Integrand size = 123, antiderivative size = 21 \[ \int \frac {-2 e^x \log (-x) \log (x)+2 \log (-x) \log (x) \log (\log (x))+\left (-4 \log (-x)+4 e^x x \log (-x) \log (x)\right ) \log \left (-e^x+\log (\log (x))\right )+\left (2 e^x \log (x)-2 \log (x) \log (\log (x))\right ) \log ^2\left (-e^x+\log (\log (x))\right )+\left (4-4 e^x x \log (x)\right ) \log ^3\left (-e^x+\log (\log (x))\right )}{-e^x x \log (x)+x \log (x) \log (\log (x))} \, dx=\left (-\log (-x)+\log ^2\left (-e^x+\log (\log (x))\right )\right )^2 \]
Time = 0.10 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-2 e^x \log (-x) \log (x)+2 \log (-x) \log (x) \log (\log (x))+\left (-4 \log (-x)+4 e^x x \log (-x) \log (x)\right ) \log \left (-e^x+\log (\log (x))\right )+\left (2 e^x \log (x)-2 \log (x) \log (\log (x))\right ) \log ^2\left (-e^x+\log (\log (x))\right )+\left (4-4 e^x x \log (x)\right ) \log ^3\left (-e^x+\log (\log (x))\right )}{-e^x x \log (x)+x \log (x) \log (\log (x))} \, dx=\left (-\log (-x)+\log ^2\left (-e^x+\log (\log (x))\right )\right )^2 \]
Integrate[(-2*E^x*Log[-x]*Log[x] + 2*Log[-x]*Log[x]*Log[Log[x]] + (-4*Log[ -x] + 4*E^x*x*Log[-x]*Log[x])*Log[-E^x + Log[Log[x]]] + (2*E^x*Log[x] - 2* Log[x]*Log[Log[x]])*Log[-E^x + Log[Log[x]]]^2 + (4 - 4*E^x*x*Log[x])*Log[- E^x + Log[Log[x]]]^3)/(-(E^x*x*Log[x]) + x*Log[x]*Log[Log[x]]),x]
Time = 0.83 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {7292, 27, 7237}
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.
\(\displaystyle \int \frac {\left (4-4 e^x x \log (x)\right ) \log ^3\left (\log (\log (x))-e^x\right )+\left (2 e^x \log (x)-2 \log (x) \log (\log (x))\right ) \log ^2\left (\log (\log (x))-e^x\right )+\left (4 e^x x \log (-x) \log (x)-4 \log (-x)\right ) \log \left (\log (\log (x))-e^x\right )-2 e^x \log (-x) \log (x)+2 \log (-x) \log (x) \log (\log (x))}{x \log (x) \log (\log (x))-e^x x \log (x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 \left (e^x \log (x)-\log (\log (x)) \log (x)-2 e^x x \log \left (\log (\log (x))-e^x\right ) \log (x)+2 \log \left (\log (\log (x))-e^x\right )\right ) \left (\log (-x)-\log ^2\left (\log (\log (x))-e^x\right )\right )}{x \log (x) \left (e^x-\log (\log (x))\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {\left (e^x \log (x)-\log (\log (x)) \log (x)-2 e^x x \log \left (\log (\log (x))-e^x\right ) \log (x)+2 \log \left (\log (\log (x))-e^x\right )\right ) \left (\log (-x)-\log ^2\left (\log (\log (x))-e^x\right )\right )}{x \log (x) \left (e^x-\log (\log (x))\right )}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \left (\log (-x)-\log ^2\left (\log (\log (x))-e^x\right )\right )^2\) |
