Integrand size = 142, antiderivative size = 26 \[ \int \frac {50 e^{2 x}+100 e^x x+50 x^2+\left (-100 e^{2 x} x-100 x^2+e^x \left (-100 x-100 x^2\right )\right ) \log (x)+\left (2 x-10 x^2+12 x^3+e^{2 x} \left (-2-4 x+8 x^2\right )+e^x \left (-12 x+8 x^2+8 x^3\right )\right ) \log ^2(x)+\left (-4 e^{2 x} x-4 x^2+e^x \left (-4 x-4 x^2\right )\right ) \log ^3(x)}{x \log ^2(x)} \, dx=2 \left (x-\left (e^x+x\right )^2 \left (2-2 x+\frac {25}{\log (x)}+\log (x)\right )\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(57\) vs. \(2(26)=52\).
Time = 0.48 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.19 \[ \int \frac {50 e^{2 x}+100 e^x x+50 x^2+\left (-100 e^{2 x} x-100 x^2+e^x \left (-100 x-100 x^2\right )\right ) \log (x)+\left (2 x-10 x^2+12 x^3+e^{2 x} \left (-2-4 x+8 x^2\right )+e^x \left (-12 x+8 x^2+8 x^3\right )\right ) \log ^2(x)+\left (-4 e^{2 x} x-4 x^2+e^x \left (-4 x-4 x^2\right )\right ) \log ^3(x)}{x \log ^2(x)} \, dx=2 \left (2 e^{2 x} (-1+x)+x+4 e^x (-1+x) x-2 x^2+2 x^3-\frac {25 \left (e^x+x\right )^2}{\log (x)}-\left (e^x+x\right )^2 \log (x)\right ) \]
Integrate[(50*E^(2*x) + 100*E^x*x + 50*x^2 + (-100*E^(2*x)*x - 100*x^2 + E ^x*(-100*x - 100*x^2))*Log[x] + (2*x - 10*x^2 + 12*x^3 + E^(2*x)*(-2 - 4*x + 8*x^2) + E^x*(-12*x + 8*x^2 + 8*x^3))*Log[x]^2 + (-4*E^(2*x)*x - 4*x^2 + E^x*(-4*x - 4*x^2))*Log[x]^3)/(x*Log[x]^2),x]
2*(2*E^(2*x)*(-1 + x) + x + 4*E^x*(-1 + x)*x - 2*x^2 + 2*x^3 - (25*(E^x + x)^2)/Log[x] - (E^x + x)^2*Log[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.
\(\displaystyle \int \frac {50 x^2+\left (-4 x^2+e^x \left (-4 x^2-4 x\right )-4 e^{2 x} x\right ) \log ^3(x)+\left (-100 x^2+e^x \left (-100 x^2-100 x\right )-100 e^{2 x} x\right ) \log (x)+\left (12 x^3-10 x^2+e^{2 x} \left (8 x^2-4 x-2\right )+e^x \left (8 x^3+8 x^2-12 x\right )+2 x\right ) \log ^2(x)+50 e^{2 x}+100 e^x x}{x \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 e^{2 x} \left (4 x^2 \log ^2(x)-2 x \log ^3(x)-2 x \log ^2(x)-\log ^2(x)-50 x \log (x)+25\right )}{x \log ^2(x)}+\frac {2 \left (6 x^2 \log ^2(x)+25 x-2 x \log ^3(x)-5 x \log ^2(x)+\log ^2(x)-50 x \log (x)\right )}{\log ^2(x)}-\frac {4 e^x \left (-2 x^2 \log ^2(x)+x \log ^3(x)+\log ^3(x)-2 x \log ^2(x)+3 \log ^2(x)+25 x \log (x)+25 \log (x)-25\right )}{\log ^2(x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 100 \int \frac {e^x}{\log ^2(x)}dx+50 \int \frac {e^{2 x}}{x \log ^2(x)}dx-100 \int \frac {e^x}{\log (x)}dx-100 \int \frac {e^{2 x}}{\log (x)}dx-100 \int \frac {e^x x}{\log (x)}dx+4 x^3+8 e^x x^2-4 x^2-2 x^2 \log (x)-\frac {50 x^2}{\log (x)}-8 e^x x+4 e^{2 x} x+2 x-4 e^{2 x}+4 e^x \log (x)-2 e^{2 x} \log (x)-4 e^x (x+1) \log (x)\) |
Int[(50*E^(2*x) + 100*E^x*x + 50*x^2 + (-100*E^(2*x)*x - 100*x^2 + E^x*(-1 00*x - 100*x^2))*Log[x] + (2*x - 10*x^2 + 12*x^3 + E^(2*x)*(-2 - 4*x + 8*x ^2) + E^x*(-12*x + 8*x^2 + 8*x^3))*Log[x]^2 + (-4*E^(2*x)*x - 4*x^2 + E^x* (-4*x - 4*x^2))*Log[x]^3)/(x*Log[x]^2),x]
3.18.89.3.1 Defintions of rubi rules used
Leaf count of result is larger than twice the leaf count of optimal. \(78\) vs. \(2(25)=50\).
