\(\int \frac {-e^x+x+3 x^2+3 x \log (x)+e^{-2+\frac {e^x-x-3 x^2-3 x \log (x)}{e^2}} (4-e^x+6 x+3 \log (x)) (e^x-x-3 x^2-3 x \log (x))}{-e^x+x+3 x^2+3 x \log (x)} \, dx\) [7716]

Optimal result

Integrand size = 94, antiderivative size = 22 \[ \int \frac {-e^x+x+3 x^2+3 x \log (x)+e^{-2+\frac {e^x-x-3 x^2-3 x \log (x)}{e^2}} \left (4-e^x+6 x+3 \log (x)\right ) \left (e^x-x-3 x^2-3 x \log (x)\right )}{-e^x+x+3 x^2+3 x \log (x)} \, dx=e^{\frac {e^x-x-3 x (x+\log (x))}{e^2}}+x \]

[Out]

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

Rubi [F]

\[ \int \frac {-e^x+x+3 x^2+3 x \log (x)+e^{-2+\frac {e^x-x-3 x^2-3 x \log (x)}{e^2}} \left (4-e^x+6 x+3 \log (x)\right ) \left (e^x-x-3 x^2-3 x \log (x)\right )}{-e^x+x+3 x^2+3 x \log (x)} \, dx=\int \frac {-e^x+x+3 x^2+3 x \log (x)+e^{-2+\frac {e^x-x-3 x^2-3 x \log (x)}{e^2}} \left (4-e^x+6 x+3 \log (x)\right ) \left (e^x-x-3 x^2-3 x \log (x)\right )}{-e^x+x+3 x^2+3 x \log (x)} \, dx \]

[In]

Int[(-E^x + x + 3*x^2 + 3*x*Log[x] + E^(-2 + (E^x - x - 3*x^2 - 3*x*Log[x])/E^2)*(4 - E^x + 6*x + 3*Log[x])*(E
^x - x - 3*x^2 - 3*x*Log[x]))/(-E^x + x + 3*x^2 + 3*x*Log[x]),x]

[Out]

x - 4*Defer[Int][E^(-2 + E^(-2 + x) - (x*(1 + 3*x))/E^2)/x^((3*x)/E^2), x] - 3*Log[x]*Defer[Int][E^(-2 + E^(-2
+ x) - (x*(1 + 3*x))/E^2)/x^((3*x)/E^2), x] + Defer[Int][E^(-2 + E^(-2 + x) + x - (x*(1 + 3*x))/E^2)/x^((3*x)
/E^2), x] - 6*Defer[Int][E^(-2 + E^(-2 + x) - (x*(1 + 3*x))/E^2)*x^(1 - (3*x)/E^2), x] + 3*Defer[Int][Defer[In
t][E^(-2 + E^(-2 + x) - (x*(1 + 3*x))/E^2)/x^((3*x)/E^2), x]/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (1+e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \left (-4+e^x-6 x-3 \log (x)\right )\right ) \, dx \\ & = x+\int e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \left (-4+e^x-6 x-3 \log (x)\right ) \, dx \\ & = x+\int \left (-4 e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}}+e^{-2+e^{-2+x}+x-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}}-6 e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{1-\frac {3 x}{e^2}}-3 e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \log (x)\right ) \, dx \\ & = x-3 \int e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \log (x) \, dx-4 \int e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \, dx-6 \int e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{1-\frac {3 x}{e^2}} \, dx+\int e^{-2+e^{-2+x}+x-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \, dx \\ & = x+3 \int \frac {\int e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \, dx}{x} \, dx-4 \int e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \, dx-6 \int e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{1-\frac {3 x}{e^2}} \, dx-(3 \log (x)) \int e^{-2+e^{-2+x}-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \, dx+\int e^{-2+e^{-2+x}+x-\frac {x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {-e^x+x+3 x^2+3 x \log (x)+e^{-2+\frac {e^x-x-3 x^2-3 x \log (x)}{e^2}} \left (4-e^x+6 x+3 \log (x)\right ) \left (e^x-x-3 x^2-3 x \log (x)\right )}{-e^x+x+3 x^2+3 x \log (x)} \, dx=x+e^{\frac {e^x-x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \]

[In]

Integrate[(-E^x + x + 3*x^2 + 3*x*Log[x] + E^(-2 + (E^x - x - 3*x^2 - 3*x*Log[x])/E^2)*(4 - E^x + 6*x + 3*Log[
x])*(E^x - x - 3*x^2 - 3*x*Log[x]))/(-E^x + x + 3*x^2 + 3*x*Log[x]),x]

[Out]

x + E^((E^x - x*(1 + 3*x))/E^2)/x^((3*x)/E^2)

Maple [A] (verified)