Int[(-2*E^x*Log[-x]*Log[x] + 2*Log[-x]*Log[x]*Log[Log[x]] + (-4*Log[-x] + 4*E^x*x*Log[-x]*Log[x])*Log[-E^x + Log[Log[x]]] + (2*E^x*Log[x] - 2*Log[x] *Log[Log[x]])*Log[-E^x + Log[Log[x]]]^2 + (4 - 4*E^x*x*Log[x])*Log[-E^x + Log[Log[x]]]^3)/(-(E^x*x*Log[x]) + x*Log[x]*Log[Log[x]]),x]
3.24.7.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Time = 85.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71
method | result | size |
parallelrisch | \(\ln \left (\ln \left (\ln \left (x \right )\right )-{\mathrm e}^{x}\right )^{4}-2 \ln \left (\ln \left (\ln \left (x \right )\right )-{\mathrm e}^{x}\right )^{2} \ln \left (-x \right )+\ln \left (-x \right )^{2}\) | \(36\) |
risch | \(\ln \left (\ln \left (\ln \left (x \right )\right )-{\mathrm e}^{x}\right )^{4}+\left (-2 i \pi \operatorname {csgn}\left (i x \right )^{3}+2 i \pi \operatorname {csgn}\left (i x \right )^{2}-2 i \pi -2 \ln \left (x \right )\right ) \ln \left (\ln \left (\ln \left (x \right )\right )-{\mathrm e}^{x}\right )^{2}+2 i \pi \ln \left (x \right ) \operatorname {csgn}\left (i x \right )^{3}-2 i \pi \ln \left (x \right ) \operatorname {csgn}\left (i x \right )^{2}+2 i \pi \ln \left (x \right )+\ln \left (x \right )^{2}\) | \(92\) |
int(((-4*x*exp(x)*ln(x)+4)*ln(ln(ln(x))-exp(x))^3+(-2*ln(x)*ln(ln(x))+2*ex p(x)*ln(x))*ln(ln(ln(x))-exp(x))^2+(4*x*exp(x)*ln(-x)*ln(x)-4*ln(-x))*ln(l n(ln(x))-exp(x))+2*ln(-x)*ln(x)*ln(ln(x))-2*exp(x)*ln(-x)*ln(x))/(x*ln(x)* ln(ln(x))-x*exp(x)*ln(x)),x,method=_RETURNVERBOSE)
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.24 \[ \int \frac {-2 e^x \log (-x) \log (x)+2 \log (-x) \log (x) \log (\log (x))+\left (-4 \log (-x)+4 e^x x \log (-x) \log (x)\right ) \log \left (-e^x+\log (\log (x))\right )+\left (2 e^x \log (x)-2 \log (x) \log (\log (x))\right ) \log ^2\left (-e^x+\log (\log (x))\right )+\left (4-4 e^x x \log (x)\right ) \log ^3\left (-e^x+\log (\log (x))\right )}{-e^x x \log (x)+x \log (x) \log (\log (x))} \, dx=\log \left (-e^{x} + \log \left (i \, \pi + \log \left (-x\right )\right )\right )^{4} - 2 \, \log \left (-x\right ) \log \left (-e^{x} + \log \left (i \, \pi + \log \left (-x\right )\right )\right )^{2} + \log \left (-x\right )^{2} \]
integrate(((-4*x*exp(x)*log(x)+4)*log(log(log(x))-exp(x))^3+(-2*log(x)*log (log(x))+2*exp(x)*log(x))*log(log(log(x))-exp(x))^2+(4*x*exp(x)*log(-x)*lo g(x)-4*log(-x))*log(log(log(x))-exp(x))+2*log(-x)*log(x)*log(log(x))-2*exp (x)*log(-x)*log(x))/(x*log(x)*log(log(x))-x*exp(x)*log(x)),x, algorithm=\
Result contains complex when optimal does not.
