Integrand size = 276, antiderivative size = 27 \[ \int \frac {-4 e^{2+\frac {4 \left (81 e^{25}-162 x\right )}{e^2 x}} x+\left (-324 e^{25+\frac {4 \left (81 e^{25}-162 x\right )}{e^2 x}}+12 e^{2+\frac {3 \left (81 e^{25}-162 x\right )}{e^2 x}} x^2\right ) \log (x)+\left (-12 e^{2+\frac {2 \left (81 e^{25}-162 x\right )}{e^2 x}} x^3+e^{\frac {3 \left (81 e^{25}-162 x\right )}{e^2 x}} \left (972 e^{25} x-4 e^2 x^2\right )\right ) \log ^2(x)+\left (4 e^{2+\frac {81 e^{25}-162 x}{e^2 x}} x^4+e^{\frac {2 \left (81 e^{25}-162 x\right )}{e^2 x}} \left (-972 e^{25} x^2+12 e^2 x^3\right )\right ) \log ^3(x)+e^{\frac {81 e^{25}-162 x}{e^2 x}} \left (324 e^{25} x^3-12 e^2 x^4\right ) \log ^4(x)+4 e^2 x^5 \log ^5(x)}{e^2 x^2 \log ^5(x)} \, dx=\left (-x+\frac {e^{\frac {81 \left (-2+\frac {e^{25}}{x}\right )}{e^2}}}{\log (x)}\right )^4 \]
Time = 0.34 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {-4 e^{2+\frac {4 \left (81 e^{25}-162 x\right )}{e^2 x}} x+\left (-324 e^{25+\frac {4 \left (81 e^{25}-162 x\right )}{e^2 x}}+12 e^{2+\frac {3 \left (81 e^{25}-162 x\right )}{e^2 x}} x^2\right ) \log (x)+\left (-12 e^{2+\frac {2 \left (81 e^{25}-162 x\right )}{e^2 x}} x^3+e^{\frac {3 \left (81 e^{25}-162 x\right )}{e^2 x}} \left (972 e^{25} x-4 e^2 x^2\right )\right ) \log ^2(x)+\left (4 e^{2+\frac {81 e^{25}-162 x}{e^2 x}} x^4+e^{\frac {2 \left (81 e^{25}-162 x\right )}{e^2 x}} \left (-972 e^{25} x^2+12 e^2 x^3\right )\right ) \log ^3(x)+e^{\frac {81 e^{25}-162 x}{e^2 x}} \left (324 e^{25} x^3-12 e^2 x^4\right ) \log ^4(x)+4 e^2 x^5 \log ^5(x)}{e^2 x^2 \log ^5(x)} \, dx=\frac {e^{-\frac {648}{e^2}} \left (e^{\frac {81 e^{23}}{x}}-e^{\frac {162}{e^2}} x \log (x)\right )^4}{\log ^4(x)} \]
Integrate[(-4*E^(2 + (4*(81*E^25 - 162*x))/(E^2*x))*x + (-324*E^(25 + (4*( 81*E^25 - 162*x))/(E^2*x)) + 12*E^(2 + (3*(81*E^25 - 162*x))/(E^2*x))*x^2) *Log[x] + (-12*E^(2 + (2*(81*E^25 - 162*x))/(E^2*x))*x^3 + E^((3*(81*E^25 - 162*x))/(E^2*x))*(972*E^25*x - 4*E^2*x^2))*Log[x]^2 + (4*E^(2 + (81*E^25 - 162*x)/(E^2*x))*x^4 + E^((2*(81*E^25 - 162*x))/(E^2*x))*(-972*E^25*x^2 + 12*E^2*x^3))*Log[x]^3 + E^((81*E^25 - 162*x)/(E^2*x))*(324*E^25*x^3 - 12 *E^2*x^4)*Log[x]^4 + 4*E^2*x^5*Log[x]^5)/(E^2*x^2*Log[x]^5),x]
Leaf count is larger than twice the leaf count of optimal. \(115\) vs. \(2(27)=54\).
