Integrand size = 162, antiderivative size = 29 \[ \int \frac {e^{-x} \left (e^x \left (-4+8 x^3\right )+e^{2 e^{-x} \left (3 x+6 x \log (x)+3 x \log ^2(x)\right )} \left (72 x^2-24 x^3+\left (96 x^2-48 x^3\right ) \log (x)+\left (24 x^2-24 x^3\right ) \log ^2(x)\right )+e^{e^{-x} \left (3 x+6 x \log (x)+3 x \log ^2(x)\right )} \left (8 e^x x^2+72 x^3-24 x^4+\left (96 x^3-48 x^4\right ) \log (x)+\left (24 x^3-24 x^4\right ) \log ^2(x)\right )\right )}{x^2} \, dx=-1+\frac {4}{x}+4 \left (e^{3 e^{-x} x (1+\log (x))^2}+x\right )^2 \] Output:
2*(x+exp(3*(1+ln(x))^2*x/exp(x)))*(2*x+2*exp(3*(1+ln(x))^2*x/exp(x)))-1+4/ x
Leaf count is larger than twice the leaf count of optimal. \(83\) vs. \(2(29)=58\).
Time = 8.94 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.86 \[ \int \frac {e^{-x} \left (e^x \left (-4+8 x^3\right )+e^{2 e^{-x} \left (3 x+6 x \log (x)+3 x \log ^2(x)\right )} \left (72 x^2-24 x^3+\left (96 x^2-48 x^3\right ) \log (x)+\left (24 x^2-24 x^3\right ) \log ^2(x)\right )+e^{e^{-x} \left (3 x+6 x \log (x)+3 x \log ^2(x)\right )} \left (8 e^x x^2+72 x^3-24 x^4+\left (96 x^3-48 x^4\right ) \log (x)+\left (24 x^3-24 x^4\right ) \log ^2(x)\right )\right )}{x^2} \, dx=\frac {4}{x}+4 x^2+4 e^{6 e^{-x} x+6 e^{-x} x \log ^2(x)} x^{12 e^{-x} x}+8 e^{3 e^{-x} x+3 e^{-x} x \log ^2(x)} x^{1+6 e^{-x} x} \] Input:
Integrate[(E^x*(-4 + 8*x^3) + E^((2*(3*x + 6*x*Log[x] + 3*x*Log[x]^2))/E^x )*(72*x^2 - 24*x^3 + (96*x^2 - 48*x^3)*Log[x] + (24*x^2 - 24*x^3)*Log[x]^2 ) + E^((3*x + 6*x*Log[x] + 3*x*Log[x]^2)/E^x)*(8*E^x*x^2 + 72*x^3 - 24*x^4 + (96*x^3 - 48*x^4)*Log[x] + (24*x^3 - 24*x^4)*Log[x]^2))/(E^x*x^2),x]
Output:
4/x + 4*x^2 + 4*E^((6*x)/E^x + (6*x*Log[x]^2)/E^x)*x^((12*x)/E^x) + 8*E^(( 3*x)/E^x + (3*x*Log[x]^2)/E^x)*x^(1 + (6*x)/E^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.
