Integrand size = 50, antiderivative size = 23 \[ \int \frac {\left (-72+120 x-56 x^2+8 x^3\right ) \log \left (\frac {e^x}{x}\right )+\left (36-48 x+12 x^2\right ) \log ^2\left (\frac {e^x}{x}\right )}{e^2} \, dx=\frac {4 (3-x)^2 x \log ^2\left (\frac {e^x}{x}\right )}{e^2} \] Output:
4*x/exp(1)^2*ln(exp(x)/x)^2*(3-x)^2
Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\left (-72+120 x-56 x^2+8 x^3\right ) \log \left (\frac {e^x}{x}\right )+\left (36-48 x+12 x^2\right ) \log ^2\left (\frac {e^x}{x}\right )}{e^2} \, dx=\frac {4 (-3+x)^2 x \log ^2\left (\frac {e^x}{x}\right )}{e^2} \] Input:
Integrate[((-72 + 120*x - 56*x^2 + 8*x^3)*Log[E^x/x] + (36 - 48*x + 12*x^2 )*Log[E^x/x]^2)/E^2,x]
Output:
(4*(-3 + x)^2*x*Log[E^x/x]^2)/E^2
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.
| \(\displaystyle \int \frac {\left (12 x^2-48 x+36\right ) \log ^2\left (\frac {e^x}{x}\right )+\left (8 x^3-56 x^2+120 x-72\right ) \log \left (\frac {e^x}{x}\right )}{e^2} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \left (12 \left (x^2-4 x+3\right ) \log ^2\left (\frac {e^x}{x}\right )-8 \left (-x^3+7 x^2-15 x+9\right ) \log \left (\frac {e^x}{x}\right )\right )dx}{e^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {12 \int x^2 \log ^2\left (\frac {e^x}{x}\right )dx+36 \int \log ^2\left (\frac {e^x}{x}\right )dx-48 \int x \log ^2\left (\frac {e^x}{x}\right )dx-\frac {2 x^5}{5}+\frac {31 x^4}{6}+2 x^4 \log \left (\frac {e^x}{x}\right )-\frac {236 x^3}{9}-\frac {56}{3} x^3 \log \left (\frac {e^x}{x}\right )+66 x^2+60 x^2 \log \left (\frac {e^x}{x}\right )-72 x-72 x \log \left (\frac {e^x}{x}\right )}{e^2}\) |
Input:
Int[((-72 + 120*x - 56*x^2 + 8*x^3)*Log[E^x/x] + (36 - 48*x + 12*x^2)*Log[ E^x/x]^2)/E^2,x]
Output:
$Aborted
Time = 0.20 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.04
| method | result | size |
| parallelrisch | \({\mathrm e}^{-2} \left (4 x^{3} \ln \left (\frac {{\mathrm e}^{x}}{x}\right )^{2}-24 \ln \left (\frac {{\mathrm e}^{x}}{x}\right )^{2} x^{2}+36 \ln \left (\frac {{\mathrm e}^{x}}{x}\right )^{2} x \right )\) | \(47\) |
| parts | \(-\frac {31 \,{\mathrm e}^{-2} x^{4}}{6}+72 \,{\mathrm e}^{-2} x -2 \,{\mathrm e}^{-2} \ln \left (\frac {{\mathrm e}^{x}}{x}\right ) x^{4}+\frac {56 \,{\mathrm e}^{-2} \ln \left (\frac {{\mathrm e}^{x}}{x}\right ) x^{3}}{3}-60 \,{\mathrm e}^{-2} \ln \left (\frac {{\mathrm e}^{x}}{x}\right ) x^{2}+72 \,{\mathrm e}^{-2} \ln \left (\frac {{\mathrm e}^{x}}{x}\right ) x +\frac {2 \,{\mathrm e}^{-2} x^{5}}{5}+\frac {236 \,{\mathrm e}^{-2} x^{3}}{9}-66 \,{\mathrm e}^{-2} x^{2}+4 \,{\mathrm e}^{-2} \ln \left (\frac {{\mathrm e}^{x}}{x}\right )^{2} x^{3}-24 \,{\mathrm e}^{-2} \ln \left (\frac {{\mathrm e}^{x}}{x}\right )^{2} x^{2}+36 \,{\mathrm e}^{-2} \ln \left (\frac {{\mathrm e}^{x}}{x}\right )^{2} x +8 \,{\mathrm e}^{-2} \left (\frac {\ln \left (\frac {{\mathrm e}^{x}}{x}\right ) x^{4}}{4}-\frac {7 \ln \left (\frac {{\mathrm e}^{x}}{x}\right ) x^{3}}{3}+\frac {15 x^{2} \ln \left (\frac {{\mathrm e}^{x}}{x}\right )}{2}-9 x \ln \left (\frac {{\mathrm e}^{x}}{x}\right )-\frac {x^{5}}{20}+\frac {31 x^{4}}{48}-\frac {59 x^{3}}{18}+\frac {33 x^{2}}{4}-9 x \right )\) | \(235\) |
