Integrand size = 180, antiderivative size = 21 \[ \int \frac {\left (12 x+18 x^2+e^x \left (12+12 x+6 x^2\right )+\left (6 x+6 e^x x\right ) \log (x)\right ) \log \left (25 e^x x^2+25 x^3+\left (50 e^x x+50 x^2\right ) \log (x)+\left (25 e^x+25 x\right ) \log ^2(x)\right )+\left (3 e^x x+3 x^2+\left (3 e^x+3 x\right ) \log (x)\right ) \log ^2\left (25 e^x x^2+25 x^3+\left (50 e^x x+50 x^2\right ) \log (x)+\left (25 e^x+25 x\right ) \log ^2(x)\right )}{2 e^x x+2 x^2+\left (2 e^x+2 x\right ) \log (x)} \, dx=\frac {3}{2} x \log ^2\left (25 \left (e^x+x\right ) (x+\log (x))^2\right ) \]
\[ \int \frac {\left (12 x+18 x^2+e^x \left (12+12 x+6 x^2\right )+\left (6 x+6 e^x x\right ) \log (x)\right ) \log \left (25 e^x x^2+25 x^3+\left (50 e^x x+50 x^2\right ) \log (x)+\left (25 e^x+25 x\right ) \log ^2(x)\right )+\left (3 e^x x+3 x^2+\left (3 e^x+3 x\right ) \log (x)\right ) \log ^2\left (25 e^x x^2+25 x^3+\left (50 e^x x+50 x^2\right ) \log (x)+\left (25 e^x+25 x\right ) \log ^2(x)\right )}{2 e^x x+2 x^2+\left (2 e^x+2 x\right ) \log (x)} \, dx=\int \frac {\left (12 x+18 x^2+e^x \left (12+12 x+6 x^2\right )+\left (6 x+6 e^x x\right ) \log (x)\right ) \log \left (25 e^x x^2+25 x^3+\left (50 e^x x+50 x^2\right ) \log (x)+\left (25 e^x+25 x\right ) \log ^2(x)\right )+\left (3 e^x x+3 x^2+\left (3 e^x+3 x\right ) \log (x)\right ) \log ^2\left (25 e^x x^2+25 x^3+\left (50 e^x x+50 x^2\right ) \log (x)+\left (25 e^x+25 x\right ) \log ^2(x)\right )}{2 e^x x+2 x^2+\left (2 e^x+2 x\right ) \log (x)} \, dx \]
Integrate[((12*x + 18*x^2 + E^x*(12 + 12*x + 6*x^2) + (6*x + 6*E^x*x)*Log[ x])*Log[25*E^x*x^2 + 25*x^3 + (50*E^x*x + 50*x^2)*Log[x] + (25*E^x + 25*x) *Log[x]^2] + (3*E^x*x + 3*x^2 + (3*E^x + 3*x)*Log[x])*Log[25*E^x*x^2 + 25* x^3 + (50*E^x*x + 50*x^2)*Log[x] + (25*E^x + 25*x)*Log[x]^2]^2)/(2*E^x*x + 2*x^2 + (2*E^x + 2*x)*Log[x]),x]
Integrate[((12*x + 18*x^2 + E^x*(12 + 12*x + 6*x^2) + (6*x + 6*E^x*x)*Log[ x])*Log[25*E^x*x^2 + 25*x^3 + (50*E^x*x + 50*x^2)*Log[x] + (25*E^x + 25*x) *Log[x]^2] + (3*E^x*x + 3*x^2 + (3*E^x + 3*x)*Log[x])*Log[25*E^x*x^2 + 25* x^3 + (50*E^x*x + 50*x^2)*Log[x] + (25*E^x + 25*x)*Log[x]^2]^2)/(2*E^x*x + 2*x^2 + (2*E^x + 2*x)*Log[x]), 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.
