Integrand size = 321, antiderivative size = 33 \[ \int \frac {e^{\frac {4 x^4}{4 e^{\frac {6 x}{10+2 \log (x)}} x^2-4 e^{\frac {3 x}{10+2 \log (x)}} x \log \left (\frac {1}{x}\right )+\log ^2\left (\frac {1}{x}\right )}} \left (-200 x^3-400 x^3 \log \left (\frac {1}{x}\right )+\left (-80 x^3-160 x^3 \log \left (\frac {1}{x}\right )\right ) \log (x)+\left (-8 x^3-16 x^3 \log \left (\frac {1}{x}\right )\right ) \log ^2(x)+e^{\frac {3 x}{10+2 \log (x)}} \left (400 x^4-96 x^5+\left (160 x^4-24 x^5\right ) \log (x)+16 x^4 \log ^2(x)\right )\right )}{-25 \log ^3\left (\frac {1}{x}\right )-10 \log ^3\left (\frac {1}{x}\right ) \log (x)-\log ^3\left (\frac {1}{x}\right ) \log ^2(x)+e^{\frac {9 x}{10+2 \log (x)}} \left (200 x^3+80 x^3 \log (x)+8 x^3 \log ^2(x)\right )+e^{\frac {6 x}{10+2 \log (x)}} \left (-300 x^2 \log \left (\frac {1}{x}\right )-120 x^2 \log \left (\frac {1}{x}\right ) \log (x)-12 x^2 \log \left (\frac {1}{x}\right ) \log ^2(x)\right )+e^{\frac {3 x}{10+2 \log (x)}} \left (150 x \log ^2\left (\frac {1}{x}\right )+60 x \log ^2\left (\frac {1}{x}\right ) \log (x)+6 x \log ^2\left (\frac {1}{x}\right ) \log ^2(x)\right )} \, dx=e^{\frac {x^2}{\left (e^{\frac {3 x}{2 (5+\log (x))}}-\frac {\log \left (\frac {1}{x}\right )}{2 x}\right )^2}} \]
Time = 0.52 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\frac {4 x^4}{4 e^{\frac {6 x}{10+2 \log (x)}} x^2-4 e^{\frac {3 x}{10+2 \log (x)}} x \log \left (\frac {1}{x}\right )+\log ^2\left (\frac {1}{x}\right )}} \left (-200 x^3-400 x^3 \log \left (\frac {1}{x}\right )+\left (-80 x^3-160 x^3 \log \left (\frac {1}{x}\right )\right ) \log (x)+\left (-8 x^3-16 x^3 \log \left (\frac {1}{x}\right )\right ) \log ^2(x)+e^{\frac {3 x}{10+2 \log (x)}} \left (400 x^4-96 x^5+\left (160 x^4-24 x^5\right ) \log (x)+16 x^4 \log ^2(x)\right )\right )}{-25 \log ^3\left (\frac {1}{x}\right )-10 \log ^3\left (\frac {1}{x}\right ) \log (x)-\log ^3\left (\frac {1}{x}\right ) \log ^2(x)+e^{\frac {9 x}{10+2 \log (x)}} \left (200 x^3+80 x^3 \log (x)+8 x^3 \log ^2(x)\right )+e^{\frac {6 x}{10+2 \log (x)}} \left (-300 x^2 \log \left (\frac {1}{x}\right )-120 x^2 \log \left (\frac {1}{x}\right ) \log (x)-12 x^2 \log \left (\frac {1}{x}\right ) \log ^2(x)\right )+e^{\frac {3 x}{10+2 \log (x)}} \left (150 x \log ^2\left (\frac {1}{x}\right )+60 x \log ^2\left (\frac {1}{x}\right ) \log (x)+6 x \log ^2\left (\frac {1}{x}\right ) \log ^2(x)\right )} \, dx=e^{\frac {4 x^4}{\left (2 e^{\frac {3 x}{2 (5+\log (x))}} x-\log \left (\frac {1}{x}\right )\right )^2}} \]
Integrate[(E^((4*x^4)/(4*E^((6*x)/(10 + 2*Log[x]))*x^2 - 4*E^((3*x)/(10 + 2*Log[x]))*x*Log[x^(-1)] + Log[x^(-1)]^2))*(-200*x^3 - 400*x^3*Log[x^(-1)] + (-80*x^3 - 160*x^3*Log[x^(-1)])*Log[x] + (-8*x^3 - 16*x^3*Log[x^(-1)])* Log[x]^2 + E^((3*x)/(10 + 2*Log[x]))*(400*x^4 - 96*x^5 + (160*x^4 - 24*x^5 )*Log[x] + 16*x^4*Log[x]^2)))/(-25*Log[x^(-1)]^3 - 10*Log[x^(-1)]^3*Log[x] - Log[x^(-1)]^3*Log[x]^2 + E^((9*x)/(10 + 2*Log[x]))*(200*x^3 + 80*x^3*Lo g[x] + 8*x^3*Log[x]^2) + E^((6*x)/(10 + 2*Log[x]))*(-300*x^2*Log[x^(-1)] - 120*x^2*Log[x^(-1)]*Log[x] - 12*x^2*Log[x^(-1)]*Log[x]^2) + E^((3*x)/(10 + 2*Log[x]))*(150*x*Log[x^(-1)]^2 + 60*x*Log[x^(-1)]^2*Log[x] + 6*x*Log[x^ (-1)]^2*Log[x]^2)),x]
Time = 21.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {7239, 27, 25, 7257}
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.