Time = 2.37 (sec) , antiderivative size = 115, normalized size of antiderivative = 4.26, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {27, 27, 7239, 27, 7293, 2009}
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.
| \(\displaystyle \int \frac {4 e^2 x^5 \log ^5(x)+\left (12 e^{\frac {3 \left (81 e^{25}-162 x\right )}{e^2 x}+2} x^2-324 e^{\frac {4 \left (81 e^{25}-162 x\right )}{e^2 x}+25}\right ) \log (x)+e^{\frac {81 e^{25}-162 x}{e^2 x}} \left (324 e^{25} x^3-12 e^2 x^4\right ) \log ^4(x)+\left (e^{\frac {3 \left (81 e^{25}-162 x\right )}{e^2 x}} \left (972 e^{25} x-4 e^2 x^2\right )-12 e^{\frac {2 \left (81 e^{25}-162 x\right )}{e^2 x}+2} x^3\right ) \log ^2(x)+\left (4 e^{\frac {81 e^{25}-162 x}{e^2 x}+2} x^4+e^{\frac {2 \left (81 e^{25}-162 x\right )}{e^2 x}} \left (12 e^2 x^3-972 e^{25} x^2\right )\right ) \log ^3(x)-4 e^{\frac {4 \left (81 e^{25}-162 x\right )}{e^2 x}+2} x}{e^2 x^2 \log ^5(x)} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {4 \left (-e^2 x^5 \log ^5(x)-3 e^{\frac {81 \left (e^{25}-2 x\right )}{e^2 x}} \left (27 e^{25} x^3-e^2 x^4\right ) \log ^4(x)-\left (e^{\frac {81 \left (e^{25}-2 x\right )}{e^2 x}+2} x^4-3 e^{\frac {162 \left (e^{25}-2 x\right )}{e^2 x}} \left (81 e^{25} x^2-e^2 x^3\right )\right ) \log ^3(x)+\left (3 e^{\frac {162 \left (e^{25}-2 x\right )}{e^2 x}+2} x^3-e^{\frac {243 \left (e^{25}-2 x\right )}{e^2 x}} \left (243 e^{25} x-e^2 x^2\right )\right ) \log ^2(x)+3 \left (27 e^{\frac {324 \left (e^{25}-2 x\right )}{e^2 x}+25}-e^{\frac {243 \left (e^{25}-2 x\right )}{e^2 x}+2} x^2\right ) \log (x)+e^{\frac {324 \left (e^{25}-2 x\right )}{e^2 x}+2} x\right )}{x^2 \log ^5(x)}dx}{e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {4 \int \frac {-e^2 x^5 \log ^5(x)-3 e^{\frac {81 \left (e^{25}-2 x\right )}{e^2 x}} \left (27 e^{25} x^3-e^2 x^4\right ) \log ^4(x)-\left (e^{\frac {81 \left (e^{25}-2 x\right )}{e^2 x}+2} x^4-3 e^{\frac {162 \left (e^{25}-2 x\right )}{e^2 x}} \left (81 e^{25} x^2-e^2 x^3\right )\right ) \log ^3(x)+\left (3 e^{\frac {162 \left (e^{25}-2 x\right )}{e^2 x}+2} x^3-e^{\frac {243 \left (e^{25}-2 x\right )}{e^2 x}} \left (243 e^{25} x-e^2 x^2\right )\right ) \log ^2(x)+3 \left (27 e^{\frac {324 \left (e^{25}-2 x\right )}{e^2 x}+25}-e^{\frac {243 \left (e^{25}-2 x\right )}{e^2 x}+2} x^2\right ) \log (x)+e^{\frac {324 \left (e^{25}-2 x\right )}{e^2 x}+2} x}{x^2 \log ^5(x)}dx}{e^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \int \frac {e^{2-\frac {648}{e^2}} \left (e^{\frac {81 e^{23}}{x}}-e^{\frac {162}{e^2}} x \log (x)\right )^3 \left (e^{\frac {162}{e^2}} x^2 \log ^2(x)+81 e^{23+\frac {81 e^{23}}{x}} \log (x)+e^{\frac {81 e^{23}}{x}} x\right )}{x^2 \log ^5(x)}dx}{e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -4 e^{-\frac {648}{e^2}} \int \frac {\left (e^{\frac {81 e^{23}}{x}}-e^{\frac {162}{e^2}} x \log (x)\right )^3 \left (e^{\frac {162}{e^2}} x^2 \log ^2(x)+81 e^{23+\frac {81 e^{23}}{x}} \log (x)+e^{\frac {81 e^{23}}{x}} x\right )}{x^2 \log ^5(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 e^{-\frac {648}{e^2}} \int \left (-e^{\frac {648}{e^2}} x^3+\frac {e^{\frac {486}{e^2}+\frac {81 e^{23}}{x}} \left (3 \log (x) x-x-81 e^{23} \log (x)\right ) x}{\log ^2(x)}-\frac {3 e^{\frac {324}{e^2}+\frac {162 e^{23}}{x}} \left (\log (x) x-x-81 e^{23} \log (x)\right )}{\log ^3(x)}+\frac {e^{\frac {162}{e^2}+\frac {243 e^{23}}{x}} \left (\log (x) x-3 x-243 e^{23} \log (x)\right )}{\log ^4(x) x}+\frac {e^{\frac {324 e^{23}}{x}} \left (x+81 e^{23} \log (x)\right )}{\log ^5(x) x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 e^{-\frac {648}{e^2}} \left (-\frac {1}{4} e^{\frac {648}{e^2}} x^4+\frac {e^{\frac {81 e^{23}}{x}+\frac {486}{e^2}} x^3}{\log (x)}-\frac {3 e^{\frac {162 e^{23}}{x}+\frac {324}{e^2}} x^2}{2 \log ^2(x)}-\frac {e^{\frac {324 e^{23}}{x}}}{4 \log ^4(x)}+\frac {e^{\frac {243 e^{23}}{x}+\frac {162}{e^2}} x}{\log ^3(x)}\right )\) |
Int[(-4*E^(2 + (4*(81*E^25 - 162*x))/(E^2*x))*x + (-324*E^(25 + (4*(81*E^2 5 - 162*x))/(E^2*x)) + 12*E^(2 + (3*(81*E^25 - 162*x))/(E^2*x))*x^2)*Log[x ] + (-12*E^(2 + (2*(81*E^25 - 162*x))/(E^2*x))*x^3 + E^((3*(81*E^25 - 162* x))/(E^2*x))*(972*E^25*x - 4*E^2*x^2))*Log[x]^2 + (4*E^(2 + (81*E^25 - 162 *x)/(E^2*x))*x^4 + E^((2*(81*E^25 - 162*x))/(E^2*x))*(-972*E^25*x^2 + 12*E ^2*x^3))*Log[x]^3 + E^((81*E^25 - 162*x)/(E^2*x))*(324*E^25*x^3 - 12*E^2*x ^4)*Log[x]^4 + 4*E^2*x^5*Log[x]^5)/(E^2*x^2*Log[x]^5),x]
(-4*(-1/4*(E^(648/E^2)*x^4) - E^((324*E^23)/x)/(4*Log[x]^4) + (E^(162/E^2 + (243*E^23)/x)*x)/Log[x]^3 - (3*E^(324/E^2 + (162*E^23)/x)*x^2)/(2*Log[x] ^2) + (E^(486/E^2 + (81*E^23)/x)*x^3)/Log[x]))/E^(648/E^2)
3.13.26.3.1 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]]
Leaf count of result is larger than twice the leaf count of optimal. \(92\) vs. \(2(26)=52\).
