\(\int \frac {e^{-x+\frac {e^{-x} (-3 x^3+3 e^x x^4+e^x (x-x^2) \log (x))}{\log (x)}} (3 x^2-3 e^x x^3+(-9 x^2+3 x^3+12 e^x x^3) \log (x)+e^x (1-2 x) \log ^2(x))}{\log ^2(x)} \, dx\) [513]

Optimal result

Integrand size = 97, antiderivative size = 30 \[ \int \frac {e^{-x+\frac {e^{-x} \left (-3 x^3+3 e^x x^4+e^x \left (x-x^2\right ) \log (x)\right )}{\log (x)}} \left (3 x^2-3 e^x x^3+\left (-9 x^2+3 x^3+12 e^x x^3\right ) \log (x)+e^x (1-2 x) \log ^2(x)\right )}{\log ^2(x)} \, dx=e^{x-x^2+\frac {3 x^2 \left (-e^{-x} x+x^2\right )}{\log (x)}} \]

[Out]

exp(3*x^2*(x^2-x/exp(x))/ln(x)+x-x^2)

Rubi [F]

\[ \int \frac {e^{-x+\frac {e^{-x} \left (-3 x^3+3 e^x x^4+e^x \left (x-x^2\right ) \log (x)\right )}{\log (x)}} \left (3 x^2-3 e^x x^3+\left (-9 x^2+3 x^3+12 e^x x^3\right ) \log (x)+e^x (1-2 x) \log ^2(x)\right )}{\log ^2(x)} \, dx=\int \frac {\exp \left (-x+\frac {e^{-x} \left (-3 x^3+3 e^x x^4+e^x \left (x-x^2\right ) \log (x)\right )}{\log (x)}\right ) \left (3 x^2-3 e^x x^3+\left (-9 x^2+3 x^3+12 e^x x^3\right ) \log (x)+e^x (1-2 x) \log ^2(x)\right )}{\log ^2(x)} \, dx \]

[In]

Int[(E^(-x + (-3*x^3 + 3*E^x*x^4 + E^x*(x - x^2)*Log[x])/(E^x*Log[x]))*(3*x^2 - 3*E^x*x^3 + (-9*x^2 + 3*x^3 +
12*E^x*x^3)*Log[x] + E^x*(1 - 2*x)*Log[x]^2))/Log[x]^2,x]

[Out]