| \(\displaystyle \int \frac {\left (3 x^2+3 e^x x+\left (3 x+3 e^x\right ) \log (x)\right ) \log ^2\left (25 x^3+25 e^x x^2+\left (50 x^2+50 e^x x\right ) \log (x)+\left (25 x+25 e^x\right ) \log ^2(x)\right )+\left (18 x^2+e^x \left (6 x^2+12 x+12\right )+12 x+\left (6 e^x x+6 x\right ) \log (x)\right ) \log \left (25 x^3+25 e^x x^2+\left (50 x^2+50 e^x x\right ) \log (x)+\left (25 x+25 e^x\right ) \log ^2(x)\right )}{2 x^2+2 e^x x+\left (2 x+2 e^x\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {3 \log \left (25 \left (x+e^x\right ) (x+\log (x))^2\right ) \left (2 e^x x^2+6 x^2+x^2 \log \left (25 \left (x+e^x\right ) (x+\log (x))^2\right )+4 e^x x+4 x+4 e^x+2 e^x x \log (x)+2 x \log (x)+e^x x \log \left (25 \left (x+e^x\right ) (x+\log (x))^2\right )+x \log (x) \log \left (25 \left (x+e^x\right ) (x+\log (x))^2\right )+e^x \log (x) \log \left (25 \left (x+e^x\right ) (x+\log (x))^2\right )\right )}{2 \left (x+e^x\right ) (x+\log (x))}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{2} \int \frac {\log \left (25 \left (x+e^x\right ) (x+\log (x))^2\right ) \left (2 e^x x^2+\log \left (25 \left (x+e^x\right ) (x+\log (x))^2\right ) x^2+6 x^2+4 e^x x+2 e^x \log (x) x+2 \log (x) x+e^x \log \left (25 \left (x+e^x\right ) (x+\log (x))^2\right ) x+\log (x) \log \left (25 \left (x+e^x\right ) (x+\log (x))^2\right ) x+4 x+4 e^x+e^x \log (x) \log \left (25 \left (x+e^x\right ) (x+\log (x))^2\right )\right )}{\left (x+e^x\right ) (x+\log (x))}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{2} \int \left (\frac {\log \left (25 \left (x+e^x\right ) (x+\log (x))^2\right ) \left (2 x^2+2 \log (x) x+\log \left (25 \left (x+e^x\right ) (x+\log (x))^2\right ) x+4 x+\log (x) \log \left (25 \left (x+e^x\right ) (x+\log (x))^2\right )+4\right )}{x+\log (x)}-\frac {2 (x-1) x \log \left (25 \left (x+e^x\right ) (x+\log (x))^2\right )}{x+e^x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{2} \left (2 \int \frac {\int \frac {x^2}{x+e^x}dx}{x+e^x}dx-2 \int \frac {x \int \frac {x^2}{x+e^x}dx}{x+e^x}dx-2 \log \left (25 \left (x+e^x\right ) (x+\log (x))^2\right ) \int \frac {x^2}{x+e^x}dx+2 \int \frac {x^2 \log \left (25 \left (x+e^x\right ) (x+\log (x))^2\right )}{x+\log (x)}dx+4 \int \frac {\int \frac {x^2}{x+e^x}dx}{x+\log (x)}dx+4 \int \frac {\int \frac {x^2}{x+e^x}dx}{x (x+\log (x))}dx+2 \int \frac {x \int \frac {x^2}{x+e^x}dx}{x+\log (x)}dx+2 \int \frac {\log (x) \int \frac {x^2}{x+e^x}dx}{x+\log (x)}dx-2 \int \frac {\int \frac {x}{x+e^x}dx}{x+e^x}dx+2 \int \frac {x \int \frac {x}{x+e^x}dx}{x+e^x}dx+\int \log ^2\left (25 \left (x+e^x\right ) (x+\log (x))^2\right )dx+2 \log \left (25 \left (x+e^x\right ) (x+\log (x))^2\right ) \int \frac {x}{x+e^x}dx+4 \int \frac {\log \left (25 \left (x+e^x\right ) (x+\log (x))^2\right )}{x+\log (x)}dx+4 \int \frac {x \log \left (25 \left (x+e^x\right ) (x+\log (x))^2\right )}{x+\log (x)}dx+2 \int \frac {x \log (x) \log \left (25 \left (x+e^x\right ) (x+\log (x))^2\right )}{x+\log (x)}dx-4 \int \frac {\int \frac {x}{x+e^x}dx}{x+\log (x)}dx-4 \int \frac {\int \frac {x}{x+e^x}dx}{x (x+\log (x))}dx-2 \int \frac {x \int \frac {x}{x+e^x}dx}{x+\log (x)}dx-2 \int \frac {\log (x) \int \frac {x}{x+e^x}dx}{x+\log (x)}dx\right )\) |
Int[((12*x + 18*x^2 + E^x*(12 + 12*x + 6*x^2) + (6*x + 6*E^x*x)*Log[x])*Lo g[25*E^x*x^2 + 25*x^3 + (50*E^x*x + 50*x^2)*Log[x] + (25*E^x + 25*x)*Log[x ]^2] + (3*E^x*x + 3*x^2 + (3*E^x + 3*x)*Log[x])*Log[25*E^x*x^2 + 25*x^3 + (50*E^x*x + 50*x^2)*Log[x] + (25*E^x + 25*x)*Log[x]^2]^2)/(2*E^x*x + 2*x^2 + (2*E^x + 2*x)*Log[x]),x]
3.6.96.3.1 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]]
Time = 7.96 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.24
| method | result | size |
| parallelrisch | \(\frac {3 x {\ln \left (\left (25 \,{\mathrm e}^{x}+25 x \right ) \ln \left (x \right )^{2}+\left (50 \,{\mathrm e}^{x} x +50 x^{2}\right ) \ln \left (x \right )+25 \,{\mathrm e}^{x} x^{2}+25 x^{3}\right )}^{2}}{2}\) | \(47\) |
| risch | \(\text {Expression too large to display}\) | \(1627\) |
int((((3*exp(x)+3*x)*ln(x)+3*exp(x)*x+3*x^2)*ln((25*exp(x)+25*x)*ln(x)^2+( 50*exp(x)*x+50*x^2)*ln(x)+25*exp(x)*x^2+25*x^3)^2+((6*exp(x)*x+6*x)*ln(x)+ (6*x^2+12*x+12)*exp(x)+18*x^2+12*x)*ln((25*exp(x)+25*x)*ln(x)^2+(50*exp(x) *x+50*x^2)*ln(x)+25*exp(x)*x^2+25*x^3))/((2*exp(x)+2*x)*ln(x)+2*exp(x)*x+2 *x^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (18) = 36\).
Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.95 \[ \int \frac {\left (12 x+18 x^2+e^x \left (12+12 x+6 x^2\right )+\left (6 x+6 e^x x\right ) \log (x)\right ) \log \left (25 e^x x^2+25 x^3+\left (50 e^x x+50 x^2\right ) \log (x)+\left (25 e^x+25 x\right ) \log ^2(x)\right )+\left (3 e^x x+3 x^2+\left (3 e^x+3 x\right ) \log (x)\right ) \log ^2\left (25 e^x x^2+25 x^3+\left (50 e^x x+50 x^2\right ) \log (x)+\left (25 e^x+25 x\right ) \log ^2(x)\right )}{2 e^x x+2 x^2+\left (2 e^x+2 x\right ) \log (x)} \, dx=\frac {3}{2} \, x \log \left (25 \, x^{3} + 25 \, x^{2} e^{x} + 25 \, {\left (x + e^{x}\right )} \log \left (x\right )^{2} + 50 \, {\left (x^{2} + x e^{x}\right )} \log \left (x\right )\right )^{2} \]