Time = 4.20 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.44
| method | result | size |
| risch | \(x^{4}-\frac {{\mathrm e}^{\frac {81 \left ({\mathrm e}^{25}-2 x \right ) {\mathrm e}^{-2}}{x}} \left (4 x^{3} \ln \left (x \right )^{3}-6 \,{\mathrm e}^{\frac {81 \left ({\mathrm e}^{25}-2 x \right ) {\mathrm e}^{-2}}{x}} x^{2} \ln \left (x \right )^{2}+4 x \,{\mathrm e}^{\frac {162 \left ({\mathrm e}^{25}-2 x \right ) {\mathrm e}^{-2}}{x}} \ln \left (x \right )-{\mathrm e}^{\frac {243 \left ({\mathrm e}^{25}-2 x \right ) {\mathrm e}^{-2}}{x}}\right )}{\ln \left (x \right )^{4}}\) | \(93\) |
| parallelrisch | \(\frac {{\mathrm e}^{-2} \left (x^{4} {\mathrm e}^{2} \ln \left (x \right )^{4}-4 \,{\mathrm e}^{2} x^{3} \ln \left (x \right )^{3} {\mathrm e}^{\frac {81 \left ({\mathrm e}^{25}-2 x \right ) {\mathrm e}^{-2}}{x}}+6 \,{\mathrm e}^{2} x^{2} \ln \left (x \right )^{2} {\mathrm e}^{\frac {162 \left ({\mathrm e}^{25}-2 x \right ) {\mathrm e}^{-2}}{x}}-4 \,{\mathrm e}^{2} x \ln \left (x \right ) {\mathrm e}^{\frac {243 \left ({\mathrm e}^{25}-2 x \right ) {\mathrm e}^{-2}}{x}}+{\mathrm e}^{2} {\mathrm e}^{\frac {324 \left ({\mathrm e}^{25}-2 x \right ) {\mathrm e}^{-2}}{x}}\right )}{\ln \left (x \right )^{4}}\) | \(123\) |
int((4*x^5*exp(2)*ln(x)^5+(324*x^3*exp(25)-12*x^4*exp(2))*exp((81*exp(25)- 162*x)/exp(2)/x)*ln(x)^4+((-972*x^2*exp(25)+12*x^3*exp(2))*exp((81*exp(25) -162*x)/exp(2)/x)^2+4*x^4*exp(2)*exp((81*exp(25)-162*x)/exp(2)/x))*ln(x)^3 +((972*x*exp(25)-4*x^2*exp(2))*exp((81*exp(25)-162*x)/exp(2)/x)^3-12*x^3*e xp(2)*exp((81*exp(25)-162*x)/exp(2)/x)^2)*ln(x)^2+(-324*exp(25)*exp((81*ex p(25)-162*x)/exp(2)/x)^4+12*x^2*exp(2)*exp((81*exp(25)-162*x)/exp(2)/x)^3) *ln(x)-4*x*exp(2)*exp((81*exp(25)-162*x)/exp(2)/x)^4)/x^2/exp(2)/ln(x)^5,x ,method=_RETURNVERBOSE)
x^4-exp(81*(exp(25)-2*x)*exp(-2)/x)*(4*x^3*ln(x)^3-6*exp(81*(exp(25)-2*x)* exp(-2)/x)*x^2*ln(x)^2+4*x*exp(162*(exp(25)-2*x)*exp(-2)/x)*ln(x)-exp(243* (exp(25)-2*x)*exp(-2)/x))/ln(x)^4
Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (23) = 46\).
Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 4.81 \[ \int \frac {-4 e^{2+\frac {4 \left (81 e^{25}-162 x\right )}{e^2 x}} x+\left (-324 e^{25+\frac {4 \left (81 e^{25}-162 x\right )}{e^2 x}}+12 e^{2+\frac {3 \left (81 e^{25}-162 x\right )}{e^2 x}} x^2\right ) \log (x)+\left (-12 e^{2+\frac {2 \left (81 e^{25}-162 x\right )}{e^2 x}} x^3+e^{\frac {3 \left (81 e^{25}-162 x\right )}{e^2 x}} \left (972 e^{25} x-4 e^2 x^2\right )\right ) \log ^2(x)+\left (4 e^{2+\frac {81 e^{25}-162 x}{e^2 x}} x^4+e^{\frac {2 \left (81 e^{25}-162 x\right )}{e^2 x}} \left (-972 e^{25} x^2+12 e^2 x^3\right )\right ) \log ^3(x)+e^{\frac {81 e^{25}-162 x}{e^2 x}} \left (324 e^{25} x^3-12 e^2 x^4\right ) \log ^4(x)+4 e^2 x^5 \log ^5(x)}{e^2 x^2 \log ^5(x)} \, dx=\frac {{\left (x^{4} e^{8} \log \left (x\right )^{4} - 4 \, x^{3} e^{\left (\frac {{\left (2 \, x e^{2} - 162 \, x + 81 \, e^{25}\right )} e^{\left (-2\right )}}{x} + 6\right )} \log \left (x\right )^{3} + 6 \, x^{2} e^{\left (\frac {2 \, {\left (2 \, x e^{2} - 162 \, x + 81 \, e^{25}\right )} e^{\left (-2\right )}}{x} + 4\right )} \log \left (x\right )^{2} - 4 \, x e^{\left (\frac {3 \, {\left (2 \, x e^{2} - 162 \, x + 81 \, e^{25}\right )} e^{\left (-2\right )}}{x} + 2\right )} \log \left (x\right ) + e^{\left (\frac {4 \, {\left (2 \, x e^{2} - 162 \, x + 81 \, e^{25}\right )} e^{\left (-2\right )}}{x}\right )}\right )} e^{\left (-8\right )}}{\log \left (x\right )^{4}} \]
integrate((4*x^5*exp(2)*log(x)^5+(324*x^3*exp(25)-12*x^4*exp(2))*exp((81*e xp(25)-162*x)/exp(2)/x)*log(x)^4+((-972*x^2*exp(25)+12*x^3*exp(2))*exp((81 *exp(25)-162*x)/exp(2)/x)^2+4*x^4*exp(2)*exp((81*exp(25)-162*x)/exp(2)/x)) *log(x)^3+((972*x*exp(25)-4*x^2*exp(2))*exp((81*exp(25)-162*x)/exp(2)/x)^3 -12*x^3*exp(2)*exp((81*exp(25)-162*x)/exp(2)/x)^2)*log(x)^2+(-324*exp(25)* exp((81*exp(25)-162*x)/exp(2)/x)^4+12*x^2*exp(2)*exp((81*exp(25)-162*x)/ex p(2)/x)^3)*log(x)-4*x*exp(2)*exp((81*exp(25)-162*x)/exp(2)/x)^4)/x^2/exp(2 )/log(x)^5,x, algorithm=\
(x^4*e^8*log(x)^4 - 4*x^3*e^((2*x*e^2 - 162*x + 81*e^25)*e^(-2)/x + 6)*log (x)^3 + 6*x^2*e^(2*(2*x*e^2 - 162*x + 81*e^25)*e^(-2)/x + 4)*log(x)^2 - 4* x*e^(3*(2*x*e^2 - 162*x + 81*e^25)*e^(-2)/x + 2)*log(x) + e^(4*(2*x*e^2 - 162*x + 81*e^25)*e^(-2)/x))*e^(-8)/log(x)^4
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (19) = 38\).
Time = 0.22 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.78 \[ \int \frac {-4 e^{2+\frac {4 \left (81 e^{25}-162 x\right )}{e^2 x}} x+\left (-324 e^{25+\frac {4 \left (81 e^{25}-162 x\right )}{e^2 x}}+12 e^{2+\frac {3 \left (81 e^{25}-162 x\right )}{e^2 x}} x^2\right ) \log (x)+\left (-12 e^{2+\frac {2 \left (81 e^{25}-162 x\right )}{e^2 x}} x^3+e^{\frac {3 \left (81 e^{25}-162 x\right )}{e^2 x}} \left (972 e^{25} x-4 e^2 x^2\right )\right ) \log ^2(x)+\left (4 e^{2+\frac {81 e^{25}-162 x}{e^2 x}} x^4+e^{\frac {2 \left (81 e^{25}-162 x\right )}{e^2 x}} \left (-972 e^{25} x^2+12 e^2 x^3\right )\right ) \log ^3(x)+e^{\frac {81 e^{25}-162 x}{e^2 x}} \left (324 e^{25} x^3-12 e^2 x^4\right ) \log ^4(x)+4 e^2 x^5 \log ^5(x)}{e^2 x^2 \log ^5(x)} \, dx=x^{4} + \frac {- 4 x^{3} e^{\frac {- 162 x + 81 e^{25}}{x e^{2}}} \log {\left (x \right )}^{9} + 6 x^{2} e^{\frac {2 \left (- 162 x + 81 e^{25}\right )}{x e^{2}}} \log {\left (x \right )}^{8} - 4 x e^{\frac {3 \left (- 162 x + 81 e^{25}\right )}{x e^{2}}} \log {\left (x \right )}^{7} + e^{\frac {4 \left (- 162 x + 81 e^{25}\right )}{x e^{2}}} \log {\left (x \right )}^{6}}{\log {\left (x \right )}^{10}} \]