Defer[Int][E^(x + (x^2*(-3*x + 3*E^x*x^2 - E^x*Log[x]))/(E^x*Log[x])), x] - 2*Defer[Int][E^(x + (x^2*(-3*x + 3
*E^x*x^2 - E^x*Log[x]))/(E^x*Log[x]))*x, x] + 3*Defer[Int][(E^((x^2*(-3*x + 3*E^x*x^2 - E^x*Log[x]))/(E^x*Log[
x]))*x^2)/Log[x]^2, x] - 3*Defer[Int][(E^(x + (x^2*(-3*x + 3*E^x*x^2 - E^x*Log[x]))/(E^x*Log[x]))*x^3)/Log[x]^
2, x] - 9*Defer[Int][(E^((x^2*(-3*x + 3*E^x*x^2 - E^x*Log[x]))/(E^x*Log[x]))*x^2)/Log[x], x] + 3*Defer[Int][(E
^((x^2*(-3*x + 3*E^x*x^2 - E^x*Log[x]))/(E^x*Log[x]))*x^3)/Log[x], x] + 12*Defer[Int][(E^(x + (x^2*(-3*x + 3*E
^x*x^2 - E^x*Log[x]))/(E^x*Log[x]))*x^3)/Log[x], x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) \left (3 x^2-3 e^x x^3+\left (-9 x^2+3 x^3+12 e^x x^3\right ) \log (x)+e^x (1-2 x) \log ^2(x)\right )}{\log ^2(x)} \, dx \\ & = \int \left (\frac {3 \exp \left (\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) x^2 (1-3 \log (x)+x \log (x))}{\log ^2(x)}-\frac {\exp \left (x+\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) \left (3 x^3-12 x^3 \log (x)-\log ^2(x)+2 x \log ^2(x)\right )}{\log ^2(x)}\right ) \, dx \\ & = 3 \int \frac {\exp \left (\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) x^2 (1-3 \log (x)+x \log (x))}{\log ^2(x)} \, dx-\int \frac {\exp \left (x+\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) \left (3 x^3-12 x^3 \log (x)-\log ^2(x)+2 x \log ^2(x)\right )}{\log ^2(x)} \, dx \\ & = 3 \int \left (\frac {\exp \left (\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) x^2}{\log ^2(x)}+\frac {\exp \left (\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) (-3+x) x^2}{\log (x)}\right ) \, dx-\int \left (-\exp \left (x+\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right )+2 \exp \left (x+\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) x+\frac {3 \exp \left (x+\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) x^3}{\log ^2(x)}-\frac {12 \exp \left (x+\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) x^3}{\log (x)}\right ) \, dx \\ & = -\left (2 \int \exp \left (x+\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) x \, dx\right )+3 \int \frac {\exp \left (\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) x^2}{\log ^2(x)} \, dx-3 \int \frac {\exp \left (x+\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) x^3}{\log ^2(x)} \, dx+3 \int \frac {\exp \left (\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) (-3+x) x^2}{\log (x)} \, dx+12 \int \frac {\exp \left (x+\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) x^3}{\log (x)} \, dx+\int \exp \left (x+\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) \, dx \\ & = -\left (2 \int \exp \left (x+\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) x \, dx\right )+3 \int \left (-\frac {3 \exp \left (\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) x^2}{\log (x)}+\frac {\exp \left (\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) x^3}{\log (x)}\right ) \, dx+3 \int \frac {\exp \left (\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) x^2}{\log ^2(x)} \, dx-3 \int \frac {\exp \left (x+\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) x^3}{\log ^2(x)} \, dx+12 \int \frac {\exp \left (x+\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) x^3}{\log (x)} \, dx+\int \exp \left (x+\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) \, dx \\ & = -\left (2 \int \exp \left (x+\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) x \, dx\right )+3 \int \frac {\exp \left (\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) x^2}{\log ^2(x)} \, dx-3 \int \frac {\exp \left (x+\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) x^3}{\log ^2(x)} \, dx+3 \int \frac {\exp \left (\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) x^3}{\log (x)} \, dx-9 \int \frac {\exp \left (\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) x^2}{\log (x)} \, dx+12 \int \frac {\exp \left (x+\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) x^3}{\log (x)} \, dx+\int \exp \left (x+\frac {e^{-x} x^2 \left (-3 x+3 e^x x^2-e^x \log (x)\right )}{\log (x)}\right ) \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 5.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-x+\frac {e^{-x} \left (-3 x^3+3 e^x x^4+e^x \left (x-x^2\right ) \log (x)\right )}{\log (x)}} \left (3 x^2-3 e^x x^3+\left (-9 x^2+3 x^3+12 e^x x^3\right ) \log (x)+e^x (1-2 x) \log ^2(x)\right )}{\log ^2(x)} \, dx=e^{x-x^2-\frac {3 \left (e^{-x}-x\right ) x^3}{\log (x)}} \]

[In]

Integrate[(E^(-x + (-3*x^3 + 3*E^x*x^4 + E^x*(x - x^2)*Log[x])/(E^x*Log[x]))*(3*x^2 - 3*E^x*x^3 + (-9*x^2 + 3*
x^3 + 12*E^x*x^3)*Log[x] + E^x*(1 - 2*x)*Log[x]^2))/Log[x]^2,x]

[Out]

E^(x - x^2 - (3*(E^(-x) - x)*x^3)/Log[x])