integrate((((3*exp(x)+3*x)*log(x)+3*exp(x)*x+3*x^2)*log((25*exp(x)+25*x)*l og(x)^2+(50*exp(x)*x+50*x^2)*log(x)+25*exp(x)*x^2+25*x^3)^2+((6*exp(x)*x+6 *x)*log(x)+(6*x^2+12*x+12)*exp(x)+18*x^2+12*x)*log((25*exp(x)+25*x)*log(x) ^2+(50*exp(x)*x+50*x^2)*log(x)+25*exp(x)*x^2+25*x^3))/((2*exp(x)+2*x)*log( x)+2*exp(x)*x+2*x^2),x, algorithm=\
Exception generated. \[ \int \frac {\left (12 x+18 x^2+e^x \left (12+12 x+6 x^2\right )+\left (6 x+6 e^x x\right ) \log (x)\right ) \log \left (25 e^x x^2+25 x^3+\left (50 e^x x+50 x^2\right ) \log (x)+\left (25 e^x+25 x\right ) \log ^2(x)\right )+\left (3 e^x x+3 x^2+\left (3 e^x+3 x\right ) \log (x)\right ) \log ^2\left (25 e^x x^2+25 x^3+\left (50 e^x x+50 x^2\right ) \log (x)+\left (25 e^x+25 x\right ) \log ^2(x)\right )}{2 e^x x+2 x^2+\left (2 e^x+2 x\right ) \log (x)} \, dx=\text {Exception raised: TypeError} \]
integrate((((3*exp(x)+3*x)*ln(x)+3*exp(x)*x+3*x**2)*ln((25*exp(x)+25*x)*ln (x)**2+(50*exp(x)*x+50*x**2)*ln(x)+25*exp(x)*x**2+25*x**3)**2+((6*exp(x)*x +6*x)*ln(x)+(6*x**2+12*x+12)*exp(x)+18*x**2+12*x)*ln((25*exp(x)+25*x)*ln(x )**2+(50*exp(x)*x+50*x**2)*ln(x)+25*exp(x)*x**2+25*x**3))/((2*exp(x)+2*x)* ln(x)+2*exp(x)*x+2*x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (18) = 36\).
Time = 0.35 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.71 \[ \int \frac {\left (12 x+18 x^2+e^x \left (12+12 x+6 x^2\right )+\left (6 x+6 e^x x\right ) \log (x)\right ) \log \left (25 e^x x^2+25 x^3+\left (50 e^x x+50 x^2\right ) \log (x)+\left (25 e^x+25 x\right ) \log ^2(x)\right )+\left (3 e^x x+3 x^2+\left (3 e^x+3 x\right ) \log (x)\right ) \log ^2\left (25 e^x x^2+25 x^3+\left (50 e^x x+50 x^2\right ) \log (x)+\left (25 e^x+25 x\right ) \log ^2(x)\right )}{2 e^x x+2 x^2+\left (2 e^x+2 x\right ) \log (x)} \, dx=6 \, x \log \left (5\right )^{2} + \frac {3}{2} \, x \log \left (x + e^{x}\right )^{2} + 12 \, x \log \left (5\right ) \log \left (x + \log \left (x\right )\right ) + 6 \, x \log \left (x + \log \left (x\right )\right )^{2} + 6 \, {\left (x \log \left (5\right ) + x \log \left (x + \log \left (x\right )\right )\right )} \log \left (x + e^{x}\right ) \]
integrate((((3*exp(x)+3*x)*log(x)+3*exp(x)*x+3*x^2)*log((25*exp(x)+25*x)*l og(x)^2+(50*exp(x)*x+50*x^2)*log(x)+25*exp(x)*x^2+25*x^3)^2+((6*exp(x)*x+6 *x)*log(x)+(6*x^2+12*x+12)*exp(x)+18*x^2+12*x)*log((25*exp(x)+25*x)*log(x) ^2+(50*exp(x)*x+50*x^2)*log(x)+25*exp(x)*x^2+25*x^3))/((2*exp(x)+2*x)*log( x)+2*exp(x)*x+2*x^2),x, algorithm=\
6*x*log(5)^2 + 3/2*x*log(x + e^x)^2 + 12*x*log(5)*log(x + log(x)) + 6*x*lo g(x + log(x))^2 + 6*(x*log(5) + x*log(x + log(x)))*log(x + e^x)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (18) = 36\).