integrate((4*x**5*exp(2)*ln(x)**5+(324*x**3*exp(25)-12*x**4*exp(2))*exp((8 1*exp(25)-162*x)/exp(2)/x)*ln(x)**4+((-972*x**2*exp(25)+12*x**3*exp(2))*ex p((81*exp(25)-162*x)/exp(2)/x)**2+4*x**4*exp(2)*exp((81*exp(25)-162*x)/exp (2)/x))*ln(x)**3+((972*x*exp(25)-4*x**2*exp(2))*exp((81*exp(25)-162*x)/exp (2)/x)**3-12*x**3*exp(2)*exp((81*exp(25)-162*x)/exp(2)/x)**2)*ln(x)**2+(-3 24*exp(25)*exp((81*exp(25)-162*x)/exp(2)/x)**4+12*x**2*exp(2)*exp((81*exp( 25)-162*x)/exp(2)/x)**3)*ln(x)-4*x*exp(2)*exp((81*exp(25)-162*x)/exp(2)/x) **4)/x**2/exp(2)/ln(x)**5,x)
x**4 + (-4*x**3*exp((-162*x + 81*exp(25))*exp(-2)/x)*log(x)**9 + 6*x**2*ex p(2*(-162*x + 81*exp(25))*exp(-2)/x)*log(x)**8 - 4*x*exp(3*(-162*x + 81*ex p(25))*exp(-2)/x)*log(x)**7 + exp(4*(-162*x + 81*exp(25))*exp(-2)/x)*log(x )**6)/log(x)**10
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (23) = 46\).
Time = 0.33 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.67 \[ \int \frac {-4 e^{2+\frac {4 \left (81 e^{25}-162 x\right )}{e^2 x}} x+\left (-324 e^{25+\frac {4 \left (81 e^{25}-162 x\right )}{e^2 x}}+12 e^{2+\frac {3 \left (81 e^{25}-162 x\right )}{e^2 x}} x^2\right ) \log (x)+\left (-12 e^{2+\frac {2 \left (81 e^{25}-162 x\right )}{e^2 x}} x^3+e^{\frac {3 \left (81 e^{25}-162 x\right )}{e^2 x}} \left (972 e^{25} x-4 e^2 x^2\right )\right ) \log ^2(x)+\left (4 e^{2+\frac {81 e^{25}-162 x}{e^2 x}} x^4+e^{\frac {2 \left (81 e^{25}-162 x\right )}{e^2 x}} \left (-972 e^{25} x^2+12 e^2 x^3\right )\right ) \log ^3(x)+e^{\frac {81 e^{25}-162 x}{e^2 x}} \left (324 e^{25} x^3-12 e^2 x^4\right ) \log ^4(x)+4 e^2 x^5 \log ^5(x)}{e^2 x^2 \log ^5(x)} \, dx={\left (x^{4} e^{2} - \frac {{\left (4 \, x^{3} e^{\left (\frac {81 \, e^{23}}{x} + 486 \, e^{\left (-2\right )} + 2\right )} \log \left (x\right )^{3} - 6 \, x^{2} e^{\left (\frac {162 \, e^{23}}{x} + 324 \, e^{\left (-2\right )} + 2\right )} \log \left (x\right )^{2} + 4 \, x e^{\left (\frac {243 \, e^{23}}{x} + 162 \, e^{\left (-2\right )} + 2\right )} \log \left (x\right ) - e^{\left (\frac {324 \, e^{23}}{x} + 2\right )}\right )} e^{\left (-648 \, e^{\left (-2\right )}\right )}}{\log \left (x\right )^{4}}\right )} e^{\left (-2\right )} \]
integrate((4*x^5*exp(2)*log(x)^5+(324*x^3*exp(25)-12*x^4*exp(2))*exp((81*e xp(25)-162*x)/exp(2)/x)*log(x)^4+((-972*x^2*exp(25)+12*x^3*exp(2))*exp((81 *exp(25)-162*x)/exp(2)/x)^2+4*x^4*exp(2)*exp((81*exp(25)-162*x)/exp(2)/x)) *log(x)^3+((972*x*exp(25)-4*x^2*exp(2))*exp((81*exp(25)-162*x)/exp(2)/x)^3 -12*x^3*exp(2)*exp((81*exp(25)-162*x)/exp(2)/x)^2)*log(x)^2+(-324*exp(25)* exp((81*exp(25)-162*x)/exp(2)/x)^4+12*x^2*exp(2)*exp((81*exp(25)-162*x)/ex p(2)/x)^3)*log(x)-4*x*exp(2)*exp((81*exp(25)-162*x)/exp(2)/x)^4)/x^2/exp(2 )/log(x)^5,x, algorithm=\
(x^4*e^2 - (4*x^3*e^(81*e^23/x + 486*e^(-2) + 2)*log(x)^3 - 6*x^2*e^(162*e ^23/x + 324*e^(-2) + 2)*log(x)^2 + 4*x*e^(243*e^23/x + 162*e^(-2) + 2)*log (x) - e^(324*e^23/x + 2))*e^(-648*e^(-2))/log(x)^4)*e^(-2)
Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (23) = 46\).