Time = 0.08 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.04
method | result | size |
risch | \(\left (-2 x^{2}-4 \,{\mathrm e}^{x} x -2 \,{\mathrm e}^{2 x}\right ) \ln \left (x \right )+4 x^{3}+8 \,{\mathrm e}^{x} x^{2}+4 x \,{\mathrm e}^{2 x}-4 x^{2}-8 \,{\mathrm e}^{x} x -4 \,{\mathrm e}^{2 x}+2 x -\frac {50 \left ({\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} x +x^{2}\right )}{\ln \left (x \right )}\) | \(79\) |
parallelrisch | \(-\frac {-4 x^{3} \ln \left (x \right )+2 x^{2} \ln \left (x \right )^{2}-8 x^{2} {\mathrm e}^{x} \ln \left (x \right )+4 x \,{\mathrm e}^{x} \ln \left (x \right )^{2}-4 \,{\mathrm e}^{2 x} \ln \left (x \right ) x +2 \,{\mathrm e}^{2 x} \ln \left (x \right )^{2}+4 x^{2} \ln \left (x \right )+8 x \,{\mathrm e}^{x} \ln \left (x \right )+4 \,{\mathrm e}^{2 x} \ln \left (x \right )+50 x^{2}-2 x \ln \left (x \right )+100 \,{\mathrm e}^{x} x +50 \,{\mathrm e}^{2 x}}{\ln \left (x \right )}\) | \(104\) |
int(((-4*x*exp(x)^2+(-4*x^2-4*x)*exp(x)-4*x^2)*ln(x)^3+((8*x^2-4*x-2)*exp( x)^2+(8*x^3+8*x^2-12*x)*exp(x)+12*x^3-10*x^2+2*x)*ln(x)^2+(-100*x*exp(x)^2 +(-100*x^2-100*x)*exp(x)-100*x^2)*ln(x)+50*exp(x)^2+100*exp(x)*x+50*x^2)/x /ln(x)^2,x,method=_RETURNVERBOSE)
(-2*x^2-4*exp(x)*x-2*exp(x)^2)*ln(x)+4*x^3+8*exp(x)*x^2+4*x*exp(x)^2-4*x^2 -8*exp(x)*x-4*exp(x)^2+2*x-50*(exp(x)^2+2*exp(x)*x+x^2)/ln(x)
Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (27) = 54\).
Time = 0.25 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.96 \[ \int \frac {50 e^{2 x}+100 e^x x+50 x^2+\left (-100 e^{2 x} x-100 x^2+e^x \left (-100 x-100 x^2\right )\right ) \log (x)+\left (2 x-10 x^2+12 x^3+e^{2 x} \left (-2-4 x+8 x^2\right )+e^x \left (-12 x+8 x^2+8 x^3\right )\right ) \log ^2(x)+\left (-4 e^{2 x} x-4 x^2+e^x \left (-4 x-4 x^2\right )\right ) \log ^3(x)}{x \log ^2(x)} \, dx=-\frac {2 \, {\left ({\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )} \log \left (x\right )^{2} + 25 \, x^{2} + 50 \, x e^{x} - {\left (2 \, x^{3} - 2 \, x^{2} + 2 \, {\left (x - 1\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{2} - x\right )} e^{x} + x\right )} \log \left (x\right ) + 25 \, e^{\left (2 \, x\right )}\right )}}{\log \left (x\right )} \]
integrate(((-4*x*exp(x)^2+(-4*x^2-4*x)*exp(x)-4*x^2)*log(x)^3+((8*x^2-4*x- 2)*exp(x)^2+(8*x^3+8*x^2-12*x)*exp(x)+12*x^3-10*x^2+2*x)*log(x)^2+(-100*x* exp(x)^2+(-100*x^2-100*x)*exp(x)-100*x^2)*log(x)+50*exp(x)^2+100*exp(x)*x+ 50*x^2)/x/log(x)^2,x, algorithm=\
-2*((x^2 + 2*x*e^x + e^(2*x))*log(x)^2 + 25*x^2 + 50*x*e^x - (2*x^3 - 2*x^ 2 + 2*(x - 1)*e^(2*x) + 4*(x^2 - x)*e^x + x)*log(x) + 25*e^(2*x))/log(x)
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (24) = 48\).
