Integrand size = 196, antiderivative size = 32 \[ \int \frac {\left (50 e^{5+2 x}+450 x^2-300 x^3+50 x^4+e^5 \left (450 x^2-300 x^3+50 x^4\right )+e^x \left (-300 x+300 x^2-50 x^3+e^5 \left (-300 x+100 x^2\right )\right )\right ) \log \left (\frac {2 e^{5+x} x-6 x^2+2 x^3+e^5 \left (-6 x^2+2 x^3\right )}{e^x-3 x+x^2}\right )}{e^{5+2 x} x+9 x^3-6 x^4+x^5+e^5 \left (9 x^3-6 x^4+x^5\right )+e^x \left (-3 x^2+x^3+e^5 \left (-6 x^2+2 x^3\right )\right )} \, dx=e+25 \log ^2\left (2 x \left (e^5+\frac {x}{-\frac {e^x}{3-x}+x}\right )\right ) \]
Time = 0.12 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19 \[ \int \frac {\left (50 e^{5+2 x}+450 x^2-300 x^3+50 x^4+e^5 \left (450 x^2-300 x^3+50 x^4\right )+e^x \left (-300 x+300 x^2-50 x^3+e^5 \left (-300 x+100 x^2\right )\right )\right ) \log \left (\frac {2 e^{5+x} x-6 x^2+2 x^3+e^5 \left (-6 x^2+2 x^3\right )}{e^x-3 x+x^2}\right )}{e^{5+2 x} x+9 x^3-6 x^4+x^5+e^5 \left (9 x^3-6 x^4+x^5\right )+e^x \left (-3 x^2+x^3+e^5 \left (-6 x^2+2 x^3\right )\right )} \, dx=25 \log ^2\left (\frac {2 x \left (e^{5+x}+(-3+x) x+e^5 (-3+x) x\right )}{e^x+(-3+x) x}\right ) \]
Integrate[((50*E^(5 + 2*x) + 450*x^2 - 300*x^3 + 50*x^4 + E^5*(450*x^2 - 3 00*x^3 + 50*x^4) + E^x*(-300*x + 300*x^2 - 50*x^3 + E^5*(-300*x + 100*x^2) ))*Log[(2*E^(5 + x)*x - 6*x^2 + 2*x^3 + E^5*(-6*x^2 + 2*x^3))/(E^x - 3*x + x^2)])/(E^(5 + 2*x)*x + 9*x^3 - 6*x^4 + x^5 + E^5*(9*x^3 - 6*x^4 + x^5) + E^x*(-3*x^2 + x^3 + E^5*(-6*x^2 + 2*x^3))),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.
| \(\displaystyle \int \frac {\left (50 x^4-300 x^3+450 x^2+e^x \left (-50 x^3+300 x^2+e^5 \left (100 x^2-300 x\right )-300 x\right )+e^5 \left (50 x^4-300 x^3+450 x^2\right )+50 e^{2 x+5}\right ) \log \left (\frac {2 x^3-6 x^2+e^5 \left (2 x^3-6 x^2\right )+2 e^{x+5} x}{x^2-3 x+e^x}\right )}{x^5-6 x^4+9 x^3+e^x \left (x^3-3 x^2+e^5 \left (2 x^3-6 x^2\right )\right )+e^5 \left (x^5-6 x^4+9 x^3\right )+e^{2 x+5} x} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {50 \left (\left (1+e^5\right ) (x-3)^2 x^2-e^x \left (x^2-6 x+6\right ) x+2 e^{x+5} (x-3) x+e^{2 x+5}\right ) \log \left (\frac {2 x \left (\left (1+e^5\right ) (x-3) x+e^{x+5}\right )}{(x-3) x+e^x}\right )}{x \left ((x-3) x+e^x\right ) \left (\left (1+e^5\right ) (x-3) x+e^{x+5}\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 50 \int \frac {\left (\left (1+e^5\right ) (3-x)^2 x^2-2 e^{x+5} (3-x) x-e^x \left (x^2-6 x+6\right ) x+e^{2 x+5}\right ) \log \left (\frac {2 x \left (e^{x+5}-\left (1+e^5\right ) (3-x) x\right )}{e^x-(3-x) x}\right )}{x \left (e^x-(3-x) x\right ) \left (e^{x+5}-\left (1+e^5\right ) (3-x) x\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 50 \int \left (\frac {\log \left (\frac {2 x \left (\left (1+e^5\right ) (x-3) x+e^{x+5}\right )}{(x-3) x+e^x}\right )}{x}+\frac {\left (x^2-5 x+3\right ) \log \left (\frac {2 x \left (\left (1+e^5\right ) (x-3) x+e^{x+5}\right )}{(x-3) x+e^x}\right )}{x^2-3 x+e^x}+\frac {\left (1+e^5\right ) \left (-x^2+5 x-3\right ) \log \left (\frac {2 x \left (\left (1+e^5\right ) (x-3) x+e^{x+5}\right )}{(x-3) x+e^x}\right )}{\left (1+e^5\right ) x^2-3 \left (1+e^5\right ) x+e^{x+5}}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle 50 \int \left (\frac {\log \left (\frac {2 x \left (\left (1+e^5\right ) (x-3) x+e^{x+5}\right )}{(x-3) x+e^x}\right )}{x}+\frac {\left (x^2-5 x+3\right ) \log \left (\frac {2 x \left (\left (1+e^5\right ) (x-3) x+e^{x+5}\right )}{(x-3) x+e^x}\right )}{x^2-3 x+e^x}+\frac {\left (1+e^5\right ) \left (-x^2+5 x-3\right ) \log \left (\frac {2 x \left (\left (1+e^5\right ) (x-3) x+e^{x+5}\right )}{(x-3) x+e^x}\right )}{\left (1+e^5\right ) x^2-3 \left (1+e^5\right ) x+e^{x+5}}\right )dx\) |
Int[((50*E^(5 + 2*x) + 450*x^2 - 300*x^3 + 50*x^4 + E^5*(450*x^2 - 300*x^3 + 50*x^4) + E^x*(-300*x + 300*x^2 - 50*x^3 + E^5*(-300*x + 100*x^2)))*Log [(2*E^(5 + x)*x - 6*x^2 + 2*x^3 + E^5*(-6*x^2 + 2*x^3))/(E^x - 3*x + x^2)] )/(E^(5 + 2*x)*x + 9*x^3 - 6*x^4 + x^5 + E^5*(9*x^3 - 6*x^4 + x^5) + E^x*( -3*x^2 + x^3 + E^5*(-6*x^2 + 2*x^3))),x]
3.6.58.3.1 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]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.72 (sec) , antiderivative size = 1829, normalized size of antiderivative = 57.16
int((50*exp(5)*exp(x)^2+((100*x^2-300*x)*exp(5)-50*x^3+300*x^2-300*x)*exp( x)+(50*x^4-300*x^3+450*x^2)*exp(5)+50*x^4-300*x^3+450*x^2)*ln((2*x*exp(5)* exp(x)+(2*x^3-6*x^2)*exp(5)+2*x^3-6*x^2)/(exp(x)+x^2-3*x))/(x*exp(5)*exp(x )^2+((2*x^3-6*x^2)*exp(5)+x^3-3*x^2)*exp(x)+(x^5-6*x^4+9*x^3)*exp(5)+x^5-6 *x^4+9*x^3),x,method=_RETURNVERBOSE)
-25*I*Pi*ln(x)*csgn(I*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x))*csgn(I/(exp(x)+x^ 2-3*x))*csgn(I*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x)/(exp(x)+x^2-3*x))-25*I*Pi *ln(x)*csgn(I*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x)/(exp(x)+x^2-3*x))*csgn(I*x /(exp(x)+x^2-3*x)*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x))*csgn(I*x)+25*I*Pi*ln( x)*csgn(I*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x))*csgn(I*((exp(x)+x^2-3*x)*exp( 5)+x^2-3*x)/(exp(x)+x^2-3*x))^2+25*I*Pi*ln(x)*csgn(I/(exp(x)+x^2-3*x))*csg n(I*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x)/(exp(x)+x^2-3*x))^2+25*I*Pi*ln(x)*cs gn(I*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x)/(exp(x)+x^2-3*x))*csgn(I*x/(exp(x)+ x^2-3*x)*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x))^2+25*I*Pi*ln(x)*csgn(I*x/(exp( x)+x^2-3*x)*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x))^2*csgn(I*x)-25*I*Pi*ln(exp( x)+x^2-3*x)*csgn(I*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x))*csgn(I*((exp(x)+x^2- 3*x)*exp(5)+x^2-3*x)/(exp(x)+x^2-3*x))^2-25*I*Pi*ln(exp(x)+x^2-3*x)*csgn(I /(exp(x)+x^2-3*x))*csgn(I*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x)/(exp(x)+x^2-3* x))^2-25*I*Pi*ln(exp(x)+x^2-3*x)*csgn(I*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x)/ (exp(x)+x^2-3*x))*csgn(I*x/(exp(x)+x^2-3*x)*((exp(x)+x^2-3*x)*exp(5)+x^2-3 *x))^2-25*I*Pi*ln(exp(x)+x^2-3*x)*csgn(I*x/(exp(x)+x^2-3*x)*((exp(x)+x^2-3 *x)*exp(5)+x^2-3*x))^2*csgn(I*x)+50*ln(2)*ln(x)+25*ln(x)^2+(50*ln(x)-50*ln (exp(x)+x^2-3*x))*ln((exp(x)+x^2-3*x)*exp(5)+x^2-3*x)-50*ln(2)*ln(exp(x)+x ^2-3*x)-50*ln(x)*ln(exp(x)+x^2-3*x)+25*I*Pi*ln(x^2-3*x+x^2*exp(-5)-3*exp(- 5)*x+exp(x))*csgn(I*((exp(x)+x^2-3*x)*exp(5)+x^2-3*x))*csgn(I*((exp(x)+...