Time = 0.34 (sec) , antiderivative size = 132, normalized size of antiderivative = 4.89 \[ \int \frac {-4 e^{2+\frac {4 \left (81 e^{25}-162 x\right )}{e^2 x}} x+\left (-324 e^{25+\frac {4 \left (81 e^{25}-162 x\right )}{e^2 x}}+12 e^{2+\frac {3 \left (81 e^{25}-162 x\right )}{e^2 x}} x^2\right ) \log (x)+\left (-12 e^{2+\frac {2 \left (81 e^{25}-162 x\right )}{e^2 x}} x^3+e^{\frac {3 \left (81 e^{25}-162 x\right )}{e^2 x}} \left (972 e^{25} x-4 e^2 x^2\right )\right ) \log ^2(x)+\left (4 e^{2+\frac {81 e^{25}-162 x}{e^2 x}} x^4+e^{\frac {2 \left (81 e^{25}-162 x\right )}{e^2 x}} \left (-972 e^{25} x^2+12 e^2 x^3\right )\right ) \log ^3(x)+e^{\frac {81 e^{25}-162 x}{e^2 x}} \left (324 e^{25} x^3-12 e^2 x^4\right ) \log ^4(x)+4 e^2 x^5 \log ^5(x)}{e^2 x^2 \log ^5(x)} \, dx=\frac {2 \, {\left (3 \, x^{2} e^{\left (\frac {2 \, {\left (x e^{2} - 162 \, x + 81 \, e^{25}\right )} e^{\left (-2\right )}}{x} + 1\right )} \log \left (x\right ) - 2 \, x e^{\left (\frac {3 \, {\left (x e^{2} - 162 \, x + 81 \, e^{25}\right )} e^{\left (-2\right )}}{x}\right )}\right )} e^{\left (-3\right )}}{\log \left (x\right )^{3}} + \frac {{\left (x^{4} e^{8} \log \left (x\right )^{4} - 4 \, x^{3} e^{\left (\frac {{\left (2 \, x e^{2} - 162 \, x + 81 \, e^{25}\right )} e^{\left (-2\right )}}{x} + 6\right )} \log \left (x\right )^{3} + e^{\left (\frac {4 \, {\left (2 \, x e^{2} - 162 \, x + 81 \, e^{25}\right )} e^{\left (-2\right )}}{x}\right )}\right )} e^{\left (-8\right )}}{\log \left (x\right )^{4}} \]
integrate((4*x^5*exp(2)*log(x)^5+(324*x^3*exp(25)-12*x^4*exp(2))*exp((81*e xp(25)-162*x)/exp(2)/x)*log(x)^4+((-972*x^2*exp(25)+12*x^3*exp(2))*exp((81 *exp(25)-162*x)/exp(2)/x)^2+4*x^4*exp(2)*exp((81*exp(25)-162*x)/exp(2)/x)) *log(x)^3+((972*x*exp(25)-4*x^2*exp(2))*exp((81*exp(25)-162*x)/exp(2)/x)^3 -12*x^3*exp(2)*exp((81*exp(25)-162*x)/exp(2)/x)^2)*log(x)^2+(-324*exp(25)* exp((81*exp(25)-162*x)/exp(2)/x)^4+12*x^2*exp(2)*exp((81*exp(25)-162*x)/ex p(2)/x)^3)*log(x)-4*x*exp(2)*exp((81*exp(25)-162*x)/exp(2)/x)^4)/x^2/exp(2 )/log(x)^5,x, algorithm=\
