Integrand size = 120, antiderivative size = 31 \begin {dmath*} \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} \left (72+e (12-3 x)-18 x+(108+18 e+90 x) \log \left (-\frac {8}{-4+x}\right )+(-180+45 x) \log ^2\left (-\frac {8}{-4+x}\right )\right )}{-4 x^2+x^3+\left (-24 x+6 x^2\right ) \log ^2\left (-\frac {8}{-4+x}\right )+(-36+9 x) \log ^4\left (-\frac {8}{-4+x}\right )} \, dx=-3+e^{\frac {6+e+5 x}{\frac {x}{3}+\log ^2\left (\frac {8}{4-x}\right )}} \end {dmath*}
Time = 0.11 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \begin {dmath*} \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} \left (72+e (12-3 x)-18 x+(108+18 e+90 x) \log \left (-\frac {8}{-4+x}\right )+(-180+45 x) \log ^2\left (-\frac {8}{-4+x}\right )\right )}{-4 x^2+x^3+\left (-24 x+6 x^2\right ) \log ^2\left (-\frac {8}{-4+x}\right )+(-36+9 x) \log ^4\left (-\frac {8}{-4+x}\right )} \, dx=e^{\frac {3 (26+e+5 (-4+x))}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} \end {dmath*}
Integrate[(E^((18 + 3*E + 15*x)/(x + 3*Log[-8/(-4 + x)]^2))*(72 + E*(12 - 3*x) - 18*x + (108 + 18*E + 90*x)*Log[-8/(-4 + x)] + (-180 + 45*x)*Log[-8/ (-4 + x)]^2))/(-4*x^2 + x^3 + (-24*x + 6*x^2)*Log[-8/(-4 + x)]^2 + (-36 + 9*x)*Log[-8/(-4 + x)]^4),x]
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.
\(\displaystyle \int \frac {e^{\frac {15 x+3 e+18}{x+3 \log ^2\left (-\frac {8}{x-4}\right )}} \left (e (12-3 x)-18 x+(45 x-180) \log ^2\left (-\frac {8}{x-4}\right )+(90 x+18 e+108) \log \left (-\frac {8}{x-4}\right )+72\right )}{x^3-4 x^2+\left (6 x^2-24 x\right ) \log ^2\left (-\frac {8}{x-4}\right )+(9 x-36) \log ^4\left (-\frac {8}{x-4}\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{\frac {15 x+3 e+18}{x+3 \log ^2\left (-\frac {8}{x-4}\right )}} \left (-e (12-3 x)+18 x-\left ((45 x-180) \log ^2\left (-\frac {8}{x-4}\right )\right )-(90 x+18 e+108) \log \left (-\frac {8}{x-4}\right )-72\right )}{(4-x) \left (x+3 \log ^2\left (-\frac {8}{x-4}\right )\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {15 e^{\frac {15 x+3 e+18}{x+3 \log ^2\left (-\frac {8}{x-4}\right )}}}{x+3 \log ^2\left (-\frac {8}{x-4}\right )}-\frac {3 (5 x+e+6) e^{\frac {15 x+3 e+18}{x+3 \log ^2\left (-\frac {8}{x-4}\right )}} \left (x-6 \log \left (-\frac {8}{x-4}\right )-4\right )}{(x-4) \left (x+3 \log ^2\left (-\frac {8}{x-4}\right )\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -3 (26+e) \int \frac {e^{\frac {15 x+3 e+18}{3 \log ^2\left (-\frac {8}{x-4}\right )+x}}}{\left (3 \log ^2\left (-\frac {8}{x-4}\right )+x\right )^2}dx+60 \int \frac {e^{\frac {15 x+3 e+18}{3 \log ^2\left (-\frac {8}{x-4}\right )+x}}}{\left (3 \log ^2\left (-\frac {8}{x-4}\right )+x\right )^2}dx-15 \int \frac {e^{\frac {15 x+3 e+18}{3 \log ^2\left (-\frac {8}{x-4}\right )+x}} x}{\left (3 \log ^2\left (-\frac {8}{x-4}\right )+x\right )^2}dx+90 \int \frac {e^{\frac {15 x+3 