Time = 14.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05

method	result	size
risch	\(x +{\mathrm e}^{\left (-3 x \ln \left (x \right )+{\mathrm e}^{x}-3 x^{2}-x \right ) {\mathrm e}^{-2}}\)	\(23\)
parallelrisch	\(\frac {18 x \,{\mathrm e}^{2 x}-72 \,{\mathrm e}^{x} x^{2}+324 x^{4} \ln \left (x \right )-108 \,{\mathrm e}^{x} x^{3}-12 \,{\mathrm e}^{x} x +162 x^{3} \ln \left (x \right )^{2}+54 x^{2} \ln \left (x \right )^{2}+6 \,{\mathrm e}^{2 x}-108 x^{2} {\mathrm e}^{x} \ln \left (x \right )-36 x \,{\mathrm e}^{x} \ln \left (x \right )+162 x^{5}+162 x^{4}+54 x^{3}+6 x^{2}+216 x^{3} \ln \left (x \right )+36 x^{2} \ln \left (x \right )-108 \,{\mathrm e}^{{\mathrm e}^{\ln \left (-3 x \ln \left (x \right )+{\mathrm e}^{x}-3 x^{2}-x \right )-2}} x \ln \left (x \right ) {\mathrm e}^{x}+18 \,{\mathrm e}^{{\mathrm e}^{\ln \left (-3 x \ln \left (x \right )+{\mathrm e}^{x}-3 x^{2}-x \right )-2}} x^{2}+108 \,{\mathrm e}^{{\mathrm e}^{\ln \left (-3 x \ln \left (x \right )+{\mathrm e}^{x}-3 x^{2}-x \right )-2}} x^{3}+162 \,{\mathrm e}^{{\mathrm e}^{\ln \left (-3 x \ln \left (x \right )+{\mathrm e}^{x}-3 x^{2}-x \right )-2}} x^{4}+18 \,{\mathrm e}^{{\mathrm e}^{\ln \left (-3 x \ln \left (x \right )+{\mathrm e}^{x}-3 x^{2}-x \right )-2}} {\mathrm e}^{2 x}+108 \,{\mathrm e}^{{\mathrm e}^{\ln \left (-3 x \ln \left (x \right )+{\mathrm e}^{x}-3 x^{2}-x \right )-2}} x^{2} \ln \left (x \right )+324 \,{\mathrm e}^{{\mathrm e}^{\ln \left (-3 x \ln \left (x \right )+{\mathrm e}^{x}-3 x^{2}-x \right )-2}} \ln \left (x \right ) x^{3}+162 \,{\mathrm e}^{{\mathrm e}^{\ln \left (-3 x \ln \left (x \right )+{\mathrm e}^{x}-3 x^{2}-x \right )-2}} \ln \left (x \right )^{2} x^{2}-36 \,{\mathrm e}^{{\mathrm e}^{\ln \left (-3 x \ln \left (x \right )+{\mathrm e}^{x}-3 x^{2}-x \right )-2}} {\mathrm e}^{x} x -108 \,{\mathrm e}^{{\mathrm e}^{\ln \left (-3 x \ln \left (x \right )+{\mathrm e}^{x}-3 x^{2}-x \right )-2}} {\mathrm e}^{x} x^{2}}{18 \left (3 x \ln \left (x \right )-{\mathrm e}^{x}+3 x^{2}+x \right )^{2}}\)	\(402\)

[In]

int(((3*ln(x)-exp(x)+6*x+4)*exp(ln(-3*x*ln(x)+exp(x)-3*x^2-x)-2)*exp(exp(ln(-3*x*ln(x)+exp(x)-3*x^2-x)-2))+3*x
*ln(x)-exp(x)+3*x^2+x)/(3*x*ln(x)-exp(x)+3*x^2+x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (20) = 40\).

Time = 0.31 (sec) , antiderivative size = 90, normalized size of antiderivative = 4.09 \[ \int \frac {-e^x+x+3 x^2+3 x \log (x)+e^{-2+\frac {e^x-x-3 x^2-3 x \log (x)}{e^2}} \left (4-e^x+6 x+3 \log (x)\right ) \left (e^x-x-3 x^2-3 x \log (x)\right )}{-e^x+x+3 x^2+3 x \log (x)} \, dx=\frac {3 \, x^{3} + 3 \, x^{2} \log \left (x\right ) + x^{2} - x e^{x} - e^{\left (-{\left (3 \, x^{2} - e^{2} \log \left (-3 \, x^{2} - 3 \, x \log \left (x\right ) - x + e^{x}\right ) + 3 \, x \log \left (x\right ) + x + 2 \, e^{2} - e^{x}\right )} e^{\left (-2\right )} + 2\right )}}{3 \, x^{2} + 3 \, x \log \left (x\right ) + x - e^{x}} \]