Time = 0.17 (sec) , antiderivative size = 104, normalized size of antiderivative = 4.00 \[ \int \frac {50 e^{2 x}+100 e^x x+50 x^2+\left (-100 e^{2 x} x-100 x^2+e^x \left (-100 x-100 x^2\right )\right ) \log (x)+\left (2 x-10 x^2+12 x^3+e^{2 x} \left (-2-4 x+8 x^2\right )+e^x \left (-12 x+8 x^2+8 x^3\right )\right ) \log ^2(x)+\left (-4 e^{2 x} x-4 x^2+e^x \left (-4 x-4 x^2\right )\right ) \log ^3(x)}{x \log ^2(x)} \, dx=4 x^{3} - 2 x^{2} \log {\left (x \right )} - 4 x^{2} - \frac {50 x^{2}}{\log {\left (x \right )}} + 2 x + \frac {\left (4 x \log {\left (x \right )}^{2} - 2 \log {\left (x \right )}^{3} - 4 \log {\left (x \right )}^{2} - 50 \log {\left (x \right )}\right ) e^{2 x} + \left (8 x^{2} \log {\left (x \right )}^{2} - 4 x \log {\left (x \right )}^{3} - 8 x \log {\left (x \right )}^{2} - 100 x \log {\left (x \right )}\right ) e^{x}}{\log {\left (x \right )}^{2}} \]
integrate(((-4*x*exp(x)**2+(-4*x**2-4*x)*exp(x)-4*x**2)*ln(x)**3+((8*x**2- 4*x-2)*exp(x)**2+(8*x**3+8*x**2-12*x)*exp(x)+12*x**3-10*x**2+2*x)*ln(x)**2 +(-100*x*exp(x)**2+(-100*x**2-100*x)*exp(x)-100*x**2)*ln(x)+50*exp(x)**2+1 00*exp(x)*x+50*x**2)/x/ln(x)**2,x)
4*x**3 - 2*x**2*log(x) - 4*x**2 - 50*x**2/log(x) + 2*x + ((4*x*log(x)**2 - 2*log(x)**3 - 4*log(x)**2 - 50*log(x))*exp(2*x) + (8*x**2*log(x)**2 - 4*x *log(x)**3 - 8*x*log(x)**2 - 100*x*log(x))*exp(x))/log(x)**2
\[ \int \frac {50 e^{2 x}+100 e^x x+50 x^2+\left (-100 e^{2 x} x-100 x^2+e^x \left (-100 x-100 x^2\right )\right ) \log (x)+\left (2 x-10 x^2+12 x^3+e^{2 x} \left (-2-4 x+8 x^2\right )+e^x \left (-12 x+8 x^2+8 x^3\right )\right ) \log ^2(x)+\left (-4 e^{2 x} x-4 x^2+e^x \left (-4 x-4 x^2\right )\right ) \log ^3(x)}{x \log ^2(x)} \, dx=\int { -\frac {2 \, {\left (2 \, {\left (x^{2} + x e^{\left (2 \, x\right )} + {\left (x^{2} + x\right )} e^{x}\right )} \log \left (x\right )^{3} - {\left (6 \, x^{3} - 5 \, x^{2} + {\left (4 \, x^{2} - 2 \, x - 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (2 \, x^{3} + 2 \, x^{2} - 3 \, x\right )} e^{x} + x\right )} \log \left (x\right )^{2} - 25 \, x^{2} - 50 \, x e^{x} + 50 \, {\left (x^{2} + x e^{\left (2 \, x\right )} + {\left (x^{2} + x\right )} e^{x}\right )} \log \left (x\right ) - 25 \, e^{\left (2 \, x\right )}\right )}}{x \log \left (x\right )^{2}} \,d x } \]
integrate(((-4*x*exp(x)^2+(-4*x^2-4*x)*exp(x)-4*x^2)*log(x)^3+((8*x^2-4*x- 2)*exp(x)^2+(8*x^3+8*x^2-12*x)*exp(x)+12*x^3-10*x^2+2*x)*log(x)^2+(-100*x* exp(x)^2+(-100*x^2-100*x)*exp(x)-100*x^2)*log(x)+50*exp(x)^2+100*exp(x)*x+ 50*x^2)/x/log(x)^2,x, algorithm=\
4*x^3 - 5*x^2 + 2*(2*x - 1)*e^(2*x) + 8*(x^2 - 2*x + 2)*e^x + 8*(x - 1)*e^ x + 2*x - (2*x^2*log(x)^2 - x^2*log(x) + 2*e^(2*x)*log(x)^2 + 4*(x*log(x)^ 2 + 25*x - log(x))*e^x)/log(x) - 2*Ei(2*x) - 2*e^(2*x) - 12*e^x + 100*gamm a(-1, -2*log(x)) - 100*integrate(x/log(x), x) + 2*integrate(-(50*x*log(x) - log(x)^2 - 25)*e^(2*x)/(x*log(x)^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (27) = 54\).
