\(\int e^{-x} (e^{2 e^{-x} x} (2-2 x)-6 e^x \log (x)-3 e^x \log ^2(x)) \, dx\) [7931]

Optimal result

Integrand size = 39, antiderivative size = 21 \[ \int e^{-x} \left (e^{2 e^{-x} x} (2-2 x)-6 e^x \log (x)-3 e^x \log ^2(x)\right ) \, dx=\frac {7}{2}+e^{2 e^{-x} x}-3 x \log ^2(x) \]

[Out]

7/2-3*x*ln(x)^2+exp(x/exp(x))^2

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int e^{-x} \left (e^{2 e^{-x} x} (2-2 x)-6 e^x \log (x)-3 e^x \log ^2(x)\right ) \, dx=e^{2 e^{-x} x}-3 x \log ^2(x) \]

[In]

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

[Out]

E^((2*x)/E^x) - 3*x*Log[x]^2

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81

method	result	size
risch	\({\mathrm e}^{2 x \,{\mathrm e}^{-x}}-3 x \ln \left (x \right )^{2}\)	\(17\)
default	\({\mathrm e}^{2 x \,{\mathrm e}^{-x}}-3 x \ln \left (x \right )^{2}\)	\(18\)
parallelrisch	\({\mathrm e}^{2 x \,{\mathrm e}^{-x}}-3 x \ln \left (x \right )^{2}\)	\(18\)
parts	\({\mathrm e}^{2 x \,{\mathrm e}^{-x}}-3 x \ln \left (x \right )^{2}\)	\(18\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int e^{-x} \left (e^{2 e^{-x} x} (2-2 x)-6 e^x \log (x)-3 e^x \log ^2(x)\right ) \, dx=-3 \, x \log \left (x\right )^{2} + e^{\left (2 \, x e^{\left (-x\right )}\right )} \]

[In]

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

[Out]

-3*x*log(x)^2 + e^(2*x*e^(-x))

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int e^{-x} \left (e^{2 e^{-x} x} (2-2 x)-6 e^x \log (x)-3 e^x \log ^2(x)\right ) \, dx=- 3 x \log {\left (x \right )}^{2} + e^{2 x e^{- x}} \]

[In]

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

[Out]

-3*x*log(x)**2 + exp(2*x*exp(-x))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int e^{-x} \left (e^{2 e^{-x} x} (2-2 x)-6 e^x \log (x)-3 e^x \log ^2(x)\right ) \, dx=-3 \, x \log \left (x\right )^{2} + e^{\left (2 \, x e^{\left (-x\right )}\right )} \]

[In]

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

[Out]

-3*x*log(x)^2 + e^(2*x*e^(-x))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43 \[ \int e^{-x} \left (e^{2 e^{-x} x} (2-2 x)-6 e^x \log (x)-3 e^x \log ^2(x)\right ) \, dx=-{\left (3 \, x e^{\left (-x\right )} \log \left (x\right )^{2} - e^{\left (2 \, x e^{\left (-x\right )} - x\right )}\right )} e^{x} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 14.46 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int e^{-x} \left (e^{2 e^{-x} x} (2-2 x)-6 e^x \log (x)-3 e^x \log ^2(x)\right ) \, dx={\mathrm {e}}^{2\,x\,{\mathrm {e}}^{-x}}-3\,x\,{\ln \left (x\right )}^2 \]

[In]

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

[Out]

exp(2*x*exp(-x)) - 3*x*log(x)^2