Time = 1.49 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.29 \[ \int \frac {\left (12 x+18 x^2+e^x \left (12+12 x+6 x^2\right )+\left (6 x+6 e^x x\right ) \log (x)\right ) \log \left (25 e^x x^2+25 x^3+\left (50 e^x x+50 x^2\right ) \log (x)+\left (25 e^x+25 x\right ) \log ^2(x)\right )+\left (3 e^x x+3 x^2+\left (3 e^x+3 x\right ) \log (x)\right ) \log ^2\left (25 e^x x^2+25 x^3+\left (50 e^x x+50 x^2\right ) \log (x)+\left (25 e^x+25 x\right ) \log ^2(x)\right )}{2 e^x x+2 x^2+\left (2 e^x+2 x\right ) \log (x)} \, dx=\frac {3}{2} \, x \log \left (25 \, x^{3} + 25 \, x^{2} e^{x} + 50 \, x^{2} \log \left (x\right ) + 50 \, x e^{x} \log \left (x\right ) + 25 \, x \log \left (x\right )^{2} + 25 \, e^{x} \log \left (x\right )^{2}\right )^{2} \]
integrate((((3*exp(x)+3*x)*log(x)+3*exp(x)*x+3*x^2)*log((25*exp(x)+25*x)*l og(x)^2+(50*exp(x)*x+50*x^2)*log(x)+25*exp(x)*x^2+25*x^3)^2+((6*exp(x)*x+6 *x)*log(x)+(6*x^2+12*x+12)*exp(x)+18*x^2+12*x)*log((25*exp(x)+25*x)*log(x) ^2+(50*exp(x)*x+50*x^2)*log(x)+25*exp(x)*x^2+25*x^3))/((2*exp(x)+2*x)*log( x)+2*exp(x)*x+2*x^2),x, algorithm=\
3/2*x*log(25*x^3 + 25*x^2*e^x + 50*x^2*log(x) + 50*x*e^x*log(x) + 25*x*log (x)^2 + 25*e^x*log(x)^2)^2
Time = 9.45 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.19 \[ \int \frac {\left (12 x+18 x^2+e^x \left (12+12 x+6 x^2\right )+\left (6 x+6 e^x x\right ) \log (x)\right ) \log \left (25 e^x x^2+25 x^3+\left (50 e^x x+50 x^2\right ) \log (x)+\left (25 e^x+25 x\right ) \log ^2(x)\right )+\left (3 e^x x+3 x^2+\left (3 e^x+3 x\right ) \log (x)\right ) \log ^2\left (25 e^x x^2+25 x^3+\left (50 e^x x+50 x^2\right ) \log (x)+\left (25 e^x+25 x\right ) \log ^2(x)\right )}{2 e^x x+2 x^2+\left (2 e^x+2 x\right ) \log (x)} \, dx=\frac {3\,x\,{\ln \left (25\,x^2\,{\mathrm {e}}^x+{\ln \left (x\right )}^2\,\left (25\,x+25\,{\mathrm {e}}^x\right )+\ln \left (x\right )\,\left (50\,x\,{\mathrm {e}}^x+50\,x^2\right )+25\,x^3\right )}^2}{2} \]
int((log(25*x^2*exp(x) + log(x)^2*(25*x + 25*exp(x)) + log(x)*(50*x*exp(x) + 50*x^2) + 25*x^3)*(12*x + log(x)*(6*x + 6*x*exp(x)) + exp(x)*(12*x + 6* x^2 + 12) + 18*x^2) + log(25*x^2*exp(x) + log(x)^2*(25*x + 25*exp(x)) + lo g(x)*(50*x*exp(x) + 50*x^2) + 25*x^3)^2*(3*x*exp(x) + log(x)*(3*x + 3*exp( x)) + 3*x^2))/(2*x*exp(x) + log(x)*(2*x + 2*exp(x)) + 2*x^2),x)