Time = 0.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.92 \[ \int \frac {50 e^{2 x}+100 e^x x+50 x^2+\left (-100 e^{2 x} x-100 x^2+e^x \left (-100 x-100 x^2\right )\right ) \log (x)+\left (2 x-10 x^2+12 x^3+e^{2 x} \left (-2-4 x+8 x^2\right )+e^x \left (-12 x+8 x^2+8 x^3\right )\right ) \log ^2(x)+\left (-4 e^{2 x} x-4 x^2+e^x \left (-4 x-4 x^2\right )\right ) \log ^3(x)}{x \log ^2(x)} \, dx=\frac {2 \, {\left (2 \, x^{3} \log \left (x\right ) + 4 \, x^{2} e^{x} \log \left (x\right ) - x^{2} \log \left (x\right )^{2} - 2 \, x e^{x} \log \left (x\right )^{2} - 2 \, x^{2} \log \left (x\right ) + 2 \, x e^{\left (2 \, x\right )} \log \left (x\right ) - 4 \, x e^{x} \log \left (x\right ) - e^{\left (2 \, x\right )} \log \left (x\right )^{2} - 25 \, x^{2} - 50 \, x e^{x} + x \log \left (x\right ) - 2 \, e^{\left (2 \, x\right )} \log \left (x\right ) - 25 \, e^{\left (2 \, x\right )}\right )}}{\log \left (x\right )} \]
integrate(((-4*x*exp(x)^2+(-4*x^2-4*x)*exp(x)-4*x^2)*log(x)^3+((8*x^2-4*x- 2)*exp(x)^2+(8*x^3+8*x^2-12*x)*exp(x)+12*x^3-10*x^2+2*x)*log(x)^2+(-100*x* exp(x)^2+(-100*x^2-100*x)*exp(x)-100*x^2)*log(x)+50*exp(x)^2+100*exp(x)*x+ 50*x^2)/x/log(x)^2,x, algorithm=\
2*(2*x^3*log(x) + 4*x^2*e^x*log(x) - x^2*log(x)^2 - 2*x*e^x*log(x)^2 - 2*x ^2*log(x) + 2*x*e^(2*x)*log(x) - 4*x*e^x*log(x) - e^(2*x)*log(x)^2 - 25*x^ 2 - 50*x*e^x + x*log(x) - 2*e^(2*x)*log(x) - 25*e^(2*x))/log(x)
Time = 11.01 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.35 \[ \int \frac {50 e^{2 x}+100 e^x x+50 x^2+\left (-100 e^{2 x} x-100 x^2+e^x \left (-100 x-100 x^2\right )\right ) \log (x)+\left (2 x-10 x^2+12 x^3+e^{2 x} \left (-2-4 x+8 x^2\right )+e^x \left (-12 x+8 x^2+8 x^3\right )\right ) \log ^2(x)+\left (-4 e^{2 x} x-4 x^2+e^x \left (-4 x-4 x^2\right )\right ) \log ^3(x)}{x \log ^2(x)} \, dx=2\,x-\frac {50\,{\left (x+{\mathrm {e}}^x\right )}^2-100\,x\,\ln \left (x\right )\,\left ({\mathrm {e}}^x+1\right )\,\left (x+{\mathrm {e}}^x\right )}{\ln \left (x\right )}-\ln \left (x\right )\,\left (2\,{\mathrm {e}}^{2\,x}+4\,x\,{\mathrm {e}}^x+2\,x^2\right )-{\mathrm {e}}^x\,\left (92\,x^2+108\,x\right )-{\mathrm {e}}^{2\,x}\,\left (96\,x+4\right )-104\,x^2+4\,x^3 \]
int((50*exp(2*x) + log(x)^2*(2*x - exp(2*x)*(4*x - 8*x^2 + 2) - 10*x^2 + 1 2*x^3 + exp(x)*(8*x^2 - 12*x + 8*x^3)) - log(x)*(100*x*exp(2*x) + exp(x)*( 100*x + 100*x^2) + 100*x^2) + 100*x*exp(x) + 50*x^2 - log(x)^3*(4*x*exp(2* x) + exp(x)*(4*x + 4*x^2) + 4*x^2))/(x*log(x)^2),x)