| \(\displaystyle \int \frac {\left (-200 x^3+\left (-8 x^3-16 x^3 \log \left (\frac {1}{x}\right )\right ) \log ^2(x)-400 x^3 \log \left (\frac {1}{x}\right )+\left (-80 x^3-160 x^3 \log \left (\frac {1}{x}\right )\right ) \log (x)+e^{\frac {3 x}{2 \log (x)+10}} \left (-96 x^5+400 x^4+16 x^4 \log ^2(x)+\left (160 x^4-24 x^5\right ) \log (x)\right )\right ) \exp \left (\frac {4 x^4}{4 x^2 e^{\frac {6 x}{2 \log (x)+10}}+\log ^2\left (\frac {1}{x}\right )-4 x e^{\frac {3 x}{2 \log (x)+10}} \log \left (\frac {1}{x}\right )}\right )}{e^{\frac {9 x}{2 \log (x)+10}} \left (200 x^3+8 x^3 \log ^2(x)+80 x^3 \log (x)\right )+e^{\frac {6 x}{2 \log (x)+10}} \left (-12 x^2 \log \left (\frac {1}{x}\right ) \log ^2(x)-300 x^2 \log \left (\frac {1}{x}\right )-120 x^2 \log \left (\frac {1}{x}\right ) \log (x)\right )-10 \log (x) \log ^3\left (\frac {1}{x}\right )-25 \log ^3\left (\frac {1}{x}\right )+e^{\frac {3 x}{2 \log (x)+10}} \left (6 x \log ^2(x) \log ^2\left (\frac {1}{x}\right )+150 x \log ^2\left (\frac {1}{x}\right )+60 x \log (x) \log ^2\left (\frac {1}{x}\right )\right )-\log ^2(x) \log ^3\left (\frac {1}{x}\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {8 x^3 \left (-12 x^2 e^{\frac {3 x}{2 (\log (x)+5)}}-\left (1-2 x e^{\frac {3 x}{2 (\log (x)+5)}}\right ) \log ^2(x)+50 x e^{\frac {3 x}{2 (\log (x)+5)}}-2 \log \left (\frac {1}{x}\right ) (\log (x)+5)^2-\left (x (3 x-20) e^{\frac {3 x}{2 (\log (x)+5)}}+10\right ) \log (x)-25\right ) \exp \left (\frac {4 x^4}{\left (\log \left (\frac {1}{x}\right )-2 x e^{\frac {3 x}{2 (\log (x)+5)}}\right )^2}\right )}{\left (2 x e^{\frac {3 x}{2 (\log (x)+5)}}-\log \left (\frac {1}{x}\right )\right )^3 (\log (x)+5)^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 8 \int -\frac {\exp \left (\frac {4 x^4}{\left (2 e^{\frac {3 x}{2 (\log (x)+5)}} x-\log \left (\frac {1}{x}\right )\right )^2}\right ) x^3 \left (12 e^{\frac {3 x}{2 (\log (x)+5)}} x^2-50 e^{\frac {3 x}{2 (\log (x)+5)}} x+\left (1-2 e^{\frac {3 x}{2 (\log (x)+5)}} x\right ) \log ^2(x)+2 \log \left (\frac {1}{x}\right ) (\log (x)+5)^2+\left (10-e^{\frac {3 x}{2 (\log (x)+5)}} (20-3 x) x\right ) \log (x)+25\right )}{\left (2 e^{\frac {3 x}{2 (\log (x)+5)}} x-\log \left (\frac {1}{x}\right )\right )^3 (\log (x)+5)^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -8 \int \frac {\exp \left (\frac {4 x^4}{\left (2 e^{\frac {3 x}{2 (\log (x)+5)}} x-\log \left (\frac {1}{x}\right )\right )^2}\right ) x^3 \left (12 e^{\frac {3 x}{2 (\log (x)+5)}} x^2-50 e^{\frac {3 x}{2 (\log (x)+5)}} x+\left (1-2 e^{\frac {3 x}{2 (\log (x)+5)}} x\right ) \log ^2(x)+2 \log \left (\frac {1}{x}\right ) (\log (x)+5)^2+\left (10-e^{\frac {3 x}{2 (\log (x)+5)}} (20-3 x) x\right ) \log (x)+25\right )}{\left (2 e^{\frac {3 x}{2 (\log (x)+5)}} x-\log \left (\frac {1}{x}\right )\right )^3 (\log (x)+5)^2}dx\) |
| \(\Big \downarrow \) 7257 |
| \(\displaystyle \exp \left (\frac {4 x^4}{\left (2 x e^{\frac {3 x}{2 (\log (x)+5)}}-\log \left (\frac {1}{x}\right )\right )^2}\right )\) |
Int[(E^((4*x^4)/(4*E^((6*x)/(10 + 2*Log[x]))*x^2 - 4*E^((3*x)/(10 + 2*Log[ x]))*x*Log[x^(-1)] + Log[x^(-1)]^2))*(-200*x^3 - 400*x^3*Log[x^(-1)] + (-8 0*x^3 - 160*x^3*Log[x^(-1)])*Log[x] + (-8*x^3 - 16*x^3*Log[x^(-1)])*Log[x] ^2 + E^((3*x)/(10 + 2*Log[x]))*(400*x^4 - 96*x^5 + (160*x^4 - 24*x^5)*Log[ x] + 16*x^4*Log[x]^2)))/(-25*Log[x^(-1)]^3 - 10*Log[x^(-1)]^3*Log[x] - Log [x^(-1)]^3*Log[x]^2 + E^((9*x)/(10 + 2*Log[x]))*(200*x^3 + 80*x^3*Log[x] + 8*x^3*Log[x]^2) + E^((6*x)/(10 + 2*Log[x]))*(-300*x^2*Log[x^(-1)] - 120*x ^2*Log[x^(-1)]*Log[x] - 12*x^2*Log[x^(-1)]*Log[x]^2) + E^((3*x)/(10 + 2*Lo g[x]))*(150*x*Log[x^(-1)]^2 + 60*x*Log[x^(-1)]^2*Log[x] + 6*x*Log[x^(-1)]^ 2*Log[x]^2)),x]
3.8.97.3.1 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_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Sim p[q*(F^v/Log[F]), x] /; !FalseQ[q]] /; FreeQ[F, x]
Time = 0.92 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33
\[{\mathrm e}^{\frac {4 x^{4}}{4 x^{2} {\mathrm e}^{\frac {3 x}{5+\ln \left (x \right )}}+4 \ln \left (x \right ) {\mathrm e}^{\frac {3 x}{2 \left (5+\ln \left (x \right )\right )}} x +\ln \left (x \right )^{2}}}\]
int(((16*x^4*ln(x)^2+(-24*x^5+160*x^4)*ln(x)-96*x^5+400*x^4)*exp(3*x/(2*ln (x)+10))+(-16*x^3*ln(1/x)-8*x^3)*ln(x)^2+(-160*x^3*ln(1/x)-80*x^3)*ln(x)-4 00*x^3*ln(1/x)-200*x^3)*exp(4*x^4/(4*x^2*exp(3*x/(2*ln(x)+10))^2-4*x*ln(1/ x)*exp(3*x/(2*ln(x)+10))+ln(1/x)^2))/((8*x^3*ln(x)^2+80*x^3*ln(x)+200*x^3) *exp(3*x/(2*ln(x)+10))^3+(-12*x^2*ln(1/x)*ln(x)^2-120*x^2*ln(1/x)*ln(x)-30 0*x^2*ln(1/x))*exp(3*x/(2*ln(x)+10))^2+(6*x*ln(1/x)^2*ln(x)^2+60*x*ln(1/x) ^2*ln(x)+150*x*ln(1/x)^2)*exp(3*x/(2*ln(x)+10))-ln(1/x)^3*ln(x)^2-10*ln(1/ x)^3*ln(x)-25*ln(1/x)^3),x)
Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.55 \[ \int \frac {e^{\frac {4 x^4}{4 e^{\frac {6 x}{10+2 \log (x)}} x^2-4 e^{\frac {3 x}{10+2 \log (x)}} x \log \left (\frac {1}{x}\right )+\log ^2\left (\frac {1}{x}\right )}} \left (-200 x^3-400 x^3 \log \left (\frac {1}{x}\right )+\left (-80 x^3-160 x^3 \log \left (\frac {1}{x}\right )\right ) \log (x)+\left (-8 x^3-16 x^3 \log \left (\frac {1}{x}\right )\right ) \log ^2(x)+e^{\frac {3 x}{10+2 \log (x)}} \left (400 x^4-96 x^5+\left (160 x^4-24 x^5\right ) \log (x)+16 x^4 \log ^2(x)\right )\right )}{-25 \log ^3\left (\frac {1}{x}\right )-10 \log ^3\left (\frac {1}{x}\right ) \log (x)-\log ^3\left (\frac {1}{x}\right ) \log ^2(x)+e^{\frac {9 x}{10+2 \log (x)}} \left (200 x^3+80 x^3 \log (x)+8 x^3 \log ^2(x)\right )+e^{\frac {6 x}{10+2 \log (x)}} \left (-300 x^2 \log \left (\frac {1}{x}\right )-120 x^2 \log \left (\frac {1}{x}\right ) \log (x)-12 x^2 \log \left (\frac {1}{x}\right ) \log ^2(x)\right )+e^{\frac {3 x}{10+2 \log (x)}} \left (150 x \log ^2\left (\frac {1}{x}\right )+60 x \log ^2\left (\frac {1}{x}\right ) \log (x)+6 x \log ^2\left (\frac {1}{x}\right ) \log ^2(x)\right )} \, dx=e^{\left (\frac {4 \, x^{4}}{4 \, x^{2} e^{\left (-\frac {3 \, x}{\log \left (\frac {1}{x}\right ) - 5}\right )} - 4 \, x e^{\left (-\frac {3 \, x}{2 \, {\left (\log \left (\frac {1}{x}\right ) - 5\right )}}\right )} \log \left (\frac {1}{x}\right ) + \log \left (\frac {1}{x}\right )^{2}}\right )} \]
integrate(((16*x^4*log(x)^2+(-24*x^5+160*x^4)*log(x)-96*x^5+400*x^4)*exp(3 *x/(2*log(x)+10))+(-16*x^3*log(1/x)-8*x^3)*log(x)^2+(-160*x^3*log(1/x)-80* x^3)*log(x)-400*x^3*log(1/x)-200*x^3)*exp(4*x^4/(4*x^2*exp(3*x/(2*log(x)+1 0))^2-4*x*log(1/x)*exp(3*x/(2*log(x)+10))+log(1/x)^2))/((8*x^3*log(x)^2+80 *x^3*log(x)+200*x^3)*exp(3*x/(2*log(x)+10))^3+(-12*x^2*log(1/x)*log(x)^2-1 20*x^2*log(1/x)*log(x)-300*x^2*log(1/x))*exp(3*x/(2*log(x)+10))^2+(6*x*log (1/x)^2*log(x)^2+60*x*log(1/x)^2*log(x)+150*x*log(1/x)^2)*exp(3*x/(2*log(x )+10))-log(1/x)^3*log(x)^2-10*log(1/x)^3*log(x)-25*log(1/x)^3),x, algorith m=\
e^(4*x^4/(4*x^2*e^(-3*x/(log(1/x) - 5)) - 4*x*e^(-3/2*x/(log(1/x) - 5))*lo g(1/x) + log(1/x)^2))
Time = 2.86 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39 \[ \int \frac {e^{\frac {4 x^4}{4 e^{\frac {6 x}{10+2 \log (x)}} x^2-4 e^{\frac {3 x}{10+2 \log (x)}} x \log \left (\frac {1}{x}\right )+\log ^2\left (\frac {1}{x}\right )}} \left (-200 x^3-400 x^3 \log \left (\frac {1}{x}\right )+\left (-80 x^3-160 x^3 \log \left (\frac {1}{x}\right )\right ) \log (x)+\left (-8 x^3-16 x^3 \log \left (\frac {1}{x}\right )\right ) \log ^2(x)+e^{\frac {3 x}{10+2 \log (x)}} \left (400 x^4-96 x^5+\left (160 x^4-24 x^5\right ) \log (x)+16 x^4 \log ^2(x)\right )\right )}{-25 \log ^3\left (\frac {1}{x}\right )-10 \log ^3\left (\frac {1}{x}\right ) \log (x)-\log ^3\left (\frac {1}{x}\right ) \log ^2(x)+e^{\frac {9 x}{10+2 \log (x)}} \left (200 x^3+80 x^3 \log (x)+8 x^3 \log ^2(x)\right )+e^{\frac {6 x}{10+2 \log (x)}} \left (-300 x^2 \log \left (\frac {1}{x}\right )-120 x^2 \log \left (\frac {1}{x}\right ) \log (x)-12 x^2 \log \left (\frac {1}{x}\right ) \log ^2(x)\right )+e^{\frac {3 x}{10+2 \log (x)}} \left (150 x \log ^2\left (\frac {1}{x}\right )+60 x \log ^2\left (\frac {1}{x}\right ) \log (x)+6 x \log ^2\left (\frac {1}{x}\right ) \log ^2(x)\right )} \, dx=e^{\frac {4 x^{4}}{4 x^{2} e^{\frac {6 x}{2 \log {\left (x \right )} + 10}} + 4 x e^{\frac {3 x}{2 \log {\left (x \right )} + 10}} \log {\left (x \right )} + \log {\left (x \right )}^{2}}} \]
integrate(((16*x**4*ln(x)**2+(-24*x**5+160*x**4)*ln(x)-96*x**5+400*x**4)*e xp(3*x/(2*ln(x)+10))+(-16*x**3*ln(1/x)-8*x**3)*ln(x)**2+(-160*x**3*ln(1/x) -80*x**3)*ln(x)-400*x**3*ln(1/x)-200*x**3)*exp(4*x**4/(4*x**2*exp(3*x/(2*l n(x)+10))**2-4*x*ln(1/x)*exp(3*x/(2*ln(x)+10))+ln(1/x)**2))/((8*x**3*ln(x) **2+80*x**3*ln(x)+200*x**3)*exp(3*x/(2*ln(x)+10))**3+(-12*x**2*ln(1/x)*ln( x)**2-120*x**2*ln(1/x)*ln(x)-300*x**2*ln(1/x))*exp(3*x/(2*ln(x)+10))**2+(6 *x*ln(1/x)**2*ln(x)**2+60*x*ln(1/x)**2*ln(x)+150*x*ln(1/x)**2)*exp(3*x/(2* ln(x)+10))-ln(1/x)**3*ln(x)**2-10*ln(1/x)**3*ln(x)-25*ln(1/x)**3),x)
exp(4*x**4/(4*x**2*exp(6*x/(2*log(x) + 10)) + 4*x*exp(3*x/(2*log(x) + 10)) *log(x) + log(x)**2))
Exception generated. \[ \int \frac {e^{\frac {4 x^4}{4 e^{\frac {6 x}{10+2 \log (x)}} x^2-4 e^{\frac {3 x}{10+2 \log (x)}} x \log \left (\frac {1}{x}\right )+\log ^2\left (\frac {1}{x}\right )}} \left (-200 x^3-400 x^3 \log \left (\frac {1}{x}\right )+\left (-80 x^3-160 x^3 \log \left (\frac {1}{x}\right )\right ) \log (x)+\left (-8 x^3-16 x^3 \log \left (\frac {1}{x}\right )\right ) \log ^2(x)+e^{\frac {3 x}{10+2 \log (x)}} \left (400 x^4-96 x^5+\left (160 x^4-24 x^5\right ) \log (x)+16 x^4 \log ^2(x)\right )\right )}{-25 \log ^3\left (\frac {1}{x}\right )-10 \log ^3\left (\frac {1}{x}\right ) \log (x)-\log ^3\left (\frac {1}{x}\right ) \log ^2(x)+e^{\frac {9 x}{10+2 \log (x)}} \left (200 x^3+80 x^3 \log (x)+8 