| default | \({\mathrm e}^{-2} \left (-6 x^{4} \ln \left (x \right )+36 x \ln \left (x \right )^{2}-72 x \ln \left (x \right )+60 x^{2} \ln \left (\frac {{\mathrm e}^{x}}{x}\right )+4 x^{3} \ln \left (x \right )^{2}-72 x \ln \left (\frac {{\mathrm e}^{x}}{x}\right )-24 x^{2} \ln \left (x \right )^{2}+\frac {88 x^{3} \ln \left (x \right )}{3}-12 x^{2} \ln \left (x \right )-\frac {16 x^{4}}{3}-24 x^{3}+72 x^{2}+2 x^{5}-\frac {56 \ln \left (\frac {{\mathrm e}^{x}}{x}\right ) x^{3}}{3}+2 \ln \left (\frac {{\mathrm e}^{x}}{x}\right ) x^{4}+4 {\left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )+\ln \left (x \right )-x \right )}^{2} x^{3}+6 \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )+\ln \left (x \right )-x \right ) x^{4}-\frac {88 x^{3} \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )+\ln \left (x \right )-x \right )}{3}+12 x^{2} \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )+\ln \left (x \right )-x \right )+72 x \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )+\ln \left (x \right )-x \right )-24 {\left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )+\ln \left (x \right )-x \right )}^{2} x^{2}+36 {\left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )+\ln \left (x \right )-x \right )}^{2} x -8 \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )+\ln \left (x \right )-x \right ) \ln \left (x \right ) x^{3}+48 \ln \left (x \right ) \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )+\ln \left (x \right )-x \right ) x^{2}-72 \ln \left (x \right ) \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )+\ln \left (x \right )-x \right ) x \right )\) | \(310\) |
| risch | \(\text {Expression too large to display}\) | \(1508\) |
Input:
int(((12*x^2-48*x+36)*ln(exp(x)/x)^2+(8*x^3-56*x^2+120*x-72)*ln(exp(x)/x)) /exp(1)^2,x,method=_RETURNVERBOSE)
Output:
1/exp(1)^2*(4*x^3*ln(exp(x)/x)^2-24*ln(exp(x)/x)^2*x^2+36*ln(exp(x)/x)^2*x )
Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\left (-72+120 x-56 x^2+8 x^3\right ) \log \left (\frac {e^x}{x}\right )+\left (36-48 x+12 x^2\right ) \log ^2\left (\frac {e^x}{x}\right )}{e^2} \, dx=4 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} e^{\left (-2\right )} \log \left (\frac {e^{x}}{x}\right )^{2} \] Input:
integrate(((12*x^2-48*x+36)*log(exp(x)/x)^2+(8*x^3-56*x^2+120*x-72)*log(ex p(x)/x))/exp(1)^2,x, algorithm="fricas")
Output:
4*(x^3 - 6*x^2 + 9*x)*e^(-2)*log(e^x/x)^2
Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-72+120 x-56 x^2+8 x^3\right ) \log \left (\frac {e^x}{x}\right )+\left (36-48 x+12 x^2\right ) \log ^2\left (\frac {e^x}{x}\right )}{e^2} \, dx=\frac {\left (4 x^{3} - 24 x^{2} + 36 x\right ) \log {\left (\frac {e^{x}}{x} \right )}^{2}}{e^{2}} \] Input:
integrate(((12*x**2-48*x+36)*ln(exp(x)/x)**2+(8*x**3-56*x**2+120*x-72)*ln( exp(x)/x))/exp(1)**2,x)
Output:
(4*x**3 - 24*x**2 + 36*x)*exp(-2)*log(exp(x)/x)**2
Time = 0.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\left (-72+120 x-56 x^2+8 x^3\right ) \log \left (\frac {e^x}{x}\right )+\left (36-48 x+12 x^2\right ) \log ^2\left (\frac {e^x}{x}\right )}{e^2} \, dx=4 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} e^{\left (-2\right )} \log \left (\frac {e^{x}}{x}\right )^{2} \] Input:
integrate(((12*x^2-48*x+36)*log(exp(x)/x)^2+(8*x^3-56*x^2+120*x-72)*log(ex p(x)/x))/exp(1)^2,x, algorithm="maxima")
Output:
4*(x^3 - 6*x^2 + 9*x)*e^(-2)*log(e^x/x)^2
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (19) = 38\).
Time = 0.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.74 \[ \int \frac {\left (-72+120 x-56 x^2+8 x^3\right ) \log \left (\frac {e^x}{x}\right )+\left (36-48 x+12 x^2\right ) \log ^2\left (\frac {e^x}{x}\right )}{e^2} \, dx=4 \, {\left (x^{5} - 2 \, x^{4} \log \left (x\right ) + x^{3} \log \left (x\right )^{2} - 6 \, x^{4} + 12 \, x^{3} \log \left (x\right ) - 6 \, x^{2} \log \left (x\right )^{2} + 9 \, x^{3} - 18 \, x^{2} \log \left (x\right ) + 9 \, x \log \left (x\right )^{2}\right )} e^{\left (-2\right )} \] Input:
integrate(((12*x^2-48*x+36)*log(exp(x)/x)^2+(8*x^3-56*x^2+120*x-72)*log(ex p(x)/x))/exp(1)^2,x, algorithm="giac")
Output:
4*(x^5 - 2*x^4*log(x) + x^3*log(x)^2 - 6*x^4 + 12*x^3*log(x) - 6*x^2*log(x )^2 + 9*x^3 - 18*x^2*log(x) + 9*x*log(x)^2)*e^(-2)
Time = 1.64 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {\left (-72+120 x-56 x^2+8 x^3\right ) \log \left (\frac {e^x}{x}\right )+\left (36-48 x+12 x^2\right ) \log ^2\left (\frac {e^x}{x}\right )}{e^2} \, dx=4\,x\,{\mathrm {e}}^{-2}\,{\ln \left (\frac {{\mathrm {e}}^x}{x}\right )}^2\,{\left (x-3\right )}^2 \] Input:
int(exp(-2)*(log(exp(x)/x)*(120*x - 56*x^2 + 8*x^3 - 72) + log(exp(x)/x)^2 *(12*x^2 - 48*x + 36)),x)
Output:
4*x*exp(-2)*log(exp(x)/x)^2*(x - 3)^2
Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-72+120 x-56 x^2+8 x^3\right ) \log \left (\frac {e^x}{x}\right )+\left (36-48 x+12 x^2\right ) \log ^2\left (\frac {e^x}{x}\right )}{e^2} \, dx=\frac {4 \mathrm {log}\left (\frac {e^{x}}{x}\right )^{2} x \left (x^{2}-6 x +9\right )}{e^{2}} \] Input:
int(((12*x^2-48*x+36)*log(exp(x)/x)^2+(8*x^3-56*x^2+120*x-72)*log(exp(x)/x ))/exp(1)^2,x)
Output:
(4*log(e**x/x)**2*x*(x**2 - 6*x + 9))/e**2