Integrand size = 106, antiderivative size = 29 \[ \int \frac {2 x-3 x^2-20 x^4+e^{2-x} \left (4+9 x+14 x^3-6 x^4\right )+\left (-5 x^2-6 x^4+e^{2-x} \left (2+4 x-3 x^2+4 x^3-2 x^4\right )\right ) \log (x)+\left (-x^2-e^{2-x} x^2\right ) \log ^2(x)}{x^2} \, dx=\left (e^{2-x}-x\right ) (3+\log (x)) \left (-\frac {2}{x}+2 x^2+\log (x)\right ) \] Output:
(exp(2-x)-x)*(3+ln(x))*(ln(x)+2*x^2-2/x)
Leaf count is larger than twice the leaf count of optimal. \(67\) vs. \(2(29)=58\).
Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.31 \[ \int \frac {2 x-3 x^2-20 x^4+e^{2-x} \left (4+9 x+14 x^3-6 x^4\right )+\left (-5 x^2-6 x^4+e^{2-x} \left (2+4 x-3 x^2+4 x^3-2 x^4\right )\right ) \log (x)+\left (-x^2-e^{2-x} x^2\right ) \log ^2(x)}{x^2} \, dx=\frac {e^{-x} \left (-6 e^x x^4+6 e^2 \left (-1+x^3\right )+\left (e^2-e^x x\right ) \left (-2+3 x+2 x^3\right ) \log (x)+x \left (e^2-e^x x\right ) \log ^2(x)\right )}{x} \] Input:
Integrate[(2*x - 3*x^2 - 20*x^4 + E^(2 - x)*(4 + 9*x + 14*x^3 - 6*x^4) + ( -5*x^2 - 6*x^4 + E^(2 - x)*(2 + 4*x - 3*x^2 + 4*x^3 - 2*x^4))*Log[x] + (-x ^2 - E^(2 - x)*x^2)*Log[x]^2)/x^2,x]
Output:
(-6*E^x*x^4 + 6*E^2*(-1 + x^3) + (E^2 - E^x*x)*(-2 + 3*x + 2*x^3)*Log[x] + x*(E^2 - E^x*x)*Log[x]^2)/(E^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.
\(\displaystyle \int \frac {-20 x^4-3 x^2+\left (-e^{2-x} x^2-x^2\right ) \log ^2(x)+e^{2-x} \left (-6 x^4+14 x^3+9 x+4\right )+\left (-6 x^4-5 x^2+e^{2-x} \left (-2 x^4+4 x^3-3 x^2+4 x+2\right )\right ) \log (x)+2 x}{x^2} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (\frac {-20 x^3-6 x^3 \log (x)-3 x-x \log ^2(x)-5 x \log (x)+2}{x}-\frac {e^{2-x} \left (6 x^4+2 x^4 \log (x)-14 x^3-4 x^3 \log (x)+x^2 \log ^2(x)+3 x^2 \log (x)-9 x-4 x \log (x)-2 \log (x)-4\right )}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int e^{2-x} \log ^2(x)dx+2 e^2 x \, _3F_3(1,1,1;2,2,2;-x)-2 e^2 \log (x) (\operatorname {ExpIntegralE}(1,x)+\operatorname {ExpIntegralEi}(-x))+2 e^2 \operatorname {ExpIntegralEi}(-x) \log (x)-6 x^3-2 x^3 \log (x)+6 e^{2-x} x^2+2 e^{2-x} x^2 \log (x)-\frac {6 e^{2-x}}{x}-x \log ^2(x)-e^2 \log ^2(x)-3 x \log (x)+3 e^{2-x} \log (x)-2 e^2 \gamma \log (x)+2 \log (x)-\frac {2 e^{2-x} \log (x)}{x}\) |
Input:
Int[(2*x - 3*x^2 - 20*x^4 + E^(2 - x)*(4 + 9*x + 14*x^3 - 6*x^4) + (-5*x^2 - 6*x^4 + E^(2 - x)*(2 + 4*x - 3*x^2 + 4*x^3 - 2*x^4))*Log[x] + (-x^2 - E ^(2 - x)*x^2)*Log[x]^2)/x^2,x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(96\) vs. \(2(28)=56\).
Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.34
\[\left ({\mathrm e}^{2-x}-x \right ) \ln \left (x \right )^{2}-\frac {\left (2 x^{4}-2 \,{\mathrm e}^{2-x} x^{3}+3 x^{2}-3 x \,{\mathrm e}^{2-x}+2 \,{\mathrm e}^{2-x}\right ) \ln \left (x \right )}{x}+\frac {-6 x^{4}+6 \,{\mathrm e}^{2-x} x^{3}+2 x \ln \left (x \right )-6 \,{\mathrm e}^{2-x}}{x}\]
Input:
int(((-x^2*exp(2-x)-x^2)*ln(x)^2+((-2*x^4+4*x^3-3*x^2+4*x+2)*exp(2-x)-6*x^ 4-5*x^2)*ln(x)+(-6*x^4+14*x^3+9*x+4)*exp(2-x)-20*x^4-3*x^2+2*x)/x^2,x)
Output:
(exp(2-x)-x)*ln(x)^2-(2*x^4-2*exp(2-x)*x^3+3*x^2-3*x*exp(2-x)+2*exp(2-x))/ x*ln(x)+2*(-3*x^4+3*exp(2-x)*x^3+x*ln(x)-3*exp(2-x))/x
Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (29) = 58\).
Time = 0.09 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.66 \[ \int \frac {2 x-3 x^2-20 x^4+e^{2-x} \left (4+9 x+14 x^3-6 x^4\right )+\left (-5 x^2-6 x^4+e^{2-x} \left (2+4 x-3 x^2+4 x^3-2 x^4\right )\right ) \log (x)+\left (-x^2-e^{2-x} x^2\right ) \log ^2(x)}{x^2} \, dx=-\frac {6 \, x^{4} + {\left (x^{2} - x e^{\left (-x + 2\right )}\right )} \log \left (x\right )^{2} - 6 \, {\left (x^{3} - 1\right )} e^{\left (-x + 2\right )} + {\left (2 \, x^{4} + 3 \, x^{2} - {\left (2 \, x^{3} + 3 \, x - 2\right )} e^{\left (-x + 2\right )} - 2 \, x\right )} \log \left (x\right )}{x} \] Input:
integrate(((-x^2*exp(2-x)-x^2)*log(x)^2+((-2*x^4+4*x^3-3*x^2+4*x+2)*exp(2- x)-6*x^4-5*x^2)*log(x)+(-6*x^4+14*x^3+9*x+4)*exp(2-x)-20*x^4-3*x^2+2*x)/x^ 2,x, algorithm="fricas")
Output:
-(6*x^4 + (x^2 - x*e^(-x + 2))*log(x)^2 - 6*(x^3 - 1)*e^(-x + 2) + (2*x^4 + 3*x^2 - (2*x^3 + 3*x - 2)*e^(-x + 2) - 2*x)*log(x))/x
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (22) = 44\).