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20

method	result	size
parallelrisch	\({\mathrm e}^{\frac {\left (\left (-x^{2}+x \right ) {\mathrm e}^{x} \ln \left (x \right )+3 \,{\mathrm e}^{x} x^{4}-3 x^{3}\right ) {\mathrm e}^{-x}}{\ln \left (x \right )}}\)	\(36\)
risch	\({\mathrm e}^{-\frac {x \left (-3 \,{\mathrm e}^{x} x^{3}+x \,{\mathrm e}^{x} \ln \left (x \right )-{\mathrm e}^{x} \ln \left (x \right )+3 x^{2}\right ) {\mathrm e}^{-x}}{\ln \left (x \right )}}\)	\(38\)

[In]

int(((1-2*x)*exp(x)*ln(x)^2+(12*exp(x)*x^3+3*x^3-9*x^2)*ln(x)-3*exp(x)*x^3+3*x^2)*exp(((-x^2+x)*exp(x)*ln(x)+3
*exp(x)*x^4-3*x^3)/exp(x)/ln(x))/exp(x)/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

exp(((-x^2+x)*exp(x)*ln(x)+3*exp(x)*x^4-3*x^3)/exp(x)/ln(x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {e^{-x+\frac {e^{-x} \left (-3 x^3+3 e^x x^4+e^x \left (x-x^2\right ) \log (x)\right )}{\log (x)}} \left (3 x^2-3 e^x x^3+\left (-9 x^2+3 x^3+12 e^x x^3\right ) \log (x)+e^x (1-2 x) \log ^2(x)\right )}{\log ^2(x)} \, dx=e^{\left (x + \frac {{\left (3 \, x^{4} e^{x} - x^{2} e^{x} \log \left (x\right ) - 3 \, x^{3}\right )} e^{\left (-x\right )}}{\log \left (x\right )}\right )} \]

[In]

integrate(((1-2*x)*exp(x)*log(x)^2+(12*exp(x)*x^3+3*x^3-9*x^2)*log(x)-3*exp(x)*x^3+3*x^2)*exp(((-x^2+x)*exp(x)
*log(x)+3*exp(x)*x^4-3*x^3)/exp(x)/log(x))/exp(x)/log(x)^2,x, algorithm="fricas")

[Out]

e^(x + (3*x^4*e^x - x^2*e^x*log(x) - 3*x^3)*e^(-x)/log(x))

Sympy [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-x+\frac {e^{-x} \left (-3 x^3+3 e^x x^4+e^x \left (x-x^2\right ) \log (x)\right )}{\log (x)}} \left (3 x^2-3 e^x x^3+\left (-9 x^2+3 x^3+12 e^x x^3\right ) \log (x)+e^x (1-2 x) \log ^2(x)\right )}{\log ^2(x)} \, dx=e^{\frac {\left (3 x^{4} e^{x} - 3 x^{3} + \left (- x^{2} + x\right ) e^{x} \log {\left (x \right )}\right ) e^{- x}}{\log {\left (x \right )}}} \]

[In]

integrate(((1-2*x)*exp(x)*ln(x)**2+(12*exp(x)*x**3+3*x**3-9*x**2)*ln(x)-3*exp(x)*x**3+3*x**2)*exp(((-x**2+x)*e
xp(x)*ln(x)+3*exp(x)*x**4-3*x**3)/exp(x)/ln(x))/exp(x)/ln(x)**2,x)

[Out]

exp((3*x**4*exp(x) - 3*x**3 + (-x**2 + x)*exp(x)*log(x))*exp(-x)/log(x))

Maxima [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-x+\frac {e^{-x} \left (-3 x^3+3 e^x x^4+e^x \left (x-x^2\right ) \log (x)\right )}{\log (x)}} \left (3 x^2-3 e^x x^3+\left (-9 x^2+3 x^3+12 e^x x^3\right ) \log (x)+e^x (1-2 x) \log ^2(x)\right )}{\log ^2(x)} \, dx=e^{\left (\frac {3 \, x^{4}}{\log \left (x\right )} - \frac {3 \, x^{3} e^{\left (-x\right )}}{\log \left (x\right )} - x^{2} + x\right )} \]