| \(\displaystyle \int \frac {e^{-x} \left (e^x \left (8 x^3-4\right )+e^{2 e^{-x} \left (3 x+3 x \log ^2(x)+6 x \log (x)\right )} \left (-24 x^3+72 x^2+\left (24 x^2-24 x^3\right ) \log ^2(x)+\left (96 x^2-48 x^3\right ) \log (x)\right )+e^{e^{-x} \left (3 x+3 x \log ^2(x)+6 x \log (x)\right )} \left (-24 x^4+72 x^3+8 e^x x^2+\left (24 x^3-24 x^4\right ) \log ^2(x)+\left (96 x^3-48 x^4\right ) \log (x)\right )\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-24 x^{12 e^{-x} x} e^{6 e^{-x} x \left (\log ^2(x)+1\right )-x} (\log (x)+1) (x+x \log (x)-\log (x)-3)+8 x^{6 e^{-x} x} e^{3 e^{-x} x \left (\log ^2(x)+1\right )-x} \left (-3 x^2-3 x^2 \log ^2(x)-6 x^2 \log (x)+9 x+e^x+3 x \log ^2(x)+12 x \log (x)\right )+\frac {4 \left (2 x^3-1\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \left (-24 x^{12 e^{-x} x} e^{-e^{-x} x \left (e^x-6 \log ^2(x)-6\right )} (\log (x)+1) (x+x \log (x)-\log (x)-3)+8 x^{6 e^{-x} x} e^{-e^{-x} x \left (e^x-3 \log ^2(x)-3\right )} \left (-3 x^2-3 x^2 \log ^2(x)-6 x^2 \log (x)+9 x+e^x+3 x \log ^2(x)+12 x \log (x)\right )+\frac {4 \left (2 x^3-1\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (-24 x^{12 e^{-x} x} e^{-e^{-x} x \left (e^x-6 \log ^2(x)-6\right )} (\log (x)+1) (x+x \log (x)-\log (x)-3)+8 x^{6 e^{-x} x} e^{-e^{-x} x \left (e^x-3 \log ^2(x)-3\right )} \left (-3 x^2-3 x^2 \log ^2(x)-6 x^2 \log (x)+9 x+e^x+3 x \log ^2(x)+12 x \log (x)\right )+\frac {4 \left (2 x^3-1\right )}{x^2}\right )dx\) |
Input:
Int[(E^x*(-4 + 8*x^3) + E^((2*(3*x + 6*x*Log[x] + 3*x*Log[x]^2))/E^x)*(72* x^2 - 24*x^3 + (96*x^2 - 48*x^3)*Log[x] + (24*x^2 - 24*x^3)*Log[x]^2) + E^ ((3*x + 6*x*Log[x] + 3*x*Log[x]^2)/E^x)*(8*E^x*x^2 + 72*x^3 - 24*x^4 + (96 *x^3 - 48*x^4)*Log[x] + (24*x^3 - 24*x^4)*Log[x]^2))/(E^x*x^2),x]
Output:
$Aborted
Time = 1.76 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55
| method | result | size |
| risch | \(\frac {4}{x}+4 x^{2}+4 \,{\mathrm e}^{6 x \left (\ln \left (x \right )+1\right )^{2} {\mathrm e}^{-x}}+8 x \,{\mathrm e}^{3 x \left (\ln \left (x \right )+1\right )^{2} {\mathrm e}^{-x}}\) | \(45\) |
| parallelrisch | \(\frac {4+4 x^{3}+8 x^{2} {\mathrm e}^{3 x \left (\ln \left (x \right )^{2}+2 \ln \left (x \right )+1\right ) {\mathrm e}^{-x}}+4 \,{\mathrm e}^{6 x \left (\ln \left (x \right )^{2}+2 \ln \left (x \right )+1\right ) {\mathrm e}^{-x}} x}{x}\) | \(58\) |
Input:
int((((-24*x^3+24*x^2)*ln(x)^2+(-48*x^3+96*x^2)*ln(x)-24*x^3+72*x^2)*exp(( 3*x*ln(x)^2+6*x*ln(x)+3*x)/exp(x))^2+((-24*x^4+24*x^3)*ln(x)^2+(-48*x^4+96 *x^3)*ln(x)+8*exp(x)*x^2-24*x^4+72*x^3)*exp((3*x*ln(x)^2+6*x*ln(x)+3*x)/ex p(x))+(8*x^3-4)*exp(x))/exp(x)/x^2,x,method=_RETURNVERBOSE)
Output:
4/x+4*x^2+4*exp(6*x*(ln(x)+1)^2*exp(-x))+8*x*exp(3*x*(ln(x)+1)^2*exp(-x))
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (27) = 54\).