Time = 0.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.72 \[ \int \frac {\left (50 e^{5+2 x}+450 x^2-300 x^3+50 x^4+e^5 \left (450 x^2-300 x^3+50 x^4\right )+e^x \left (-300 x+300 x^2-50 x^3+e^5 \left (-300 x+100 x^2\right )\right )\right ) \log \left (\frac {2 e^{5+x} x-6 x^2+2 x^3+e^5 \left (-6 x^2+2 x^3\right )}{e^x-3 x+x^2}\right )}{e^{5+2 x} x+9 x^3-6 x^4+x^5+e^5 \left (9 x^3-6 x^4+x^5\right )+e^x \left (-3 x^2+x^3+e^5 \left (-6 x^2+2 x^3\right )\right )} \, dx=25 \, \log \left (\frac {2 \, {\left ({\left (x^{3} - 3 \, x^{2}\right )} e^{10} + {\left (x^{3} - 3 \, x^{2}\right )} e^{5} + x e^{\left (x + 10\right )}\right )}}{{\left (x^{2} - 3 \, x\right )} e^{5} + e^{\left (x + 5\right )}}\right )^{2} \]
integrate((50*exp(5)*exp(x)^2+((100*x^2-300*x)*exp(5)-50*x^3+300*x^2-300*x )*exp(x)+(50*x^4-300*x^3+450*x^2)*exp(5)+50*x^4-300*x^3+450*x^2)*log((2*x* exp(5)*exp(x)+(2*x^3-6*x^2)*exp(5)+2*x^3-6*x^2)/(exp(x)+x^2-3*x))/(x*exp(5 )*exp(x)^2+((2*x^3-6*x^2)*exp(5)+x^3-3*x^2)*exp(x)+(x^5-6*x^4+9*x^3)*exp(5 )+x^5-6*x^4+9*x^3),x, algorithm=\
25*log(2*((x^3 - 3*x^2)*e^10 + (x^3 - 3*x^2)*e^5 + x*e^(x + 10))/((x^2 - 3 *x)*e^5 + e^(x + 5)))^2
Time = 0.50 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50 \[ \int \frac {\left (50 e^{5+2 x}+450 x^2-300 x^3+50 x^4+e^5 \left (450 x^2-300 x^3+50 x^4\right )+e^x \left (-300 x+300 x^2-50 x^3+e^5 \left (-300 x+100 x^2\right )\right )\right ) \log \left (\frac {2 e^{5+x} x-6 x^2+2 x^3+e^5 \left (-6 x^2+2 x^3\right )}{e^x-3 x+x^2}\right )}{e^{5+2 x} x+9 x^3-6 x^4+x^5+e^5 \left (9 x^3-6 x^4+x^5\right )+e^x \left (-3 x^2+x^3+e^5 \left (-6 x^2+2 x^3\right )\right )} \, dx=25 \log {\left (\frac {2 x^{3} - 6 x^{2} + 2 x e^{5} e^{x} + \left (2 x^{3} - 6 x^{2}\right ) e^{5}}{x^{2} - 3 x + e^{x}} \right )}^{2} \]
integrate((50*exp(5)*exp(x)**2+((100*x**2-300*x)*exp(5)-50*x**3+300*x**2-3 00*x)*exp(x)+(50*x**4-300*x**3+450*x**2)*exp(5)+50*x**4-300*x**3+450*x**2) *ln((2*x*exp(5)*exp(x)+(2*x**3-6*x**2)*exp(5)+2*x**3-6*x**2)/(exp(x)+x**2- 3*x))/(x*exp(5)*exp(x)**2+((2*x**3-6*x**2)*exp(5)+x**3-3*x**2)*exp(x)+(x** 5-6*x**4+9*x**3)*exp(5)+x**5-6*x**4+9*x**3),x)
25*log((2*x**3 - 6*x**2 + 2*x*exp(5)*exp(x) + (2*x**3 - 6*x**2)*exp(5))/(x **2 - 3*x + exp(x)))**2
Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (28) = 56\).