[In]

integrate(((1-2*x)*exp(x)*log(x)^2+(12*exp(x)*x^3+3*x^3-9*x^2)*log(x)-3*exp(x)*x^3+3*x^2)*exp(((-x^2+x)*exp(x)
*log(x)+3*exp(x)*x^4-3*x^3)/exp(x)/log(x))/exp(x)/log(x)^2,x, algorithm="maxima")

[Out]

e^(3*x^4/log(x) - 3*x^3*e^(-x)/log(x) - x^2 + x)

Giac [F]

\[ \int \frac {e^{-x+\frac {e^{-x} \left (-3 x^3+3 e^x x^4+e^x \left (x-x^2\right ) \log (x)\right )}{\log (x)}} \left (3 x^2-3 e^x x^3+\left (-9 x^2+3 x^3+12 e^x x^3\right ) \log (x)+e^x (1-2 x) \log ^2(x)\right )}{\log ^2(x)} \, dx=\int { -\frac {{\left (3 \, x^{3} e^{x} + {\left (2 \, x - 1\right )} e^{x} \log \left (x\right )^{2} - 3 \, x^{2} - 3 \, {\left (4 \, x^{3} e^{x} + x^{3} - 3 \, x^{2}\right )} \log \left (x\right )\right )} e^{\left (-x + \frac {{\left (3 \, x^{4} e^{x} - 3 \, x^{3} - {\left (x^{2} - x\right )} e^{x} \log \left (x\right )\right )} e^{\left (-x\right )}}{\log \left (x\right )}\right )}}{\log \left (x\right )^{2}} \,d x } \]

[In]

integrate(((1-2*x)*exp(x)*log(x)^2+(12*exp(x)*x^3+3*x^3-9*x^2)*log(x)-3*exp(x)*x^3+3*x^2)*exp(((-x^2+x)*exp(x)
*log(x)+3*exp(x)*x^4-3*x^3)/exp(x)/log(x))/exp(x)/log(x)^2,x, algorithm="giac")

[Out]

integrate(-(3*x^3*e^x + (2*x - 1)*e^x*log(x)^2 - 3*x^2 - 3*(4*x^3*e^x + x^3 - 3*x^2)*log(x))*e^(-x + (3*x^4*e^
x - 3*x^3 - (x^2 - x)*e^x*log(x))*e^(-x)/log(x))/log(x)^2, x)

Mupad [B] (verification not implemented)

Time = 7.14 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10 \[ \int \frac {e^{-x+\frac {e^{-x} \left (-3 x^3+3 e^x x^4+e^x \left (x-x^2\right ) \log (x)\right )}{\log (x)}} \left (3 x^2-3 e^x x^3+\left (-9 x^2+3 x^3+12 e^x x^3\right ) \log (x)+e^x (1-2 x) \log ^2(x)\right )}{\log ^2(x)} \, dx={\mathrm {e}}^{-\frac {3\,x^3\,{\mathrm {e}}^{-x}}{\ln \left (x\right )}}\,{\mathrm {e}}^{-x^2}\,{\mathrm {e}}^x\,{\mathrm {e}}^{\frac {3\,x^4}{\ln \left (x\right )}} \]

[In]

int(-(exp((exp(-x)*(3*x^4*exp(x) - 3*x^3 + exp(x)*log(x)*(x - x^2)))/log(x))*exp(-x)*(3*x^3*exp(x) - log(x)*(1
2*x^3*exp(x) - 9*x^2 + 3*x^3) - 3*x^2 + exp(x)*log(x)^2*(2*x - 1)))/log(x)^2,x)

[Out]

exp(-(3*x^3*exp(-x))/log(x))*exp(-x^2)*exp(x)*exp((3*x^4)/log(x))