Time = 0.13 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.97 \[ \int \frac {e^{-x} \left (e^x \left (-4+8 x^3\right )+e^{2 e^{-x} \left (3 x+6 x \log (x)+3 x \log ^2(x)\right )} \left (72 x^2-24 x^3+\left (96 x^2-48 x^3\right ) \log (x)+\left (24 x^2-24 x^3\right ) \log ^2(x)\right )+e^{e^{-x} \left (3 x+6 x \log (x)+3 x \log ^2(x)\right )} \left (8 e^x x^2+72 x^3-24 x^4+\left (96 x^3-48 x^4\right ) \log (x)+\left (24 x^3-24 x^4\right ) \log ^2(x)\right )\right )}{x^2} \, dx=\frac {4 \, {\left (x^{3} + 2 \, x^{2} e^{\left (3 \, {\left (x \log \left (x\right )^{2} + 2 \, x \log \left (x\right ) + x\right )} e^{\left (-x\right )}\right )} + x e^{\left (6 \, {\left (x \log \left (x\right )^{2} + 2 \, x \log \left (x\right ) + x\right )} e^{\left (-x\right )}\right )} + 1\right )}}{x} \] Input:
integrate((((-24*x^3+24*x^2)*log(x)^2+(-48*x^3+96*x^2)*log(x)-24*x^3+72*x^ 2)*exp((3*x*log(x)^2+6*x*log(x)+3*x)/exp(x))^2+((-24*x^4+24*x^3)*log(x)^2+ (-48*x^4+96*x^3)*log(x)+8*exp(x)*x^2-24*x^4+72*x^3)*exp((3*x*log(x)^2+6*x* log(x)+3*x)/exp(x))+(8*x^3-4)*exp(x))/exp(x)/x^2,x, algorithm="fricas")
Output:
4*(x^3 + 2*x^2*e^(3*(x*log(x)^2 + 2*x*log(x) + x)*e^(-x)) + x*e^(6*(x*log( x)^2 + 2*x*log(x) + x)*e^(-x)) + 1)/x
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (26) = 52\).
Time = 81.62 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.07 \[ \int \frac {e^{-x} \left (e^x \left (-4+8 x^3\right )+e^{2 e^{-x} \left (3 x+6 x \log (x)+3 x \log ^2(x)\right )} \left (72 x^2-24 x^3+\left (96 x^2-48 x^3\right ) \log (x)+\left (24 x^2-24 x^3\right ) \log ^2(x)\right )+e^{e^{-x} \left (3 x+6 x \log (x)+3 x \log ^2(x)\right )} \left (8 e^x x^2+72 x^3-24 x^4+\left (96 x^3-48 x^4\right ) \log (x)+\left (24 x^3-24 x^4\right ) \log ^2(x)\right )\right )}{x^2} \, dx=4 x^{2} + 8 x e^{\left (3 x \log {\left (x \right )}^{2} + 6 x \log {\left (x \right )} + 3 x\right ) e^{- x}} + 4 e^{2 \cdot \left (3 x \log {\left (x \right )}^{2} + 6 x \log {\left (x \right )} + 3 x\right ) e^{- x}} + \frac {4}{x} \] Input:
integrate((((-24*x**3+24*x**2)*ln(x)**2+(-48*x**3+96*x**2)*ln(x)-24*x**3+7 2*x**2)*exp((3*x*ln(x)**2+6*x*ln(x)+3*x)/exp(x))**2+((-24*x**4+24*x**3)*ln (x)**2+(-48*x**4+96*x**3)*ln(x)+8*exp(x)*x**2-24*x**4+72*x**3)*exp((3*x*ln (x)**2+6*x*ln(x)+3*x)/exp(x))+(8*x**3-4)*exp(x))/exp(x)/x**2,x)
Output:
4*x**2 + 8*x*exp((3*x*log(x)**2 + 6*x*log(x) + 3*x)*exp(-x)) + 4*exp(2*(3* x*log(x)**2 + 6*x*log(x) + 3*x)*exp(-x)) + 4/x
