\(\int e^{-2 x} (-e^{2 x} x+e^{e^{-2 x} (x^2+e^{2 x} (2+4 x))} (2 x^3-2 x^4+e^{2 x} (2 x+4 x^2))-2 e^{2 x} x \log (x)) \, dx\) [1591]

Optimal result

Integrand size = 75, antiderivative size = 25 \[ \int e^{-2 x} \left (-e^{2 x} x+e^{e^{-2 x} \left (x^2+e^{2 x} (2+4 x)\right )} \left (2 x^3-2 x^4+e^{2 x} \left (2 x+4 x^2\right )\right )-2 e^{2 x} x \log (x)\right ) \, dx=x^2 \left (e^{2+4 x+e^{-2 x} x^2}-\log (x)\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int e^{-2 x} \left (-e^{2 x} x+e^{e^{-2 x} \left (x^2+e^{2 x} (2+4 x)\right )} \left (2 x^3-2 x^4+e^{2 x} \left (2 x+4 x^2\right )\right )-2 e^{2 x} x \log (x)\right ) \, dx=x^2 \left (e^{2+4 x+e^{-2 x} x^2}-\log (x)\right ) \]

[In]

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

[Out]

x^2*(E^(2 + 4*x + x^2/E^(2*x)) - Log[x])

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32

method	result	size
parallelrisch	\(-x^{2} \ln \left (x \right )+x^{2} {\mathrm e}^{\left (\left (4 x +2\right ) {\mathrm e}^{2 x}+x^{2}\right ) {\mathrm e}^{-2 x}}\)	\(33\)
risch	\(-x^{2} \ln \left (x \right )+{\mathrm e}^{\left (4 x \,{\mathrm e}^{2 x}+x^{2}+2 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{-2 x}} x^{2}\)	\(36\)
default	\(-x^{2} \ln \left (x \right )+{\mathrm e}^{\left (4 x \,{\mathrm e}^{2 x}+x^{2}+2 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{-2 x}} x^{2}\)	\(40\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int e^{-2 x} \left (-e^{2 x} x+e^{e^{-2 x} \left (x^2+e^{2 x} (2+4 x)\right )} \left (2 x^3-2 x^4+e^{2 x} \left (2 x+4 x^2\right )\right )-2 e^{2 x} x \log (x)\right ) \, dx=x^{2} e^{\left ({\left (x^{2} + 2 \, {\left (2 \, x + 1\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x\right )}\right )} - x^{2} \log \left (x\right ) \]

[In]

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

[Out]

x^2*e^((x^2 + 2*(2*x + 1)*e^(2*x))*e^(-2*x)) - x^2*log(x)

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int e^{-2 x} \left (-e^{2 x} x+e^{e^{-2 x} \left (x^2+e^{2 x} (2+4 x)\right )} \left (2 x^3-2 x^4+e^{2 x} \left (2 x+4 x^2\right )\right )-2 e^{2 x} x \log (x)\right ) \, dx=x^{2} e^{\left (x^{2} + \left (4 x + 2\right ) e^{2 x}\right ) e^{- 2 x}} - x^{2} \log {\left (x \right )} \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 8.65 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int e^{-2 x} \left (-e^{2 x} x+e^{e^{-2 x} \left (x^2+e^{2 x} (2+4 x)\right )} \left (2 x^3-2 x^4+e^{2 x} \left (2 x+4 x^2\right )\right )-2 e^{2 x} x \log (x)\right ) \, dx=x^2\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^2\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{-2\,x}}-x^2\,\ln \left (x\right ) \]

[In]

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

[Out]

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