Integrand size = 146, antiderivative size = 30 \[ \int \frac {4 e^x+\left (e^x \left (-4 x+x^2\right )+4 e^x x^2 (i \pi +\log (\log (4)))+e^x x^2 \log ^2(\log (4)) (i \pi +\log (\log (4)))\right ) \log \left (\frac {-4+x+4 x (i \pi +\log (\log (4)))+x \log ^2(\log (4)) (i \pi +\log (\log (4)))}{x (i \pi +\log (\log (4)))}\right )}{-4 x+x^2+4 x^2 (i \pi +\log (\log (4)))+x^2 \log ^2(\log (4)) (i \pi +\log (\log (4)))} \, dx=e^x \log \left (4+\log ^2(\log (4))+\frac {-4+x}{x (i \pi +\log (\log (4)))}\right ) \] Output:
ln(ln(2*ln(2))^2+4+(-4+x)/ln(-2*ln(2))/x)*exp(x)
Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {4 e^x+\left (e^x \left (-4 x+x^2\right )+4 e^x x^2 (i \pi +\log (\log (4)))+e^x x^2 \log ^2(\log (4)) (i \pi +\log (\log (4)))\right ) \log \left (\frac {-4+x+4 x (i \pi +\log (\log (4)))+x \log ^2(\log (4)) (i \pi +\log (\log (4)))}{x (i \pi +\log (\log (4)))}\right )}{-4 x+x^2+4 x^2 (i \pi +\log (\log (4)))+x^2 \log ^2(\log (4)) (i \pi +\log (\log (4)))} \, dx=e^x \log \left (4+\log ^2(\log (4))+\frac {1}{i \pi +\log (\log (4))}-\frac {4}{x (i \pi +\log (\log (4)))}\right ) \] Input:
Integrate[(4*E^x + (E^x*(-4*x + x^2) + 4*E^x*x^2*(I*Pi + Log[Log[4]]) + E^ x*x^2*Log[Log[4]]^2*(I*Pi + Log[Log[4]]))*Log[(-4 + x + 4*x*(I*Pi + Log[Lo g[4]]) + x*Log[Log[4]]^2*(I*Pi + Log[Log[4]]))/(x*(I*Pi + Log[Log[4]]))])/ (-4*x + x^2 + 4*x^2*(I*Pi + Log[Log[4]]) + x^2*Log[Log[4]]^2*(I*Pi + Log[L og[4]])),x]
Output:
E^x*Log[4 + Log[Log[4]]^2 + (I*Pi + Log[Log[4]])^(-1) - 4/(x*(I*Pi + Log[L og[4]]))]
Time = 1.40 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {6, 6, 2026, 7239, 2726}
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.
| \(\displaystyle \int \frac {4 e^x+\left (e^x \left (x^2-4 x\right )+e^x x^2 \log ^2(\log (4)) (\log (\log (4))+i \pi )+4 e^x x^2 (\log (\log (4))+i \pi )\right ) \log \left (\frac {x+x \log ^2(\log (4)) (\log (\log (4))+i \pi )+4 x (\log (\log (4))+i \pi )-4}{x (\log (\log (4))+i \pi )}\right )}{x^2+x^2 \log ^2(\log (4)) (\log (\log (4))+i \pi )+4 x^2 (\log (\log (4))+i \pi )-4 x} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {4 e^x+\left (e^x \left (x^2-4 x\right )+e^x x^2 \log ^2(\log (4)) (\log (\log (4))+i \pi )+4 e^x x^2 (\log (\log (4))+i \pi )\right ) \log \left (\frac {x+x \log ^2(\log (4)) (\log (\log (4))+i \pi )+4 x (\log (\log (4))+i \pi )-4}{x (\log (\log (4))+i \pi )}\right )}{x^2 \log ^2(\log (4)) (\log (\log (4))+i \pi )+x^2 (1+4 (\log (\log (4))+i \pi ))-4 x}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {4 e^x+\left (e^x \left (x^2-4 x\right )+e^x x^2 \log ^2(\log (4)) (\log (\log (4))+i \pi )+4 e^x x^2 (\log (\log (4))+i \pi )\right ) \log \left (\frac {x+x \log ^2(\log (4)) (\log (\log (4))+i \pi )+4 x (\log (\log (4))+i \pi )-4}{x (\log (\log (4))+i \pi )}\right )}{-4 x+x^2 \left (1+\log ^2(\log (4)) (\log (\log (4))+i \pi )+4 (\log (\log (4))+i \pi )\right )}dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {4 e^x+\left (e^x \left (x^2-4 x\right )+e^x x^2 \log ^2(\log (4)) (\log (\log (4))+i \pi )+4 e^x x^2 (\log (\log (4))+i \pi )\right ) \log \left (\frac {x+x \log ^2(\log (4)) (\log (\log (4))+i \pi )+4 x (\log (\log (4))+i \pi )-4}{x (\log (\log (4))+i \pi )}\right )}{x \left (-4+x \left (1+\log ^3(\log (4))+i \pi \left (4+\log ^2(\log (4))\right )+4 \log (\log (4))\right )\right )}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int e^x \left (\log \left (-\frac {4}{x (\log (\log (4))+i \pi )}+4+\log ^2(\log (4))+\frac {1}{\log (\log (4))+i \pi }\right )+\frac {4}{x \left (-4+x \left (1+\log ^3(\log (4))+i \pi \left (4+\log ^2(\log (4))\right )+4 \log (\log (4))\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 2726 |
| \(\displaystyle e^x \log \left (-\frac {4}{x (\log (\log (4))+i \pi )}+4+\log ^2(\log (4))+\frac {1}{\log (\log (4))+i \pi }\right )\) |
Input:
Int[(4*E^x + (E^x*(-4*x + x^2) + 4*E^x*x^2*(I*Pi + Log[Log[4]]) + E^x*x^2* Log[Log[4]]^2*(I*Pi + Log[Log[4]]))*Log[(-4 + x + 4*x*(I*Pi + Log[Log[4]]) + x*Log[Log[4]]^2*(I*Pi + Log[Log[4]]))/(x*(I*Pi + Log[Log[4]]))])/(-4*x + x^2 + 4*x^2*(I*Pi + Log[Log[4]]) + x^2*Log[Log[4]]^2*(I*Pi + Log[Log[4]] )),x]
Output:
E^x*Log[4 + Log[Log[4]]^2 + (I*Pi + Log[Log[4]])^(-1) - 4/(x*(I*Pi + Log[L og[4]]))]
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 43.13 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37
| method | result | size |
| default | \({\mathrm e}^{x} \ln \left (\frac {x \ln \left (-2 \ln \left (2\right )\right ) \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \ln \left (-2 \ln \left (2\right )\right )+x -4}{x \ln \left (-2 \ln \left (2\right )\right )}\right )\) | \(41\) |
| norman | \({\mathrm e}^{x} \ln \left (\frac {x \ln \left (-2 \ln \left (2\right )\right ) \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \ln \left (-2 \ln \left (2\right )\right )+x -4}{x \ln \left (-2 \ln \left (2\right )\right )}\right )\) | \(41\) |
| parallelrisch | \({\mathrm e}^{x} \ln \left (\frac {x \ln \left (-2 \ln \left (2\right )\right ) \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \ln \left (-2 \ln \left (2\right )\right )+x -4}{x \ln \left (-2 \ln \left (2\right )\right )}\right )\) | \(41\) |
| risch | \({\mathrm e}^{x} \ln \left (\left (x \left (\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )\right )^{2}+4 x \right ) \left (\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )+i \pi \right )+x -4\right )-{\mathrm e}^{x} \ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )\right ) \operatorname {csgn}\left (\frac {i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )}{x}\right ) {\mathrm e}^{x}}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) {\operatorname {csgn}\left (\frac {i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )}{x}\right )}^{2} {\mathrm e}^{x}}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )\right ) {\operatorname {csgn}\left (\frac {i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )}{x}\right )}^{2} {\mathrm e}^{x}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )}{x}\right ) \operatorname {csgn}\left (\frac {i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )}{\ln \left (-2 \ln \left (2\right )\right ) x}\right ) {\mathrm e}^{x}}{2}+\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )}{\ln \left (-2 \ln \left (2\right )\right ) x}\right )}^{2} {\mathrm e}^{x}}{2}-\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )}{x}\right )}^{3} {\mathrm e}^{x}}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )}{x}\right ) {\operatorname {csgn}\left (\frac {i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )}{\ln \left (-2 \ln \left (2\right )\right ) x}\right )}^{2} {\mathrm e}^{x}}{2}-\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (\left (x \ln \left (2 \ln \left (2\right )\right )^{2}+4 x \right ) \ln \left (-2 \ln \left (2\right )\right )+x -4\right )}{\ln \left (-2 \ln \left (2\right )\right ) x}\right )}^{3} {\mathrm e}^{x}}{2}-\ln \left (\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )+i \pi \right ) {\mathrm e}^{x}\) | \(499\) |
Input:
int(((x^2*exp(x)*ln(-2*ln(2))*ln(2*ln(2))^2+4*x^2*exp(x)*ln(-2*ln(2))+(x^2 -4*x)*exp(x))*ln((x*ln(-2*ln(2))*ln(2*ln(2))^2+4*x*ln(-2*ln(2))+x-4)/x/ln( -2*ln(2)))+4*exp(x))/(x^2*ln(-2*ln(2))*ln(2*ln(2))^2+4*x^2*ln(-2*ln(2))+x^ 2-4*x),x,method=_RETURNVERBOSE)
Output:
exp(x)*ln((x*ln(-2*ln(2))*ln(2*ln(2))^2+4*x*ln(-2*ln(2))+x-4)/x/ln(-2*ln(2 )))
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80 \[ \int \frac {4 e^x+\left (e^x \left (-4 x+x^2\right )+4 e^x x^2 (i \pi +\log (\log (4)))+e^x x^2 \log ^2(\log (4)) (i \pi +\log (\log (4)))\right ) \log \left (\frac {-4+x+4 x (i \pi +\log (\log (4)))+x \log ^2(\log (4)) (i \pi +\log (\log (4)))}{x (i \pi +\log (\log (4)))}\right )}{-4 x+x^2+4 x^2 (i \pi +\log (\log (4)))+x^2 \log ^2(\log (4)) (i \pi +\log (\log (4)))} \, dx=e^{x} \log \left (\frac {2 i \, \pi x \log \left (-2 \, \log \left (2\right )\right )^{2} + x \log \left (-2 \, \log \left (2\right )\right )^{3} - {\left (\pi ^{2} x - 4 \, x\right )} \log \left (-2 \, \log \left (2\right )\right ) + x - 4}{x \log \left (-2 \, \log \left (2\right )\right )}\right ) \] Input:
integrate(((x^2*exp(x)*log(-2*log(2))*log(2*log(2))^2+4*x^2*exp(x)*log(-2* log(2))+(x^2-4*x)*exp(x))*log((x*log(-2*log(2))*log(2*log(2))^2+4*x*log(-2 *log(2))+x-4)/x/log(-2*log(2)))+4*exp(x))/(x^2*log(-2*log(2))*log(2*log(2) )^2+4*x^2*log(-2*log(2))+x^2-4*x),x, algorithm="fricas")
Output:
e^x*log((2*I*pi*x*log(-2*log(2))^2 + x*log(-2*log(2))^3 - (pi^2*x - 4*x)*l og(-2*log(2)) + x - 4)/(x*log(-2*log(2))))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (29) = 58\).