Time = 0.17 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.41 \[ \int \frac {2 x-3 x^2-20 x^4+e^{2-x} \left (4+9 x+14 x^3-6 x^4\right )+\left (-5 x^2-6 x^4+e^{2-x} \left (2+4 x-3 x^2+4 x^3-2 x^4\right )\right ) \log (x)+\left (-x^2-e^{2-x} x^2\right ) \log ^2(x)}{x^2} \, dx=- 6 x^{3} - x \log {\left (x \right )}^{2} + \left (- 2 x^{3} - 3 x\right ) \log {\left (x \right )} + 2 \log {\left (x \right )} + \frac {\left (2 x^{3} \log {\left (x \right )} + 6 x^{3} + x \log {\left (x \right )}^{2} + 3 x \log {\left (x \right )} - 2 \log {\left (x \right )} - 6\right ) e^{2 - x}}{x} \] Input:
integrate(((-x**2*exp(2-x)-x**2)*ln(x)**2+((-2*x**4+4*x**3-3*x**2+4*x+2)*e xp(2-x)-6*x**4-5*x**2)*ln(x)+(-6*x**4+14*x**3+9*x+4)*exp(2-x)-20*x**4-3*x* *2+2*x)/x**2,x)
Output:
-6*x**3 - x*log(x)**2 + (-2*x**3 - 3*x)*log(x) + 2*log(x) + (2*x**3*log(x) + 6*x**3 + x*log(x)**2 + 3*x*log(x) - 2*log(x) - 6)*exp(2 - x)/x
\[ \int \frac {2 x-3 x^2-20 x^4+e^{2-x} \left (4+9 x+14 x^3-6 x^4\right )+\left (-5 x^2-6 x^4+e^{2-x} \left (2+4 x-3 x^2+4 x^3-2 x^4\right )\right ) \log (x)+\left (-x^2-e^{2-x} x^2\right ) \log ^2(x)}{x^2} \, dx=\int { -\frac {20 \, x^{4} + {\left (x^{2} e^{\left (-x + 2\right )} + x^{2}\right )} \log \left (x\right )^{2} + 3 \, x^{2} + {\left (6 \, x^{4} - 14 \, x^{3} - 9 \, x - 4\right )} e^{\left (-x + 2\right )} + {\left (6 \, x^{4} + 5 \, x^{2} + {\left (2 \, x^{4} - 4 \, x^{3} + 3 \, x^{2} - 4 \, x - 2\right )} e^{\left (-x + 2\right )}\right )} \log \left (x\right ) - 2 \, x}{x^{2}} \,d x } \] Input:
integrate(((-x^2*exp(2-x)-x^2)*log(x)^2+((-2*x^4+4*x^3-3*x^2+4*x+2)*exp(2- x)-6*x^4-5*x^2)*log(x)+(-6*x^4+14*x^3+9*x+4)*exp(2-x)-20*x^4-3*x^2+2*x)/x^ 2,x, algorithm="maxima")
Output:
-2*x^3*log(x) - 6*x^3 + 6*Ei(-x)*e^2 + 6*(x^2*e^2 + 2*x*e^2 + 2*e^2)*e^(-x ) - 14*(x*e^2 + e^2)*e^(-x) - 4*e^2*gamma(-1, x) - 5*x*log(x) + 3*e^(-x + 2)*log(x) + 2*x - (x^2*log(x)^2 - 2*x^2*log(x) + 2*x^2 - (x*e^2*log(x)^2 + 2*(x^3*e^2 - e^2)*log(x))*e^(-x))/x - integrate(2*(x^3*e^2 - e^2)*e^(-x)/ x^2, x) + 2*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (29) = 58\).
Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.59 \[ \int \frac {2 x-3 x^2-20 x^4+e^{2-x} \left (4+9 x+14 x^3-6 x^4\right )+\left (-5 x^2-6 x^4+e^{2-x} \left (2+4 x-3 x^2+4 x^3-2 x^4\right )\right ) \log (x)+\left (-x^2-e^{2-x} x^2\right ) \log ^2(x)}{x^2} \, dx=-\frac {2 \, x^{4} \log \left (x\right ) - 2 \, x^{3} e^{\left (-x + 2\right )} \log \left (x\right ) + 6 \, x^{4} - 6 \, x^{3} e^{\left (-x + 2\right )} + x^{2} \log \left (x\right )^{2} - x e^{\left (-x + 2\right )} \log \left (x\right )^{2} + 3 \, x^{2} \log \left (x\right ) - 3 \, x e^{\left (-x + 2\right )} \log \left (x\right ) - 2 \, x \log \left (x\right ) + 2 \, e^{\left (-x + 2\right )} \log \left (x\right ) + 6 \, e^{\left (-x + 2\right )}}{x} \] Input:
integrate(((-x^2*exp(2-x)-x^2)*log(x)^2+((-2*x^4+4*x^3-3*x^2+4*x+2)*exp(2- x)-6*x^4-5*x^2)*log(x)+(-6*x^4+14*x^3+9*x+4)*exp(2-x)-20*x^4-3*x^2+2*x)/x^ 2,x, algorithm="giac")
Output:
-(2*x^4*log(x) - 2*x^3*e^(-x + 2)*log(x) + 6*x^4 - 6*x^3*e^(-x + 2) + x^2* log(x)^2 - x*e^(-x + 2)*log(x)^2 + 3*x^2*log(x) - 3*x*e^(-x + 2)*log(x) - 2*x*log(x) + 2*e^(-x + 2)*log(x) + 6*e^(-x + 2))/x
Time = 2.98 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.86 \[ \int \frac {2 x-3 x^2-20 x^4+e^{2-x} \left (4+9 x+14 x^3-6 x^4\right )+\left (-5 x^2-6 x^4+e^{2-x} \left (2+4 x-3 x^2+4 x^3-2 x^4\right )\right ) \log (x)+\left (-x^2-e^{2-x} x^2\right ) \log ^2(x)}{x^2} \, dx=2\,\ln \left (x\right )-{\ln \left (x\right )}^2\,\left (x-{\mathrm {e}}^{2-x}\right )-6\,x^3-\ln \left (x\right )\,\left (3\,x-{\mathrm {e}}^{2-x}\,\left (\frac {2\,x^3+3\,x}{x}-\frac {2}{x}\right )+2\,x^3\right )+\frac {{\mathrm {e}}^{2-x}\,\left (6\,x^3-6\right )}{x} \] Input:
int(-(log(x)^2*(x^2*exp(2 - x) + x^2) - exp(2 - x)*(9*x + 14*x^3 - 6*x^4 + 4) - 2*x + 3*x^2 + 20*x^4 + log(x)*(5*x^2 - exp(2 - x)*(4*x - 3*x^2 + 4*x ^3 - 2*x^4 + 2) + 6*x^4))/x^2,x)
Output:
2*log(x) - log(x)^2*(x - exp(2 - x)) - 6*x^3 - log(x)*(3*x - exp(2 - x)*(( 3*x + 2*x^3)/x - 2/x) + 2*x^3) + (exp(2 - x)*(6*x^3 - 6))/x
Time = 0.19 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.62 \[ \int \frac {2 x-3 x^2-20 x^4+e^{2-x} \left (4+9 x+14 x^3-6 x^4\right )+\left (-5 x^2-6 x^4+e^{2-x} \left (2+4 x-3 x^2+4 x^3-2 x^4\right )\right ) \log (x)+\left (-x^2-e^{2-x} x^2\right ) \log ^2(x)}{x^2} \, dx=\frac {-e^{x} \mathrm {log}\left (x \right )^{2} x^{2}-2 e^{x} \mathrm {log}\left (x \right ) x^{4}-3 e^{x} \mathrm {log}\left (x \right ) x^{2}+2 e^{x} \mathrm {log}\left (x \right ) x -6 e^{x} x^{4}+\mathrm {log}\left (x \right )^{2} e^{2} x +2 \,\mathrm {log}\left (x \right ) e^{2} x^{3}+3 \,\mathrm {log}\left (x \right ) e^{2} x -2 \,\mathrm {log}\left (x \right ) e^{2}+6 e^{2} x^{3}-6 e^{2}}{e^{x} x} \] Input:
int(((-x^2*exp(2-x)-x^2)*log(x)^2+((-2*x^4+4*x^3-3*x^2+4*x+2)*exp(2-x)-6*x ^4-5*x^2)*log(x)+(-6*x^4+14*x^3+9*x+4)*exp(2-x)-20*x^4-3*x^2+2*x)/x^2,x)
Output:
( - e**x*log(x)**2*x**2 - 2*e**x*log(x)*x**4 - 3*e**x*log(x)*x**2 + 2*e**x *log(x)*x - 6*e**x*x**4 + log(x)**2*e**2*x + 2*log(x)*e**2*x**3 + 3*log(x) *e**2*x - 2*log(x)*e**2 + 6*e**2*x**3 - 6*e**2)/(e**x*x)