\[ \int \frac {e^{-x} \left (e^x \left (-4+8 x^3\right )+e^{2 e^{-x} \left (3 x+6 x \log (x)+3 x \log ^2(x)\right )} \left (72 x^2-24 x^3+\left (96 x^2-48 x^3\right ) \log (x)+\left (24 x^2-24 x^3\right ) \log ^2(x)\right )+e^{e^{-x} \left (3 x+6 x \log (x)+3 x \log ^2(x)\right )} \left (8 e^x x^2+72 x^3-24 x^4+\left (96 x^3-48 x^4\right ) \log (x)+\left (24 x^3-24 x^4\right ) \log ^2(x)\right )\right )}{x^2} \, dx=\int { -\frac {4 \, {\left (6 \, {\left (x^{3} + {\left (x^{3} - x^{2}\right )} \log \left (x\right )^{2} - 3 \, x^{2} + 2 \, {\left (x^{3} - 2 \, x^{2}\right )} \log \left (x\right )\right )} e^{\left (6 \, {\left (x \log \left (x\right )^{2} + 2 \, x \log \left (x\right ) + x\right )} e^{\left (-x\right )}\right )} + 2 \, {\left (3 \, x^{4} - 9 \, x^{3} - x^{2} e^{x} + 3 \, {\left (x^{4} - x^{3}\right )} \log \left (x\right )^{2} + 6 \, {\left (x^{4} - 2 \, x^{3}\right )} \log \left (x\right )\right )} e^{\left (3 \, {\left (x \log \left (x\right )^{2} + 2 \, x \log \left (x\right ) + x\right )} e^{\left (-x\right )}\right )} - {\left (2 \, x^{3} - 1\right )} e^{x}\right )} e^{\left (-x\right )}}{x^{2}} \,d x } \] Input:
integrate((((-24*x^3+24*x^2)*log(x)^2+(-48*x^3+96*x^2)*log(x)-24*x^3+72*x^ 2)*exp((3*x*log(x)^2+6*x*log(x)+3*x)/exp(x))^2+((-24*x^4+24*x^3)*log(x)^2+ (-48*x^4+96*x^3)*log(x)+8*exp(x)*x^2-24*x^4+72*x^3)*exp((3*x*log(x)^2+6*x* log(x)+3*x)/exp(x))+(8*x^3-4)*exp(x))/exp(x)/x^2,x, algorithm="maxima")
Output:
4*x^2 + 4/x + 4*e^(6*x*e^(-x)*log(x)^2 + 12*x*e^(-x)*log(x) + 6*x*e^(-x)) + 4*integrate(-2*(3*(x^2 - x)*log(x)^2 + 3*x^2 + 6*(x^2 - 2*x)*log(x) - 9* x - e^x)*e^(3*x*e^(-x)*log(x)^2 + 6*x*e^(-x)*log(x) + 3*x*e^(-x) - x), x)
\[ \int \frac {e^{-x} \left (e^x \left (-4+8 x^3\right )+e^{2 e^{-x} \left (3 x+6 x \log (x)+3 x \log ^2(x)\right )} \left (72 x^2-24 x^3+\left (96 x^2-48 x^3\right ) \log (x)+\left (24 x^2-24 x^3\right ) \log ^2(x)\right )+e^{e^{-x} \left (3 x+6 x \log (x)+3 x \log ^2(x)\right )} \left (8 e^x x^2+72 x^3-24 x^4+\left (96 x^3-48 x^4\right ) \log (x)+\left (24 x^3-24 x^4\right ) \log ^2(x)\right )\right )}{x^2} \, dx=\int { -\frac {4 \, {\left (6 \, {\left (x^{3} + {\left (x^{3} - x^{2}\right )} \log \left (x\right )^{2} - 3 \, x^{2} + 2 \, {\left (x^{3} - 2 \, x^{2}\right )} \log \left (x\right )\right )} e^{\left (6 \, {\left (x \log \left (x\right )^{2} + 2 \, x \log \left (x\right ) + x\right )} e^{\left (-x\right )}\right )} + 2 \, {\left (3 \, x^{4} - 9 \, x^{3} - x^{2} e^{x} + 3 \, {\left (x^{4} - x^{3}\right )} \log \left (x\right )^{2} + 6 \, {\left (x^{4} - 2 \, x^{3}\right )} \log \left (x\right )\right )} e^{\left (3 \, {\left (x \log \left (x\right )^{2} + 2 \, x \log \left (x\right ) + x\right )} e^{\left (-x\right )}\right )} - {\left (2 \, x^{3} - 1\right )} e^{x}\right )} e^{\left (-x\right )}}{x^{2}} \,d x } \] Input:
integrate((((-24*x^3+24*x^2)*log(x)^2+(-48*x^3+96*x^2)*log(x)-24*x^3+72*x^ 2)*exp((3*x*log(x)^2+6*x*log(x)+3*x)/exp(x))^2+((-24*x^4+24*x^3)*log(x)^2+ (-48*x^4+96*x^3)*log(x)+8*exp(x)*x^2-24*x^4+72*x^3)*exp((3*x*log(x)^2+6*x* log(x)+3*x)/exp(x))+(8*x^3-4)*exp(x))/exp(x)/x^2,x, algorithm="giac")