Time = 33.82 (sec) , antiderivative size = 313, normalized size of antiderivative = 10.43 \[ \int \frac {4 e^x+\left (e^x \left (-4 x+x^2\right )+4 e^x x^2 (i \pi +\log (\log (4)))+e^x x^2 \log ^2(\log (4)) (i \pi +\log (\log (4)))\right ) \log \left (\frac {-4+x+4 x (i \pi +\log (\log (4)))+x \log ^2(\log (4)) (i \pi +\log (\log (4)))}{x (i \pi +\log (\log (4)))}\right )}{-4 x+x^2+4 x^2 (i \pi +\log (\log (4)))+x^2 \log ^2(\log (4)) (i \pi +\log (\log (4)))} \, dx=e^{x} \log {\left (\frac {4 x \log {\left (\log {\left (2 \right )} \right )}}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} + \frac {3 x \log {\left (2 \right )}^{2} \log {\left (\log {\left (2 \right )} \right )}}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} + \frac {x \log {\left (\log {\left (2 \right )} \right )}^{3}}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} + \frac {3 x \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )}^{2}}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} + \frac {x \log {\left (2 \right )}^{3}}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} + \frac {x}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} + \frac {4 x \log {\left (2 \right )}}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} + \frac {2 i \pi x \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )}}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} + \frac {i \pi x \log {\left (\log {\left (2 \right )} \right )}^{2}}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} + \frac {i \pi x \log {\left (2 \right )}^{2}}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} + \frac {4 i \pi x}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} - \frac {4}{x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + i \pi x} \right )} \] Input:
integrate(((x**2*exp(x)*ln(-2*ln(2))*ln(2*ln(2))**2+4*x**2*exp(x)*ln(-2*ln (2))+(x**2-4*x)*exp(x))*ln((x*ln(-2*ln(2))*ln(2*ln(2))**2+4*x*ln(-2*ln(2)) +x-4)/x/ln(-2*ln(2)))+4*exp(x))/(x**2*ln(-2*ln(2))*ln(2*ln(2))**2+4*x**2*l n(-2*ln(2))+x**2-4*x),x)
Output:
exp(x)*log(4*x*log(log(2))/(x*log(log(2)) + x*log(2) + I*pi*x) + 3*x*log(2 )**2*log(log(2))/(x*log(log(2)) + x*log(2) + I*pi*x) + x*log(log(2))**3/(x *log(log(2)) + x*log(2) + I*pi*x) + 3*x*log(2)*log(log(2))**2/(x*log(log(2 )) + x*log(2) + I*pi*x) + x*log(2)**3/(x*log(log(2)) + x*log(2) + I*pi*x) + x/(x*log(log(2)) + x*log(2) + I*pi*x) + 4*x*log(2)/(x*log(log(2)) + x*lo g(2) + I*pi*x) + 2*I*pi*x*log(2)*log(log(2))/(x*log(log(2)) + x*log(2) + I *pi*x) + I*pi*x*log(log(2))**2/(x*log(log(2)) + x*log(2) + I*pi*x) + I*pi* x*log(2)**2/(x*log(log(2)) + x*log(2) + I*pi*x) + 4*I*pi*x/(x*log(log(2)) + x*log(2) + I*pi*x) - 4/(x*log(log(2)) + x*log(2) + I*pi*x))