2*(3*x^2*e^(2*(x*e^2 - 162*x + 81*e^25)*e^(-2)/x + 1)*log(x) - 2*x*e^(3*(x *e^2 - 162*x + 81*e^25)*e^(-2)/x))*e^(-3)/log(x)^3 + (x^4*e^8*log(x)^4 - 4 *x^3*e^((2*x*e^2 - 162*x + 81*e^25)*e^(-2)/x + 6)*log(x)^3 + e^(4*(2*x*e^2 - 162*x + 81*e^25)*e^(-2)/x))*e^(-8)/log(x)^4
Time = 13.96 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.19 \[ \int \frac {-4 e^{2+\frac {4 \left (81 e^{25}-162 x\right )}{e^2 x}} x+\left (-324 e^{25+\frac {4 \left (81 e^{25}-162 x\right )}{e^2 x}}+12 e^{2+\frac {3 \left (81 e^{25}-162 x\right )}{e^2 x}} x^2\right ) \log (x)+\left (-12 e^{2+\frac {2 \left (81 e^{25}-162 x\right )}{e^2 x}} x^3+e^{\frac {3 \left (81 e^{25}-162 x\right )}{e^2 x}} \left (972 e^{25} x-4 e^2 x^2\right )\right ) \log ^2(x)+\left (4 e^{2+\frac {81 e^{25}-162 x}{e^2 x}} x^4+e^{\frac {2 \left (81 e^{25}-162 x\right )}{e^2 x}} \left (-972 e^{25} x^2+12 e^2 x^3\right )\right ) \log ^3(x)+e^{\frac {81 e^{25}-162 x}{e^2 x}} \left (324 e^{25} x^3-12 e^2 x^4\right ) \log ^4(x)+4 e^2 x^5 \log ^5(x)}{e^2 x^2 \log ^5(x)} \, dx=\frac {{\mathrm {e}}^{\frac {324\,{\mathrm {e}}^{23}}{x}-648\,{\mathrm {e}}^{-2}}}{{\ln \left (x\right )}^4}+x^4-\frac {4\,x\,{\mathrm {e}}^{\frac {243\,{\mathrm {e}}^{23}}{x}-486\,{\mathrm {e}}^{-2}}}{{\ln \left (x\right )}^3}-\frac {4\,x^3\,{\mathrm {e}}^{\frac {81\,{\mathrm {e}}^{23}}{x}-162\,{\mathrm {e}}^{-2}}}{\ln \left (x\right )}+\frac {6\,x^2\,{\mathrm {e}}^{\frac {162\,{\mathrm {e}}^{23}}{x}-324\,{\mathrm {e}}^{-2}}}{{\ln \left (x\right )}^2} \]
int((exp(-2)*(log(x)^2*(exp(-(3*exp(-2)*(162*x - 81*exp(25)))/x)*(972*x*ex p(25) - 4*x^2*exp(2)) - 12*x^3*exp(2)*exp(-(2*exp(-2)*(162*x - 81*exp(25)) )/x)) + log(x)^3*(exp(-(2*exp(-2)*(162*x - 81*exp(25)))/x)*(12*x^3*exp(2) - 972*x^2*exp(25)) + 4*x^4*exp(2)*exp(-(exp(-2)*(162*x - 81*exp(25)))/x)) - log(x)*(324*exp(25)*exp(-(4*exp(-2)*(162*x - 81*exp(25)))/x) - 12*x^2*ex p(2)*exp(-(3*exp(-2)*(162*x - 81*exp(25)))/x)) - exp(-(exp(-2)*(162*x - 81 *exp(25)))/x)*log(x)^4*(12*x^4*exp(2) - 324*x^3*exp(25)) - 4*x*exp(2)*exp( -(4*exp(-2)*(162*x - 81*exp(25)))/x) + 4*x^5*exp(2)*log(x)^5))/(x^2*log(x) ^5),x)