e+18}{3 \log ^2\left (-\frac {8}{x-4}\right )+x}} \log \left (-\frac {8}{x-4}\right )}{\left (3 \log ^2\left (-\frac {8}{x-4}\right )+x\right )^2}dx+18 (26+e) \int \frac {e^{\frac {15 x+3 e+18}{3 \log ^2\left (-\frac {8}{x-4}\right )+x}} \log \left (-\frac {8}{x-4}\right )}{(x-4) \left (3 \log ^2\left (-\frac {8}{x-4}\right )+x\right )^2}dx+15 \int \frac {e^{\frac {15 x+3 e+18}{3 \log ^2\left (-\frac {8}{x-4}\right )+x}}}{3 \log ^2\left (-\frac {8}{x-4}\right )+x}dx\) |
Int[(E^((18 + 3*E + 15*x)/(x + 3*Log[-8/(-4 + x)]^2))*(72 + E*(12 - 3*x) - 18*x + (108 + 18*E + 90*x)*Log[-8/(-4 + x)] + (-180 + 45*x)*Log[-8/(-4 + x)]^2))/(-4*x^2 + x^3 + (-24*x + 6*x^2)*Log[-8/(-4 + x)]^2 + (-36 + 9*x)*L og[-8/(-4 + x)]^4),x]
3.10.9.3.1 Defintions of rubi rules used
Time = 5.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87
method | result | size |
risch | \({\mathrm e}^{\frac {3 \,{\mathrm e}+15 x +18}{3 \ln \left (-\frac {8}{x -4}\right )^{2}+x}}\) | \(27\) |
parallelrisch | \({\mathrm e}^{\frac {3 \,{\mathrm e}+15 x +18}{3 \ln \left (-\frac {8}{x -4}\right )^{2}+x}}\) | \(27\) |
int(((45*x-180)*ln(-8/(x-4))^2+(18*exp(1)+90*x+108)*ln(-8/(x-4))+(-3*x+12) *exp(1)-18*x+72)*exp((3*exp(1)+15*x+18)/(3*ln(-8/(x-4))^2+x))/((9*x-36)*ln (-8/(x-4))^4+(6*x^2-24*x)*ln(-8/(x-4))^2+x^3-4*x^2),x,method=_RETURNVERBOS E)
Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \begin {dmath*} \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} \left (72+e (12-3 x)-18 x+(108+18 e+90 x) \log \left (-\frac {8}{-4+x}\right )+(-180+45 x) \log ^2\left (-\frac {8}{-4+x}\right )\right )}{-4 x^2+x^3+\left (-24 x+6 x^2\right ) \log ^2\left (-\frac {8}{-4+x}\right )+(-36+9 x) \log ^4\left (-\frac {8}{-4+x}\right )} \, dx=e^{\left (\frac {3 \, {\left (5 \, x + e + 6\right )}}{3 \, \log \left (-\frac {8}{x - 4}\right )^{2} + x}\right )} \end {dmath*}
integrate(((45*x-180)*log(-8/(x-4))^2+(18*exp(1)+90*x+108)*log(-8/(x-4))+( -3*x+12)*exp(1)-18*x+72)*exp((3*exp(1)+15*x+18)/(3*log(-8/(x-4))^2+x))/((9 *x-36)*log(-8/(x-4))^4+(6*x^2-24*x)*log(-8/(x-4))^2+x^3-4*x^2),x, algorith m=\
Time = 0.44 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \begin {dmath*} \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} \left (72+e (12-3 x)-18 x+(108+18 e+90 x) \log \left (-\frac {8}{-4+x}\right )+(-180+45 x) \log ^2\left (-\frac {8}{-4+x}\right )\right )}{-4 x^2+x^3+\left (-24 x+6 x^2\right ) \log ^2\left (-\frac {8}{-4+x}\right )+(-36+9 x) \log ^4\left (-\frac {8}{-4+x}\right )} \, dx=e^{\frac {15 x + 3 e + 18}{x + 3 \log {\left (- \frac {8}{x - 4} \right )}^{2}}} \end {dmath*}
integrate(((45*x-180)*ln(-8/(x-4))**2+(18*exp(1)+90*x+108)*ln(-8/(x-4))+(- 3*x+12)*exp(1)-18*x+72)*exp((3*exp(1)+15*x+18)/(3*ln(-8/(x-4))**2+x))/((9* x-36)*ln(-8/(x-4))**4+(6*x**2-24*x)*ln(-8/(x-4))**2+x**3-4*x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (28) = 56\).