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.73 \[ \int \frac {4 e^x+\left (e^x \left (-4 x+x^2\right )+4 e^x x^2 (i \pi +\log (\log (4)))+e^x x^2 \log ^2(\log (4)) (i \pi +\log (\log (4)))\right ) \log \left (\frac {-4+x+4 x (i \pi +\log (\log (4)))+x \log ^2(\log (4)) (i \pi +\log (\log (4)))}{x (i \pi +\log (\log (4)))}\right )}{-4 x+x^2+4 x^2 (i \pi +\log (\log (4)))+x^2 \log ^2(\log (4)) (i \pi +\log (\log (4)))} \, dx=-{\left (\log \left (i \, \pi + \log \left (2\right ) + \log \left (\log \left (2\right )\right )\right ) + \log \left (x\right )\right )} e^{x} + e^{x} \log \left ({\left (4 i \, \pi + i \, \pi \log \left (2\right )^{2} + \log \left (2\right )^{3} + {\left (i \, \pi + 3 \, \log \left (2\right )\right )} \log \left (\log \left (2\right )\right )^{2} + \log \left (\log \left (2\right )\right )^{3} + {\left (2 i \, \pi \log \left (2\right ) + 3 \, \log \left (2\right )^{2} + 4\right )} \log \left (\log \left (2\right )\right ) + 4 \, \log \left (2\right ) + 1\right )} x - 4\right ) \] Input:
integrate(((x^2*exp(x)*log(-2*log(2))*log(2*log(2))^2+4*x^2*exp(x)*log(-2* log(2))+(x^2-4*x)*exp(x))*log((x*log(-2*log(2))*log(2*log(2))^2+4*x*log(-2 *log(2))+x-4)/x/log(-2*log(2)))+4*exp(x))/(x^2*log(-2*log(2))*log(2*log(2) )^2+4*x^2*log(-2*log(2))+x^2-4*x),x, algorithm="maxima")
Output:
-(log(I*pi + log(2) + log(log(2))) + log(x))*e^x + e^x*log((4*I*pi + I*pi* log(2)^2 + log(2)^3 + (I*pi + 3*log(2))*log(log(2))^2 + log(log(2))^3 + (2 *I*pi*log(2) + 3*log(2)^2 + 4)*log(log(2)) + 4*log(2) + 1)*x - 4)
Leaf count of result is larger than twice the leaf count of optimal. 443 vs. \(2 (27) = 54\).
Time = 1.12 (sec) , antiderivative size = 443, normalized size of antiderivative = 14.77 \[ \int \frac {4 e^x+\left (e^x \left (-4 x+x^2\right )+4 e^x x^2 (i \pi +\log (\log (4)))+e^x x^2 \log ^2(\log (4)) (i \pi +\log (\log (4)))\right ) \log \left (\frac {-4+x+4 x (i \pi +\log (\log (4)))+x \log ^2(\log (4)) (i \pi +\log (\log (4)))}{x (i \pi +\log (\log (4)))}\right )}{-4 x+x^2+4 x^2 (i \pi +\log (\log (4)))+x^2 \log ^2(\log (4)) (i \pi +\log (\log (4)))} \, dx =\text {Too large to display} \] Input:
integrate(((x^2*exp(x)*log(-2*log(2))*log(2*log(2))^2+4*x^2*exp(x)*log(-2* log(2))+(x^2-4*x)*exp(x))*log((x*log(-2*log(2))*log(2*log(2))^2+4*x*log(-2 *log(2))+x-4)/x/log(-2*log(2)))+4*exp(x))/(x^2*log(-2*log(2))*log(2*log(2) )^2+4*x^2*log(-2*log(2))+x^2-4*x),x, algorithm="giac")
Output:
1/2*e^x*log(pi^2*x^2*log(2)^4 + x^2*log(2)^6 + 4*pi^2*x^2*log(2)^3*log(log (2)) + 6*x^2*log(2)^5*log(log(2)) + 6*pi^2*x^2*log(2)^2*log(log(2))^2 + 15 *x^2*log(2)^4*log(log(2))^2 + 4*pi^2*x^2*log(2)*log(log(2))^3 + 20*x^2*log (2)^3*log(log(2))^3 + pi^2*x^2*log(log(2))^4 + 15*x^2*log(2)^2*log(log(2)) ^4 + 6*x^2*log(2)*log(log(2))^5 + x^2*log(log(2))^6 + 8*pi^2*x^2*log(2)^2 + 8*x^2*log(2)^4 + 16*pi^2*x^2*log(2)*log(log(2)) + 32*x^2*log(2)^3*log(lo g(2)) + 8*pi^2*x^2*log(log(2))^2 + 48*x^2*log(2)^2*log(log(2))^2 + 32*x^2* log(2)*log(log(2))^3 + 8*x^2*log(log(2))^4 + 2*x^2*log(2)^3 + 6*x^2*log(2) ^2*log(log(2)) + 6*x^2*log(2)*log(log(2))^2 + 2*x^2*log(log(2))^3 + 16*pi^ 2*x^2 + 16*x^2*log(2)^2 - 8*x*log(2)^3 + 32*x^2*log(2)*log(log(2)) - 24*x* log(2)^2*log(log(2)) + 16*x^2*log(log(2))^2 - 24*x*log(2)*log(log(2))^2 - 8*x*log(log(2))^3 + 8*x^2*log(2) + 8*x^2*log(log(2)) + x^2 - 32*x*log(2) - 32*x*log(log(2)) - 8*x + 16) - 1/2*e^x*log(pi^2*x^2 + x^2*log(2)^2 + 2*x^ 2*log(2)*log(log(2)) + x^2*log(log(2))^2)
Time = 5.70 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \[ \int \frac {4 e^x+\left (e^x \left (-4 x+x^2\right )+4 e^x x^2 (i \pi +\log (\log (4)))+e^x x^2 \log ^2(\log (4)) (i \pi +\log (\log (4)))\right ) \log \left (\frac {-4+x+4 x (i \pi +\log (\log (4)))+x \log ^2(\log (4)) (i \pi +\log (\log (4)))}{x (i \pi +\log (\log (4)))}\right )}{-4 x+x^2+4 x^2 (i \pi +\log (\log (4)))+x^2 \log ^2(\log (4)) (i \pi +\log (\log (4)))} \, dx={\mathrm {e}}^x\,\ln \left (\frac {x+4\,x\,\ln \left (-\ln \left (4\right )\right )+x\,\ln \left (-\ln \left (4\right )\right )\,{\ln \left (\ln \left (4\right )\right )}^2-4}{x\,\ln \left (-\ln \left (4\right )\right )}\right ) \] Input:
int((4*exp(x) + log((x + 4*x*log(-2*log(2)) + x*log(-2*log(2))*log(2*log(2 ))^2 - 4)/(x*log(-2*log(2))))*(4*x^2*log(-2*log(2))*exp(x) - exp(x)*(4*x - x^2) + x^2*log(-2*log(2))*log(2*log(2))^2*exp(x)))/(4*x^2*log(-2*log(2)) - 4*x + x^2 + x^2*log(-2*log(2))*log(2*log(2))^2),x)
Output:
exp(x)*log((x + 4*x*log(-log(4)) + x*log(-log(4))*log(log(4))^2 - 4)/(x*lo g(-log(4))))
Time = 0.15 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {4 e^x+\left (e^x \left (-4 x+x^2\right )+4 e^x x^2 (i \pi +\log (\log (4)))+e^x x^2 \log ^2(\log (4)) (i \pi +\log (\log (4)))\right ) \log \left (\frac {-4+x+4 x (i \pi +\log (\log (4)))+x \log ^2(\log (4)) (i \pi +\log (\log (4)))}{x (i \pi +\log (\log (4)))}\right )}{-4 x+x^2+4 x^2 (i \pi +\log (\log (4)))+x^2 \log ^2(\log (4)) (i \pi +\log (\log (4)))} \, dx=e^{x} \mathrm {log}\left (\frac {\mathrm {log}\left (-2 \,\mathrm {log}\left (2\right )\right ) \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )^{2} x +4 \,\mathrm {log}\left (-2 \,\mathrm {log}\left (2\right )\right ) x +x -4}{\mathrm {log}\left (-2 \,\mathrm {log}\left (2\right )\right ) x}\right ) \] Input:
int(((x^2*exp(x)*log(-2*log(2))*log(2*log(2))^2+4*x^2*exp(x)*log(-2*log(2) )+(x^2-4*x)*exp(x))*log((x*log(-2*log(2))*log(2*log(2))^2+4*x*log(-2*log(2 ))+x-4)/x/log(-2*log(2)))+4*exp(x))/(x^2*log(-2*log(2))*log(2*log(2))^2+4* x^2*log(-2*log(2))+x^2-4*x),x)
Output:
e**x*log((log( - 2*log(2))*log(2*log(2))**2*x + 4*log( - 2*log(2))*x + x - 4)/(log( - 2*log(2))*x))