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

Optimal. Leaf size=21 \[ \frac {7}{2}+e^{2 e^{-x} x}-3 x \log ^2(x) \]

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Rubi [F]  time = 0.37, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 18, normalized size = 0.86 \begin {gather*} e^{2 e^{-x} x}-3 x \log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [A]  time = 1.19, size = 16, normalized size = 0.76 \begin {gather*} -3 \, x \log \relax (x)^{2} + e^{\left (2 \, x e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+2)*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))

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giac [A]  time = 0.19, size = 30, normalized size = 1.43 \begin {gather*} -{\left (3 \, x e^{\left (-x\right )} \log \relax (x)^{2} - e^{\left (2 \, x e^{\left (-x\right )} - x\right )}\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+2)*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

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maple [A]  time = 0.04, size = 17, normalized size = 0.81

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x+2)*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

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maxima [A]  time = 0.40, size = 16, normalized size = 0.76 \begin {gather*} -3 \, x \log \relax (x)^{2} + e^{\left (2 \, x e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+2)*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))

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mupad [B]  time = 4.75, size = 16, normalized size = 0.76 \begin {gather*} {\mathrm {e}}^{2\,x\,{\mathrm {e}}^{-x}}-3\,x\,{\ln \relax (x)}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.34, size = 15, normalized size = 0.71 \begin {gather*} - 3 x \log {\relax (x )}^{2} + e^{2 x e^{- x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+2)*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))

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