Time = 1.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.29 \[ \int \frac {-2 e^x \log (-x) \log (x)+2 \log (-x) \log (x) \log (\log (x))+\left (-4 \log (-x)+4 e^x x \log (-x) \log (x)\right ) \log \left (-e^x+\log (\log (x))\right )+\left (2 e^x \log (x)-2 \log (x) \log (\log (x))\right ) \log ^2\left (-e^x+\log (\log (x))\right )+\left (4-4 e^x x \log (x)\right ) \log ^3\left (-e^x+\log (\log (x))\right )}{-e^x x \log (x)+x \log (x) \log (\log (x))} \, dx=\left (- 2 \log {\left (x \right )} - 2 i \pi \right ) \log {\left (- e^{x} + \log {\left (\log {\left (x \right )} \right )} \right )}^{2} + \log {\left (x \right )}^{2} + 2 i \pi \log {\left (x \right )} + \log {\left (- e^{x} + \log {\left (\log {\left (x \right )} \right )} \right )}^{4} \]
integrate(((-4*x*exp(x)*ln(x)+4)*ln(ln(ln(x))-exp(x))**3+(-2*ln(x)*ln(ln(x ))+2*exp(x)*ln(x))*ln(ln(ln(x))-exp(x))**2+(4*x*exp(x)*ln(-x)*ln(x)-4*ln(- x))*ln(ln(ln(x))-exp(x))+2*ln(-x)*ln(x)*ln(ln(x))-2*exp(x)*ln(-x)*ln(x))/( x*ln(x)*ln(ln(x))-x*exp(x)*ln(x)),x)
(-2*log(x) - 2*I*pi)*log(-exp(x) + log(log(x)))**2 + log(x)**2 + 2*I*pi*lo g(x) + log(-exp(x) + log(log(x)))**4
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.90 \[ \int \frac {-2 e^x \log (-x) \log (x)+2 \log (-x) \log (x) \log (\log (x))+\left (-4 \log (-x)+4 e^x x \log (-x) \log (x)\right ) \log \left (-e^x+\log (\log (x))\right )+\left (2 e^x \log (x)-2 \log (x) \log (\log (x))\right ) \log ^2\left (-e^x+\log (\log (x))\right )+\left (4-4 e^x x \log (x)\right ) \log ^3\left (-e^x+\log (\log (x))\right )}{-e^x x \log (x)+x \log (x) \log (\log (x))} \, dx=\log \left (-e^{x} + \log \left (\log \left (x\right )\right )\right )^{4} - 2 \, {\left (i \, \pi + \log \left (x\right )\right )} \log \left (-e^{x} + \log \left (\log \left (x\right )\right )\right )^{2} + 2 i \, \pi \log \left (x\right ) + \log \left (x\right )^{2} \]
integrate(((-4*x*exp(x)*log(x)+4)*log(log(log(x))-exp(x))^3+(-2*log(x)*log (log(x))+2*exp(x)*log(x))*log(log(log(x))-exp(x))^2+(4*x*exp(x)*log(-x)*lo g(x)-4*log(-x))*log(log(log(x))-exp(x))+2*log(-x)*log(x)*log(log(x))-2*exp (x)*log(-x)*log(x))/(x*log(x)*log(log(x))-x*exp(x)*log(x)),x, algorithm=\
log(-e^x + log(log(x)))^4 - 2*(I*pi + log(x))*log(-e^x + log(log(x)))^2 + 2*I*pi*log(x) + log(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 3110 vs. \(2 (20) = 40\).