Time = 0.45 (sec) , antiderivative size = 185, normalized size of antiderivative = 5.97 \begin {dmath*} \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} \left (72+e (12-3 x)-18 x+(108+18 e+90 x) \log \left (-\frac {8}{-4+x}\right )+(-180+45 x) \log ^2\left (-\frac {8}{-4+x}\right )\right )}{-4 x^2+x^3+\left (-24 x+6 x^2\right ) \log ^2\left (-\frac {8}{-4+x}\right )+(-36+9 x) \log ^4\left (-\frac {8}{-4+x}\right )} \, dx=e^{\left (-\frac {405 \, \log \left (2\right )^{2}}{27 \, \log \left (2\right )^{2} - 18 \, \log \left (2\right ) \log \left (-x + 4\right ) + 3 \, \log \left (-x + 4\right )^{2} + x} + \frac {270 \, \log \left (2\right ) \log \left (-x + 4\right )}{27 \, \log \left (2\right )^{2} - 18 \, \log \left (2\right ) \log \left (-x + 4\right ) + 3 \, \log \left (-x + 4\right )^{2} + x} - \frac {45 \, \log \left (-x + 4\right )^{2}}{27 \, \log \left (2\right )^{2} - 18 \, \log \left (2\right ) \log \left (-x + 4\right ) + 3 \, \log \left (-x + 4\right )^{2} + x} + \frac {3 \, e}{27 \, \log \left (2\right )^{2} - 18 \, \log \left (2\right ) \log \left (-x + 4\right ) + 3 \, \log \left (-x + 4\right )^{2} + x} + \frac {18}{27 \, \log \left (2\right )^{2} - 18 \, \log \left (2\right ) \log \left (-x + 4\right ) + 3 \, \log \left (-x + 4\right )^{2} + x} + 15\right )} \end {dmath*}
integrate(((45*x-180)*log(-8/(x-4))^2+(18*exp(1)+90*x+108)*log(-8/(x-4))+( -3*x+12)*exp(1)-18*x+72)*exp((3*exp(1)+15*x+18)/(3*log(-8/(x-4))^2+x))/((9 *x-36)*log(-8/(x-4))^4+(6*x^2-24*x)*log(-8/(x-4))^2+x^3-4*x^2),x, algorith m=\
e^(-405*log(2)^2/(27*log(2)^2 - 18*log(2)*log(-x + 4) + 3*log(-x + 4)^2 + x) + 270*log(2)*log(-x + 4)/(27*log(2)^2 - 18*log(2)*log(-x + 4) + 3*log(- x + 4)^2 + x) - 45*log(-x + 4)^2/(27*log(2)^2 - 18*log(2)*log(-x + 4) + 3* log(-x + 4)^2 + x) + 3*e/(27*log(2)^2 - 18*log(2)*log(-x + 4) + 3*log(-x + 4)^2 + x) + 18/(27*log(2)^2 - 18*log(2)*log(-x + 4) + 3*log(-x + 4)^2 + x ) + 15)
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (28) = 56\).
Time = 1.68 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.90 \begin {dmath*} \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} \left (72+e (12-3 x)-18 x+(108+18 e+90 x) \log \left (-\frac {8}{-4+x}\right )+(-180+45 x) \log ^2\left (-\frac {8}{-4+x}\right )\right )}{-4 x^2+x^3+\left (-24 x+6 x^2\right ) \log ^2\left (-\frac {8}{-4+x}\right )+(-36+9 x) \log ^4\left (-\frac {8}{-4+x}\right )} \, dx=e^{\left (\frac {15 \, x}{3 \, \log \left (-\frac {8}{x - 4}\right )^{2} + x} + \frac {3 \, e}{3 \, \log \left (-\frac {8}{x - 4}\right )^{2} + x} + \frac {18}{3 \, \log \left (-\frac {8}{x - 4}\right )^{2} + x}\right )} \end {dmath*}
integrate(((45*x-180)*log(-8/(x-4))^2+(18*exp(1)+90*x+108)*log(-8/(x-4))+( -3*x+12)*exp(1)-18*x+72)*exp((3*exp(1)+15*x+18)/(3*log(-8/(x-4))^2+x))/((9 *x-36)*log(-8/(x-4))^4+(6*x^2-24*x)*log(-8/(x-4))^2+x^3-4*x^2),x, algorith m=\
e^(15*x/(3*log(-8/(x - 4))^2 + x) + 3*e/(3*log(-8/(x - 4))^2 + x) + 18/(3* log(-8/(x - 4))^2 + x))
Time = 17.85 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.97 \begin {dmath*} \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} \left (72+e (12-3 x)-18 x+(108+18 e+90 x) \log \left (-\frac {8}{-4+x}\right )+(-180+45 x) \log ^2\left (-\frac {8}{-4+x}\right )\right )}{-4 x^2+x^3+\left (-24 x+6 x^2\right ) \log ^2\left (-\frac {8}{-4+x}\right )+(-36+9 x) \log ^4\left (-\frac {8}{-4+x}\right )} \, dx={\mathrm {e}}^{\frac {3\,\mathrm {e}}{3\,{\ln \left (-\frac {8}{x-4}\right )}^2+x}}\,{\mathrm {e}}^{\frac {15\,x}{3\,{\ln \left (-\frac {8}{x-4}\right )}^2+x}}\,{\mathrm {e}}^{\frac {18}{3\,{\ln \left (-\frac {8}{x-4}\right )}^2+x}} \end {dmath*}
int(-(exp((15*x + 3*exp(1) + 18)/(x + 3*log(-8/(x - 4))^2))*(log(-8/(x - 4 ))*(90*x + 18*exp(1) + 108) - 18*x + log(-8/(x - 4))^2*(45*x - 180) - exp( 1)*(3*x - 12) + 72))/(log(-8/(x - 4))^2*(24*x - 6*x^2) + 4*x^2 - x^3 - log (-8/(x - 4))^4*(9*x - 36)),x)