x^3 \log ^2(x)\right )+e^{\frac {6 x}{10+2 \log (x)}} \left (-300 x^2 \log \left (\frac {1}{x}\right )-120 x^2 \log \left (\frac {1}{x}\right ) \log (x)-12 x^2 \log \left (\frac {1}{x}\right ) \log ^2(x)\right )+e^{\frac {3 x}{10+2 \log (x)}} \left (150 x \log ^2\left (\frac {1}{x}\right )+60 x \log ^2\left (\frac {1}{x}\right ) \log (x)+6 x \log ^2\left (\frac {1}{x}\right ) \log ^2(x)\right )} \, dx=\text {Exception raised: RuntimeError} \]
integrate(((16*x^4*log(x)^2+(-24*x^5+160*x^4)*log(x)-96*x^5+400*x^4)*exp(3 *x/(2*log(x)+10))+(-16*x^3*log(1/x)-8*x^3)*log(x)^2+(-160*x^3*log(1/x)-80* x^3)*log(x)-400*x^3*log(1/x)-200*x^3)*exp(4*x^4/(4*x^2*exp(3*x/(2*log(x)+1 0))^2-4*x*log(1/x)*exp(3*x/(2*log(x)+10))+log(1/x)^2))/((8*x^3*log(x)^2+80 *x^3*log(x)+200*x^3)*exp(3*x/(2*log(x)+10))^3+(-12*x^2*log(1/x)*log(x)^2-1 20*x^2*log(1/x)*log(x)-300*x^2*log(1/x))*exp(3*x/(2*log(x)+10))^2+(6*x*log (1/x)^2*log(x)^2+60*x*log(1/x)^2*log(x)+150*x*log(1/x)^2)*exp(3*x/(2*log(x )+10))-log(1/x)^3*log(x)^2-10*log(1/x)^3*log(x)-25*log(1/x)^3),x, algorith m=\
Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is 0which is not of the expected type LIST
\[ \int \frac {e^{\frac {4 x^4}{4 e^{\frac {6 x}{10+2 \log (x)}} x^2-4 e^{\frac {3 x}{10+2 \log (x)}} x \log \left (\frac {1}{x}\right )+\log ^2\left (\frac {1}{x}\right )}} \left (-200 x^3-400 x^3 \log \left (\frac {1}{x}\right )+\left (-80 x^3-160 x^3 \log \left (\frac {1}{x}\right )\right ) \log (x)+\left (-8 x^3-16 x^3 \log \left (\frac {1}{x}\right )\right ) \log ^2(x)+e^{\frac {3 x}{10+2 \log (x)}} \left (400 x^4-96 x^5+\left (160 x^4-24 x^5\right ) \log (x)+16 x^4 \log ^2(x)\right )\right )}{-25 \log ^3\left (\frac {1}{x}\right )-10 \log ^3\left (\frac {1}{x}\right ) \log (x)-\log ^3\left (\frac {1}{x}\right ) \log ^2(x)+e^{\frac {9 x}{10+2 \log (x)}} \left (200 x^3+80 x^3 \log (x)+8 x^3 \log ^2(x)\right )+e^{\frac {6 x}{10+2 \log (x)}} \left (-300 x^2 \log \left (\frac {1}{x}\right )-120 x^2 \log \left (\frac {1}{x}\right ) \log (x)-12 x^2 \log \left (\frac {1}{x}\right ) \log ^2(x)\right )+e^{\frac {3 x}{10+2 \log (x)}} \left (150 x \log ^2\left (\frac {1}{x}\right )+60 x \log ^2\left (\frac {1}{x}\right ) \log (x)+6 x \log ^2\left (\frac {1}{x}\right ) \log ^2(x)\right )} \, dx=\int { \frac {8 \, {\left (50 \, x^{3} \log \left (\frac {1}{x}\right ) + 25 \, x^{3} + {\left (2 \, x^{3} \log \left (\frac {1}{x}\right ) + x^{3}\right )} \log \left (x\right )^{2} - {\left (2 \, x^{4} \log \left (x\right )^{2} - 12 \, x^{5} + 50 \, x^{4} - {\left (3 \, x^{5} - 20 \, x^{4}\right )} \log \left (x\right )\right )} e^{\left (\frac {3 \, x}{2 \, {\left (\log \left (x\right ) + 5\right )}}\right )} + 10 \, {\left (2 \, x^{3} \log \left (\frac {1}{x}\right ) + x^{3}\right )} \log \left (x\right )\right )} e^{\left (\frac {4 \, x^{4}}{4 \, x^{2} e^{\left (\frac {3 \, x}{\log \left (x\right ) + 5}\right )} - 4 \, x e^{\left (\frac {3 \, x}{2 \, {\left (\log \left (x\right ) + 5\right )}}\right )} \log \left (\frac {1}{x}\right ) + \log \left (\frac {1}{x}\right )^{2}}\right )}}{\log \left (x\right )^{2} \log \left (\frac {1}{x}\right )^{3} + 10 \, \log \left (x\right ) \log \left (\frac {1}{x}\right )^{3} + 25 \, \log \left (\frac {1}{x}\right )^{3} - 8 \, {\left (x^{3} \log \left (x\right )^{2} + 10 \, x^{3} \log \left (x\right ) + 25 \, x^{3}\right )} e^{\left (\frac {9 \, x}{2 \, {\left (\log \left (x\right ) + 5\right )}}\right )} + 12 \, {\left (x^{2} \log \left (x\right )^{2} \log \left (\frac {1}{x}\right ) + 10 \, x^{2} \log \left (x\right ) \log \left (\frac {1}{x}\right ) + 25 \, x^{2} \log \left (\frac {1}{x}\right )\right )} e^{\left (\frac {3 \, x}{\log \left (x\right ) + 5}\right )} - 6 \, {\left (x \log \left (x\right )^{2} \log \left (\frac {1}{x}\right )^{2} + 10 \, x \log \left (x\right ) \log \left (\frac {1}{x}\right )^{2} + 25 \, x \log \left (\frac {1}{x}\right )^{2}\right )} e^{\left (\frac {3 \, x}{2 \, {\left (\log \left (x\right ) + 5\right )}}\right )}} \,d x } \]
integrate(((16*x^4*log(x)^2+(-24*x^5+160*x^4)*log(x)-96*x^5+400*x^4)*exp(3 *x/(2*log(x)+10))+(-16*x^3*log(1/x)-8*x^3)*log(x)^2+(-160*x^3*log(1/x)-80* x^3)*log(x)-400*x^3*log(1/x)-200*x^3)*exp(4*x^4/(4*x^2*exp(3*x/(2*log(x)+1 0))^2-4*x*log(1/x)*exp(3*x/(2*log(x)+10))+log(1/x)^2))/((8*x^3*log(x)^2+80 *x^3*log(x)+200*x^3)*exp(3*x/(2*log(x)+10))^3+(-12*x^2*log(1/x)*log(x)^2-1 20*x^2*log(1/x)*log(x)-300*x^2*log(1/x))*exp(3*x/(2*log(x)+10))^2+(6*x*log (1/x)^2*log(x)^2+60*x*log(1/x)^2*log(x)+150*x*log(1/x)^2)*exp(3*x/(2*log(x )+10))-log(1/x)^3*log(x)^2-10*log(1/x)^3*log(x)-25*log(1/x)^3),x, algorith m=\
