}{\ln \left (1-{\mathrm e}^{2}\right )^{2}+2 \ln \left (1-{\mathrm e}^{2}\right )+1}+\frac {\ln \left ({\mathrm e}^{2 \left (1-{\mathrm e}^{2}\right )^{2 x} {\mathrm e}^{x \left (\ln \left (1-{\mathrm e}^{2}\right )^{2}+1\right )}+4}-1\right )}{\ln \left (1-{\mathrm e}^{2}\right )^{2}+2 \ln \left (1-{\mathrm e}^{2}\right )+1}\) | \(501\) |
int((2*ln(1-exp(2))^2+4*ln(1-exp(2))+2)*exp(x*ln(1-exp(2))^2+2*x*ln(1-exp( 2))+x)*exp(exp(x*ln(1-exp(2))^2+2*x*ln(1-exp(2))+x)+2)^2/(exp(exp(x*ln(1-e xp(2))^2+2*x*ln(1-exp(2))+x)+2)^2-1),x,method=_RETURNVERBOSE)
ln(exp(exp(x*ln(1-exp(2))^2+2*x*ln(1-exp(2))+x)+2)-1)+ln(exp(exp(x*ln(1-ex p(2))^2+2*x*ln(1-exp(2))+x)+2)+1)
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (24) = 48\).
Time = 0.32 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.50 \[ \int \frac {e^{4+2 e^{x+2 x \left (i \pi +\log \left (-1+e^2\right )\right )+x \left (i \pi +\log \left (-1+e^2\right )\right )^2}+x+2 x \left (i \pi +\log \left (-1+e^2\right )\right )+x \left (i \pi +\log \left (-1+e^2\right )\right )^2} \left (2+4 \left (i \pi +\log \left (-1+e^2\right )\right )+2 \left (i \pi +\log \left (-1+e^2\right )\right )^2\right )}{-1+e^{4+2 e^{x+2 x \left (i \pi +\log \left (-1+e^2\right )\right )+x \left (i \pi +\log \left (-1+e^2\right )\right )^2}}} \, dx=-x \log \left (-e^{2} + 1\right )^{2} - 2 \, x \log \left (-e^{2} + 1\right ) - x + \log \left (e^{\left (x \log \left (-e^{2} + 1\right )^{2} + 2 \, x \log \left (-e^{2} + 1\right ) + x + 2 \, e^{\left (x \log \left (-e^{2} + 1\right )^{2} + 2 \, x \log \left (-e^{2} + 1\right ) + x\right )} + 4\right )} - e^{\left (x \log \left (-e^{2} + 1\right )^{2} + 2 \, x \log \left (-e^{2} + 1\right ) + x\right )}\right ) \]
integrate((2*log(1-exp(2))^2+4*log(1-exp(2))+2)*exp(x*log(1-exp(2))^2+2*x* log(1-exp(2))+x)*exp(exp(x*log(1-exp(2))^2+2*x*log(1-exp(2))+x)+2)^2/(exp( exp(x*log(1-exp(2))^2+2*x*log(1-exp(2))+x)+2)^2-1),x, algorithm=\
-x*log(-e^2 + 1)^2 - 2*x*log(-e^2 + 1) - x + log(e^(x*log(-e^2 + 1)^2 + 2* x*log(-e^2 + 1) + x + 2*e^(x*log(-e^2 + 1)^2 + 2*x*log(-e^2 + 1) + x) + 4) - e^(x*log(-e^2 + 1)^2 + 2*x*log(-e^2 + 1) + x))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (24) = 48\).