Output:
integrate(-4*(6*(x^3 + (x^3 - x^2)*log(x)^2 - 3*x^2 + 2*(x^3 - 2*x^2)*log( x))*e^(6*(x*log(x)^2 + 2*x*log(x) + x)*e^(-x)) + 2*(3*x^4 - 9*x^3 - x^2*e^ x + 3*(x^4 - x^3)*log(x)^2 + 6*(x^4 - 2*x^3)*log(x))*e^(3*(x*log(x)^2 + 2* x*log(x) + x)*e^(-x)) - (2*x^3 - 1)*e^x)*e^(-x)/x^2, x)
Time = 3.53 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.55 \[ \int \frac {e^{-x} \left (e^x \left (-4+8 x^3\right )+e^{2 e^{-x} \left (3 x+6 x \log (x)+3 x \log ^2(x)\right )} \left (72 x^2-24 x^3+\left (96 x^2-48 x^3\right ) \log (x)+\left (24 x^2-24 x^3\right ) \log ^2(x)\right )+e^{e^{-x} \left (3 x+6 x \log (x)+3 x \log ^2(x)\right )} \left (8 e^x x^2+72 x^3-24 x^4+\left (96 x^3-48 x^4\right ) \log (x)+\left (24 x^3-24 x^4\right ) \log ^2(x)\right )\right )}{x^2} \, dx=4\,x^{12\,x\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{6\,x\,{\mathrm {e}}^{-x}\,{\ln \left (x\right )}^2+6\,x\,{\mathrm {e}}^{-x}}+\frac {4}{x}+4\,x^2+8\,x\,x^{6\,x\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{3\,x\,{\mathrm {e}}^{-x}\,{\ln \left (x\right )}^2+3\,x\,{\mathrm {e}}^{-x}} \] Input:
int((exp(-x)*(exp(2*exp(-x)*(3*x + 3*x*log(x)^2 + 6*x*log(x)))*(log(x)*(96 *x^2 - 48*x^3) + log(x)^2*(24*x^2 - 24*x^3) + 72*x^2 - 24*x^3) + exp(exp(- x)*(3*x + 3*x*log(x)^2 + 6*x*log(x)))*(log(x)*(96*x^3 - 48*x^4) + 8*x^2*ex p(x) + log(x)^2*(24*x^3 - 24*x^4) + 72*x^3 - 24*x^4) + exp(x)*(8*x^3 - 4)) )/x^2,x)
Output:
4*x^(12*x*exp(-x))*exp(6*x*exp(-x) + 6*x*exp(-x)*log(x)^2) + 4/x + 4*x^2 + 8*x*x^(6*x*exp(-x))*exp(3*x*exp(-x) + 3*x*exp(-x)*log(x)^2)
Time = 0.50 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.24 \[ \int \frac {e^{-x} \left (e^x \left (-4+8 x^3\right )+e^{2 e^{-x} \left (3 x+6 x \log (x)+3 x \log ^2(x)\right )} \left (72 x^2-24 x^3+\left (96 x^2-48 x^3\right ) \log (x)+\left (24 x^2-24 x^3\right ) \log ^2(x)\right )+e^{e^{-x} \left (3 x+6 x \log (x)+3 x \log ^2(x)\right )} \left (8 e^x x^2+72 x^3-24 x^4+\left (96 x^3-48 x^4\right ) \log (x)+\left (24 x^3-24 x^4\right ) \log ^2(x)\right )\right )}{x^2} \, dx=\frac {4 e^{\frac {6 \mathrm {log}\left (x \right )^{2} x +12 \,\mathrm {log}\left (x \right ) x +6 x}{e^{x}}} x +8 e^{\frac {3 \mathrm {log}\left (x \right )^{2} x +6 \,\mathrm {log}\left (x \right ) x +3 x}{e^{x}}} x^{2}+4 x^{3}+4}{x} \] Input:
int((((-24*x^3+24*x^2)*log(x)^2+(-48*x^3+96*x^2)*log(x)-24*x^3+72*x^2)*exp ((3*x*log(x)^2+6*x*log(x)+3*x)/exp(x))^2+((-24*x^4+24*x^3)*log(x)^2+(-48*x ^4+96*x^3)*log(x)+8*exp(x)*x^2-24*x^4+72*x^3)*exp((3*x*log(x)^2+6*x*log(x) +3*x)/exp(x))+(8*x^3-4)*exp(x))/exp(x)/x^2,x)
Output:
(4*(e**((6*log(x)**2*x + 12*log(x)*x + 6*x)/e**x)*x + 2*e**((3*log(x)**2*x + 6*log(x)*x + 3*x)/e**x)*x**2 + x**3 + 1))/x