[In]

integrate(((3*log(x)-exp(x)+6*x+4)*exp(log(-3*x*log(x)+exp(x)-3*x^2-x)-2)*exp(exp(log(-3*x*log(x)+exp(x)-3*x^2
-x)-2))+3*x*log(x)-exp(x)+3*x^2+x)/(3*x*log(x)-exp(x)+3*x^2+x),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-e^x+x+3 x^2+3 x \log (x)+e^{-2+\frac {e^x-x-3 x^2-3 x \log (x)}{e^2}} \left (4-e^x+6 x+3 \log (x)\right ) \left (e^x-x-3 x^2-3 x \log (x)\right )}{-e^x+x+3 x^2+3 x \log (x)} \, dx=x + e^{\frac {- 3 x^{2} - 3 x \log {\left (x \right )} - x + e^{x}}{e^{2}}} \]

[In]

integrate(((3*ln(x)-exp(x)+6*x+4)*exp(ln(-3*x*ln(x)+exp(x)-3*x**2-x)-2)*exp(exp(ln(-3*x*ln(x)+exp(x)-3*x**2-x)
-2))+3*x*ln(x)-exp(x)+3*x**2+x)/(3*x*ln(x)-exp(x)+3*x**2+x),x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).

Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.00 \[ \int \frac {-e^x+x+3 x^2+3 x \log (x)+e^{-2+\frac {e^x-x-3 x^2-3 x \log (x)}{e^2}} \left (4-e^x+6 x+3 \log (x)\right ) \left (e^x-x-3 x^2-3 x \log (x)\right )}{-e^x+x+3 x^2+3 x \log (x)} \, dx={\left (x e^{\left (3 \, x^{2} e^{\left (-2\right )} + x e^{\left (-2\right )}\right )} + e^{\left (-3 \, x e^{\left (-2\right )} \log \left (x\right ) + e^{\left (x - 2\right )}\right )}\right )} e^{\left (-3 \, x^{2} e^{\left (-2\right )} - x e^{\left (-2\right )}\right )} \]

[In]

integrate(((3*log(x)-exp(x)+6*x+4)*exp(log(-3*x*log(x)+exp(x)-3*x^2-x)-2)*exp(exp(log(-3*x*log(x)+exp(x)-3*x^2
-x)-2))+3*x*log(x)-exp(x)+3*x^2+x)/(3*x*log(x)-exp(x)+3*x^2+x),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {-e^x+x+3 x^2+3 x \log (x)+e^{-2+\frac {e^x-x-3 x^2-3 x \log (x)}{e^2}} \left (4-e^x+6 x+3 \log (x)\right ) \left (e^x-x-3 x^2-3 x \log (x)\right )}{-e^x+x+3 x^2+3 x \log (x)} \, dx=\int { \frac {3 \, x^{2} + {\left (6 \, x - e^{x} + 3 \, \log \left (x\right ) + 4\right )} e^{\left (e^{\left (\log \left (-3 \, x^{2} - 3 \, x \log \left (x\right ) - x + e^{x}\right ) - 2\right )} + \log \left (-3 \, x^{2} - 3 \, x \log \left (x\right ) - x + e^{x}\right ) - 2\right )} + 3 \, x \log \left (x\right ) + x - e^{x}}{3 \, x^{2} + 3 \, x \log \left (x\right ) + x - e^{x}} \,d x } \]

[In]

integrate(((3*log(x)-exp(x)+6*x+4)*exp(log(-3*x*log(x)+exp(x)-3*x^2-x)-2)*exp(exp(log(-3*x*log(x)+exp(x)-3*x^2
-x)-2))+3*x*log(x)-exp(x)+3*x^2+x)/(3*x*log(x)-exp(x)+3*x^2+x),x, algorithm="giac")

[Out]

integrate((3*x^2 + (6*x - e^x + 3*log(x) + 4)*e^(e^(log(-3*x^2 - 3*x*log(x) - x + e^x) - 2) + log(-3*x^2 - 3*x
*log(x) - x + e^x) - 2) + 3*x*log(x) + x - e^x)/(3*x^2 + 3*x*log(x) + x - e^x), x)

Mupad [B] (verification not implemented)

Time = 15.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {-e^x+x+3 x^2+3 x \log (x)+e^{-2+\frac {e^x-x-3 x^2-3 x \log (x)}{e^2}} \left (4-e^x+6 x+3 \log (x)\right ) \left (e^x-x-3 x^2-3 x \log (x)\right )}{-e^x+x+3 x^2+3 x \log (x)} \, dx=x+\frac {{\mathrm {e}}^{-3\,x^2\,{\mathrm {e}}^{-2}}\,{\mathrm {e}}^{{\mathrm {e}}^{-2}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{-2}}}{x^{3\,x\,{\mathrm {e}}^{-2}}} \]

[In]

int((x - exp(x) + 3*x*log(x) + 3*x^2 + exp(log(exp(x) - x - 3*x*log(x) - 3*x^2) - 2)*exp(exp(log(exp(x) - x -
3*x*log(x) - 3*x^2) - 2))*(6*x - exp(x) + 3*log(x) + 4))/(x - exp(x) + 3*x*log(x) + 3*x^2),x)

[Out]

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