Time = 66.95 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.73 \[ \int \frac {e^{4+2 e^{x+2 x \left (i \pi +\log \left (-1+e^2\right )\right )+x \left (i \pi +\log \left (-1+e^2\right )\right )^2}+x+2 x \left (i \pi +\log \left (-1+e^2\right )\right )+x \left (i \pi +\log \left (-1+e^2\right )\right )^2} \left (2+4 \left (i \pi +\log \left (-1+e^2\right )\right )+2 \left (i \pi +\log \left (-1+e^2\right )\right )^2\right )}{-1+e^{4+2 e^{x+2 x \left (i \pi +\log \left (-1+e^2\right )\right )+x \left (i \pi +\log \left (-1+e^2\right )\right )^2}}} \, dx=\frac {\left (2 + 4 \log {\left (-1 + e^{2} \right )} + 4 i \pi + 2 \left (\log {\left (-1 + e^{2} \right )} + i \pi \right )^{2}\right ) \log {\left (e^{2 e^{x + 2 x \left (\log {\left (-1 + e^{2} \right )} + i \pi \right ) + x \left (\log {\left (-1 + e^{2} \right )} + i \pi \right )^{2}} + 4} - 1 \right )}}{2 \left (1 + \log {\left (-1 + e^{2} \right )} + i \pi \right )^{2}} \]
integrate((2*ln(1-exp(2))**2+4*ln(1-exp(2))+2)*exp(x*ln(1-exp(2))**2+2*x*l n(1-exp(2))+x)*exp(exp(x*ln(1-exp(2))**2+2*x*ln(1-exp(2))+x)+2)**2/(exp(ex p(x*ln(1-exp(2))**2+2*x*ln(1-exp(2))+x)+2)**2-1),x)
(2 + 4*log(-1 + exp(2)) + 4*I*pi + 2*(log(-1 + exp(2)) + I*pi)**2)*log(exp (2*exp(x + 2*x*(log(-1 + exp(2)) + I*pi) + x*(log(-1 + exp(2)) + I*pi)**2) + 4) - 1)/(2*(1 + log(-1 + exp(2)) + I*pi)**2)
Time = 0.18 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {e^{4+2 e^{x+2 x \left (i \pi +\log \left (-1+e^2\right )\right )+x \left (i \pi +\log \left (-1+e^2\right )\right )^2}+x+2 x \left (i \pi +\log \left (-1+e^2\right )\right )+x \left (i \pi +\log \left (-1+e^2\right )\right )^2} \left (2+4 \left (i \pi +\log \left (-1+e^2\right )\right )+2 \left (i \pi +\log \left (-1+e^2\right )\right )^2\right )}{-1+e^{4+2 e^{x+2 x \left (i \pi +\log \left (-1+e^2\right )\right )+x \left (i \pi +\log \left (-1+e^2\right )\right )^2}}} \, dx=\log \left (e^{\left (2 \, e^{\left (x \log \left (-e^{2} + 1\right )^{2} + 2 \, x \log \left (-e^{2} + 1\right ) + x\right )} + 4\right )} - 1\right ) \]
integrate((2*log(1-exp(2))^2+4*log(1-exp(2))+2)*exp(x*log(1-exp(2))^2+2*x* log(1-exp(2))+x)*exp(exp(x*log(1-exp(2))^2+2*x*log(1-exp(2))+x)+2)^2/(exp( exp(x*log(1-exp(2))^2+2*x*log(1-exp(2))+x)+2)^2-1),x, algorithm=\
Time = 0.69 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {e^{4+2 e^{x+2 x \left (i \pi +\log \left (-1+e^2\right )\right )+x \left (i \pi +\log \left (-1+e^2\right )\right )^2}+x+2 x \left (i \pi +\log \left (-1+e^2\right )\right )+x \left (i \pi +\log \left (-1+e^2\right )\right )^2} \left (2+4 \left (i \pi +\log \left (-1+e^2\right )\right )+2 \left (i \pi +\log \left (-1+e^2\right )\right )^2\right )}{-1+e^{4+2 e^{x+2 x \left (i \pi +\log \left (-1+e^2\right )\right )+x \left (i \pi +\log \left (-1+e^2\right )\right )^2}}} \, dx=\log \left (e^{\left (2 \, e^{\left (x \log \left (-e^{2} + 1\right )^{2} + 2 \, x \log \left (-e^{2} + 1\right ) + x\right )} + 4\right )} - 1\right ) \]
integrate((2*log(1-exp(2))^2+4*log(1-exp(2))+2)*exp(x*log(1-exp(2))^2+2*x* log(1-exp(2))+x)*exp(exp(x*log(1-exp(2))^2+2*x*log(1-exp(2))+x)+2)^2/(exp( exp(x*log(1-exp(2))^2+2*x*log(1-exp(2))+x)+2)^2-1),x, algorithm=\
Time = 12.68 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {e^{4+2 e^{x+2 x \left (i \pi +\log \left (-1+e^2\right )\right )+x \left (i \pi +\log \left (-1+e^2\right )\right )^2}+x+2 x \left (i \pi +\log \left (-1+e^2\right )\right )+x \left (i \pi +\log \left (-1+e^2\right )\right )^2} \left (2+4 \left (i \pi +\log \left (-1+e^2\right )\right )+2 \left (i \pi +\log \left (-1+e^2\right )\right )^2\right )}{-1+e^{4+2 e^{x+2 x \left (i \pi +\log \left (-1+e^2\right )\right )+x \left (i \pi +\log \left (-1+e^2\right )\right )^2}}} \, dx=\ln \left ({\mathrm {e}}^{2\,{\mathrm {e}}^{x\,{\ln \left (1-{\mathrm {e}}^2\right )}^2}\,{\mathrm {e}}^x\,{\left (1-{\mathrm {e}}^2\right )}^{2\,x}+4}-1\right ) \]