Time = 1.20 (sec) , antiderivative size = 3110, normalized size of antiderivative = 148.10 \[ \int \frac {-2 e^x \log (-x) \log (x)+2 \log (-x) \log (x) \log (\log (x))+\left (-4 \log (-x)+4 e^x x \log (-x) \log (x)\right ) \log \left (-e^x+\log (\log (x))\right )+\left (2 e^x \log (x)-2 \log (x) \log (\log (x))\right ) \log ^2\left (-e^x+\log (\log (x))\right )+\left (4-4 e^x x \log (x)\right ) \log ^3\left (-e^x+\log (\log (x))\right )}{-e^x x \log (x)+x \log (x) \log (\log (x))} \, dx=\text {Too large to display} \]
integrate(((-4*x*exp(x)*log(x)+4)*log(log(log(x))-exp(x))^3+(-2*log(x)*log (log(x))+2*exp(x)*log(x))*log(log(log(x))-exp(x))^2+(4*x*exp(x)*log(-x)*lo g(x)-4*log(-x))*log(log(log(x))-exp(x))+2*log(-x)*log(x)*log(log(x))-2*exp (x)*log(-x)*log(x))/(x*log(x)*log(log(x))-x*exp(x)*log(x)),x, algorithm=\
-2*pi^3*arctan(1/2*(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x))))/(e^x - log(abs (log(x)))))*sgn(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi*sgn(-pi + pi *sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x))))*sgn(e^x - log(abs( log(x)))) - 2*pi*arctan(1/2*(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi *sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x))))/(e^x - log(abs(log(x)))))^3*sgn(pi*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) - pi*s gn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(abs(x))))*sgn(e^x - log(abs(log(x)))) + 3/2*pi*arctan(1/2*(pi*sgn(-pi + pi*sgn(x))*sgn(log( abs(x))) - pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(ab s(x))))/(e^x - log(abs(log(x)))))*log(-pi*arctan(-1/2*(pi - pi*sgn(x))/log (abs(x)))*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) + pi*arctan(-1/2*(pi - pi* sgn(x))/log(abs(x)))*sgn(-pi + pi*sgn(x)) - 1/2*pi^2*sgn(log(abs(x))) + 1/ 2*pi^2 + arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))^2 - 2*e^x*log(abs(log(x ))) + log(abs(log(x)))^2 + e^(2*x))^2*sgn(pi*sgn(-pi + pi*sgn(x))*sgn(log( abs(x))) - pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x))/log(ab s(x))))*sgn(e^x - log(abs(log(x)))) - 1/2*pi^2*log(-pi*arctan(-1/2*(pi - p i*sgn(x))/log(abs(x)))*sgn(-pi + pi*sgn(x))*sgn(log(abs(x))) + pi*arctan(- 1/2*(pi - pi*sgn(x))/log(abs(x)))*sgn(-pi + pi*sgn(x)) - 1/2*pi^2*sgn(log( abs(x))) + 1/2*pi^2 + arctan(-1/2*(pi - pi*sgn(x))/log(abs(x)))^2 - 2*e...
Time = 8.55 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.05 \[ \int \frac {-2 e^x \log (-x) \log (x)+2 \log (-x) \log (x) \log (\log (x))+\left (-4 \log (-x)+4 e^x x \log (-x) \log (x)\right ) \log \left (-e^x+\log (\log (x))\right )+\left (2 e^x \log (x)-2 \log (x) \log (\log (x))\right ) \log ^2\left (-e^x+\log (\log (x))\right )+\left (4-4 e^x x \log (x)\right ) \log ^3\left (-e^x+\log (\log (x))\right )}{-e^x x \log (x)+x \log (x) \log (\log (x))} \, dx={\ln \left (\ln \left (\ln \left (x\right )\right )-{\mathrm {e}}^x\right )}^4-2\,\ln \left (-x\right )\,{\ln \left (\ln \left (\ln \left (x\right )\right )-{\mathrm {e}}^x\right )}^2-{\ln \left (x\right )}^2+2\,\ln \left (-x\right )\,\ln \left (x\right ) \]
int(-(log(log(log(x)) - exp(x))^3*(4*x*exp(x)*log(x) - 4) - log(log(log(x) ) - exp(x))^2*(2*exp(x)*log(x) - 2*log(log(x))*log(x)) + log(log(log(x)) - exp(x))*(4*log(-x) - 4*x*log(-x)*exp(x)*log(x)) + 2*log(-x)*exp(x)*log(x) - 2*log(-x)*log(log(x))*log(x))/(x*log(log(x))*log(x) - x*exp(x)*log(x)), x)