Time = 0.27 (sec) , antiderivative size = 189, normalized size of antiderivative = 5.91 \[ \int \frac {\left (50 e^{5+2 x}+450 x^2-300 x^3+50 x^4+e^5 \left (450 x^2-300 x^3+50 x^4\right )+e^x \left (-300 x+300 x^2-50 x^3+e^5 \left (-300 x+100 x^2\right )\right )\right ) \log \left (\frac {2 e^{5+x} x-6 x^2+2 x^3+e^5 \left (-6 x^2+2 x^3\right )}{e^x-3 x+x^2}\right )}{e^{5+2 x} x+9 x^3-6 x^4+x^5+e^5 \left (9 x^3-6 x^4+x^5\right )+e^x \left (-3 x^2+x^3+e^5 \left (-6 x^2+2 x^3\right )\right )} \, dx=50 \, {\left (\log \left (x^{2} - 3 \, x + e^{x}\right ) - \log \left (x\right ) + 5\right )} \log \left (x^{2} {\left (e^{5} + 1\right )} - 3 \, x {\left (e^{5} + 1\right )} + e^{\left (x + 5\right )}\right ) - 25 \, \log \left (x^{2} {\left (e^{5} + 1\right )} - 3 \, x {\left (e^{5} + 1\right )} + e^{\left (x + 5\right )}\right )^{2} + 50 \, {\left (\log \left (x\right ) - 5\right )} \log \left (x^{2} - 3 \, x + e^{x}\right ) - 25 \, \log \left (x^{2} - 3 \, x + e^{x}\right )^{2} - 25 \, \log \left (x\right )^{2} - 50 \, {\left (\log \left (x^{2} - 3 \, x + e^{x}\right ) - \log \left ({\left (x^{2} {\left (e^{5} + 1\right )} - 3 \, x {\left (e^{5} + 1\right )} + e^{\left (x + 5\right )}\right )} e^{\left (-5\right )}\right ) - \log \left (x\right )\right )} \log \left (\frac {2 \, {\left (x^{3} - 3 \, x^{2} + {\left (x^{3} - 3 \, x^{2}\right )} e^{5} + x e^{\left (x + 5\right )}\right )}}{x^{2} - 3 \, x + e^{x}}\right ) + 250 \, \log \left (x\right ) \]
integrate((50*exp(5)*exp(x)^2+((100*x^2-300*x)*exp(5)-50*x^3+300*x^2-300*x )*exp(x)+(50*x^4-300*x^3+450*x^2)*exp(5)+50*x^4-300*x^3+450*x^2)*log((2*x* exp(5)*exp(x)+(2*x^3-6*x^2)*exp(5)+2*x^3-6*x^2)/(exp(x)+x^2-3*x))/(x*exp(5 )*exp(x)^2+((2*x^3-6*x^2)*exp(5)+x^3-3*x^2)*exp(x)+(x^5-6*x^4+9*x^3)*exp(5 )+x^5-6*x^4+9*x^3),x, algorithm=\
50*(log(x^2 - 3*x + e^x) - log(x) + 5)*log(x^2*(e^5 + 1) - 3*x*(e^5 + 1) + e^(x + 5)) - 25*log(x^2*(e^5 + 1) - 3*x*(e^5 + 1) + e^(x + 5))^2 + 50*(lo g(x) - 5)*log(x^2 - 3*x + e^x) - 25*log(x^2 - 3*x + e^x)^2 - 25*log(x)^2 - 50*(log(x^2 - 3*x + e^x) - log((x^2*(e^5 + 1) - 3*x*(e^5 + 1) + e^(x + 5) )*e^(-5)) - log(x))*log(2*(x^3 - 3*x^2 + (x^3 - 3*x^2)*e^5 + x*e^(x + 5))/ (x^2 - 3*x + e^x)) + 250*log(x)