integrate(8*(50*x^3*log(1/x) + 25*x^3 + (2*x^3*log(1/x) + x^3)*log(x)^2 - (2*x^4*log(x)^2 - 12*x^5 + 50*x^4 - (3*x^5 - 20*x^4)*log(x))*e^(3/2*x/(log (x) + 5)) + 10*(2*x^3*log(1/x) + x^3)*log(x))*e^(4*x^4/(4*x^2*e^(3*x/(log( x) + 5)) - 4*x*e^(3/2*x/(log(x) + 5))*log(1/x) + log(1/x)^2))/(log(x)^2*lo g(1/x)^3 + 10*log(x)*log(1/x)^3 + 25*log(1/x)^3 - 8*(x^3*log(x)^2 + 10*x^3 *log(x) + 25*x^3)*e^(9/2*x/(log(x) + 5)) + 12*(x^2*log(x)^2*log(1/x) + 10* x^2*log(x)*log(1/x) + 25*x^2*log(1/x))*e^(3*x/(log(x) + 5)) - 6*(x*log(x)^ 2*log(1/x)^2 + 10*x*log(x)*log(1/x)^2 + 25*x*log(1/x)^2)*e^(3/2*x/(log(x) + 5))), x)
Time = 9.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \frac {e^{\frac {4 x^4}{4 e^{\frac {6 x}{10+2 \log (x)}} x^2-4 e^{\frac {3 x}{10+2 \log (x)}} x \log \left (\frac {1}{x}\right )+\log ^2\left (\frac {1}{x}\right )}} \left (-200 x^3-400 x^3 \log \left (\frac {1}{x}\right )+\left (-80 x^3-160 x^3 \log \left (\frac {1}{x}\right )\right ) \log (x)+\left (-8 x^3-16 x^3 \log \left (\frac {1}{x}\right )\right ) \log ^2(x)+e^{\frac {3 x}{10+2 \log (x)}} \left (400 x^4-96 x^5+\left (160 x^4-24 x^5\right ) \log (x)+16 x^4 \log ^2(x)\right )\right )}{-25 \log ^3\left (\frac {1}{x}\right )-10 \log ^3\left (\frac {1}{x}\right ) \log (x)-\log ^3\left (\frac {1}{x}\right ) \log ^2(x)+e^{\frac {9 x}{10+2 \log (x)}} \left (200 x^3+80 x^3 \log (x)+8 x^3 \log ^2(x)\right )+e^{\frac {6 x}{10+2 \log (x)}} \left (-300 x^2 \log \left (\frac {1}{x}\right )-120 x^2 \log \left (\frac {1}{x}\right ) \log (x)-12 x^2 \log \left (\frac {1}{x}\right ) \log ^2(x)\right )+e^{\frac {3 x}{10+2 \log (x)}} \left (150 x \log ^2\left (\frac {1}{x}\right )+60 x \log ^2\left (\frac {1}{x}\right ) \log (x)+6 x \log ^2\left (\frac {1}{x}\right ) \log ^2(x)\right )} \, dx={\mathrm {e}}^{\frac {4\,x^4}{4\,x^2\,{\mathrm {e}}^{\frac {3\,x}{\ln \left (x\right )+5}}+{\ln \left (\frac {1}{x}\right )}^2-4\,x\,\ln \left (\frac {1}{x}\right )\,{\mathrm {e}}^{\frac {3\,x}{2\,\ln \left (x\right )+10}}}} \]
int((exp((4*x^4)/(4*x^2*exp((6*x)/(2*log(x) + 10)) + log(1/x)^2 - 4*x*log( 1/x)*exp((3*x)/(2*log(x) + 10))))*(log(x)*(160*x^3*log(1/x) + 80*x^3) + 40 0*x^3*log(1/x) + log(x)^2*(16*x^3*log(1/x) + 8*x^3) + 200*x^3 - exp((3*x)/ (2*log(x) + 10))*(log(x)*(160*x^4 - 24*x^5) + 16*x^4*log(x)^2 + 400*x^4 - 96*x^5)))/(log(1/x)^3*log(x)^2 - exp((3*x)/(2*log(x) + 10))*(150*x*log(1/x )^2 + 60*x*log(1/x)^2*log(x) + 6*x*log(1/x)^2*log(x)^2) + exp((6*x)/(2*log (x) + 10))*(300*x^2*log(1/x) + 120*x^2*log(1/x)*log(x) + 12*x^2*log(1/x)*l og(x)^2) + 25*log(1/x)^3 - exp((9*x)/(2*log(x) + 10))*(80*x^3*log(x) + 8*x ^3*log(x)^2 + 200*x^3) + 10*log(1/x)^3*log(x)),x)