\[ \int \frac {\left (50 e^{5+2 x}+450 x^2-300 x^3+50 x^4+e^5 \left (450 x^2-300 x^3+50 x^4\right )+e^x \left (-300 x+300 x^2-50 x^3+e^5 \left (-300 x+100 x^2\right )\right )\right ) \log \left (\frac {2 e^{5+x} x-6 x^2+2 x^3+e^5 \left (-6 x^2+2 x^3\right )}{e^x-3 x+x^2}\right )}{e^{5+2 x} x+9 x^3-6 x^4+x^5+e^5 \left (9 x^3-6 x^4+x^5\right )+e^x \left (-3 x^2+x^3+e^5 \left (-6 x^2+2 x^3\right )\right )} \, dx=\int { \frac {50 \, {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2} + {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} e^{5} - {\left (x^{3} - 6 \, x^{2} - 2 \, {\left (x^{2} - 3 \, x\right )} e^{5} + 6 \, x\right )} e^{x} + e^{\left (2 \, x + 5\right )}\right )} \log \left (\frac {2 \, {\left (x^{3} - 3 \, x^{2} + {\left (x^{3} - 3 \, x^{2}\right )} e^{5} + x e^{\left (x + 5\right )}\right )}}{x^{2} - 3 \, x + e^{x}}\right )}{x^{5} - 6 \, x^{4} + 9 \, x^{3} + {\left (x^{5} - 6 \, x^{4} + 9 \, x^{3}\right )} e^{5} + x e^{\left (2 \, x + 5\right )} + {\left (x^{3} - 3 \, x^{2} + 2 \, {\left (x^{3} - 3 \, x^{2}\right )} e^{5}\right )} e^{x}} \,d x } \]
integrate((50*exp(5)*exp(x)^2+((100*x^2-300*x)*exp(5)-50*x^3+300*x^2-300*x )*exp(x)+(50*x^4-300*x^3+450*x^2)*exp(5)+50*x^4-300*x^3+450*x^2)*log((2*x* exp(5)*exp(x)+(2*x^3-6*x^2)*exp(5)+2*x^3-6*x^2)/(exp(x)+x^2-3*x))/(x*exp(5 )*exp(x)^2+((2*x^3-6*x^2)*exp(5)+x^3-3*x^2)*exp(x)+(x^5-6*x^4+9*x^3)*exp(5 )+x^5-6*x^4+9*x^3),x, algorithm=\
integrate(50*(x^4 - 6*x^3 + 9*x^2 + (x^4 - 6*x^3 + 9*x^2)*e^5 - (x^3 - 6*x ^2 - 2*(x^2 - 3*x)*e^5 + 6*x)*e^x + e^(2*x + 5))*log(2*(x^3 - 3*x^2 + (x^3 - 3*x^2)*e^5 + x*e^(x + 5))/(x^2 - 3*x + e^x))/(x^5 - 6*x^4 + 9*x^3 + (x^ 5 - 6*x^4 + 9*x^3)*e^5 + x*e^(2*x + 5) + (x^3 - 3*x^2 + 2*(x^3 - 3*x^2)*e^ 5)*e^x), x)
Time = 13.05 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.56 \[ \int \frac {\left (50 e^{5+2 x}+450 x^2-300 x^3+50 x^4+e^5 \left (450 x^2-300 x^3+50 x^4\right )+e^x \left (-300 x+300 x^2-50 x^3+e^5 \left (-300 x+100 x^2\right )\right )\right ) \log \left (\frac {2 e^{5+x} x-6 x^2+2 x^3+e^5 \left (-6 x^2+2 x^3\right )}{e^x-3 x+x^2}\right )}{e^{5+2 x} x+9 x^3-6 x^4+x^5+e^5 \left (9 x^3-6 x^4+x^5\right )+e^x \left (-3 x^2+x^3+e^5 \left (-6 x^2+2 x^3\right )\right )} \, dx=25\,{\ln \left (-\frac {{\mathrm {e}}^5\,\left (6\,x^2-2\,x^3\right )+6\,x^2-2\,x^3-2\,x\,{\mathrm {e}}^5\,{\mathrm {e}}^x}{{\mathrm {e}}^x-3\,x+x^2}\right )}^2 \]
int((log(-(exp(5)*(6*x^2 - 2*x^3) + 6*x^2 - 2*x^3 - 2*x*exp(5)*exp(x))/(ex p(x) - 3*x + x^2))*(50*exp(2*x)*exp(5) - exp(x)*(300*x + exp(5)*(300*x - 1 00*x^2) - 300*x^2 + 50*x^3) + exp(5)*(450*x^2 - 300*x^3 + 50*x^4) + 450*x^ 2 - 300*x^3 + 50*x^4))/(exp(5)*(9*x^3 - 6*x^4 + x^5) - exp(x)*(exp(5)*(6*x ^2 - 2*x^3) + 3*x^2 - x^3) + 9*x^3 - 6*x^4 + x^5 + x*exp(2*x)*exp(5)),x)