3.18.84 \(\int e^{-4+\frac {2-4 e^{4+x}-\log (x)}{e^4}} (-x+2 e^4 x+(1-e^4) \log (20)+e^x (-4 e^4 x^2+4 e^4 x \log (20))) \, dx\)

Optimal. Leaf size=26 \[ e^{-4 e^x+\frac {2-\log (x)}{e^4}} x (x-\log (20)) \]

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Rubi [F]  time = 2.22, 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^{-4+\frac {2-4 e^{4+x}-\log (x)}{e^4}} \left (-x+2 e^4 x+\left (1-e^4\right ) \log (20)+e^x \left (-4 e^4 x^2+4 e^4 x \log (20)\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(-4 + (2 - 4*E^(4 + x) - Log[x])/E^4)*(-x + 2*E^4*x + (1 - E^4)*Log[20] + E^x*(-4*E^4*x^2 + 4*E^4*x*Log[
20])),x]

[Out]

-((1 - 2*E^4)*Defer[Int][x^(1 - E^(-4))/E^((2*(-1 + 2*E^4 + 2*E^(4 + x)))/E^4), x]) + 4*Log[20]*Defer[Int][E^(
2/E^4 - 4*E^x + x)*x^(1 - E^(-4)), x] - 4*Defer[Int][E^(2/E^4 - 4*E^x + x)*x^(2 - E^(-4)), x] + (1 - E^4)*Log[
20]*Defer[Int][1/(E^((2*(-1 + 2*E^4 + 2*E^(4 + x)))/E^4)*x^E^(-4)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{-4+\frac {2-4 e^{4+x}-\log (x)}{e^4}} \left (\left (-1+2 e^4\right ) x+\left (1-e^4\right ) \log (20)+e^x \left (-4 e^4 x^2+4 e^4 x \log (20)\right )\right ) \, dx\\ &=\int e^{-\frac {4 e^{4+x}-2 \left (1-2 e^4\right )+\log (x)}{e^4}} \left (\left (-1+2 e^4\right ) x+\left (1-e^4\right ) \log (20)+e^x \left (-4 e^4 x^2+4 e^4 x \log (20)\right )\right ) \, dx\\ &=\int e^{-\frac {4 e^{4+x}-2 \left (1-2 e^4\right )}{e^4}} x^{-\frac {1}{e^4}} \left (\left (-1+2 e^4\right ) x+\left (1-e^4\right ) \log (20)+e^x \left (-4 e^4 x^2+4 e^4 x \log (20)\right )\right ) \, dx\\ &=\int e^{-\frac {2 \left (-1+2 e^4+2 e^{4+x}\right )}{e^4}} x^{-\frac {1}{e^4}} \left (\left (-1+2 e^4\right ) x+\left (1-e^4\right ) \log (20)+e^x \left (-4 e^4 x^2+4 e^4 x \log (20)\right )\right ) \, dx\\ &=\int \left (e^{-\frac {2 \left (-1+2 e^4+2 e^{4+x}\right )}{e^4}} \left (-1+2 e^4\right ) x^{1-\frac {1}{e^4}}-4 e^{4-\frac {2 \left (-1+2 e^4+2 e^{4+x}\right )}{e^4}+x} x^{1-\frac {1}{e^4}} (x-\log (20))-e^{-\frac {2 \left (-1+2 e^4+2 e^{4+x}\right )}{e^4}} \left (-1+e^4\right ) x^{-\frac {1}{e^4}} \log (20)\right ) \, dx\\ &=-\left (4 \int e^{4-\frac {2 \left (-1+2 e^4+2 e^{4+x}\right )}{e^4}+x} x^{1-\frac {1}{e^4}} (x-\log (20)) \, dx\right )+\left (-1+2 e^4\right ) \int e^{-\frac {2 \left (-1+2 e^4+2 e^{4+x}\right )}{e^4}} x^{1-\frac {1}{e^4}} \, dx+\left (\left (1-e^4\right ) \log (20)\right ) \int e^{-\frac {2 \left (-1+2 e^4+2 e^{4+x}\right )}{e^4}} x^{-\frac {1}{e^4}} \, dx\\ &=-\left (4 \int e^{\frac {2}{e^4}-4 e^x+x} x^{1-\frac {1}{e^4}} (x-\log (20)) \, dx\right )+\left (-1+2 e^4\right ) \int e^{-\frac {2 \left (-1+2 e^4+2 e^{4+x}\right )}{e^4}} x^{1-\frac {1}{e^4}} \, dx+\left (\left (1-e^4\right ) \log (20)\right ) \int e^{-\frac {2 \left (-1+2 e^4+2 e^{4+x}\right )}{e^4}} x^{-\frac {1}{e^4}} \, dx\\ &=-\left (4 \int \left (e^{\frac {2}{e^4}-4 e^x+x} x^{2-\frac {1}{e^4}}-e^{\frac {2}{e^4}-4 e^x+x} x^{1-\frac {1}{e^4}} \log (20)\right ) \, dx\right )+\left (-1+2 e^4\right ) \int e^{-\frac {2 \left (-1+2 e^4+2 e^{4+x}\right )}{e^4}} x^{1-\frac {1}{e^4}} \, dx+\left (\left (1-e^4\right ) \log (20)\right ) \int e^{-\frac {2 \left (-1+2 e^4+2 e^{4+x}\right )}{e^4}} x^{-\frac {1}{e^4}} \, dx\\ &=-\left (4 \int e^{\frac {2}{e^4}-4 e^x+x} x^{2-\frac {1}{e^4}} \, dx\right )+\left (-1+2 e^4\right ) \int e^{-\frac {2 \left (-1+2 e^4+2 e^{4+x}\right )}{e^4}} x^{1-\frac {1}{e^4}} \, dx+(4 \log (20)) \int e^{\frac {2}{e^4}-4 e^x+x} x^{1-\frac {1}{e^4}} \, dx+\left (\left (1-e^4\right ) \log (20)\right ) \int e^{-\frac {2 \left (-1+2 e^4+2 e^{4+x}\right )}{e^4}} x^{-\frac {1}{e^4}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.71, size = 29, normalized size = 1.12 \begin {gather*} e^{\frac {2}{e^4}-4 e^x} x^{1-\frac {1}{e^4}} (x-\log (20)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-4 + (2 - 4*E^(4 + x) - Log[x])/E^4)*(-x + 2*E^4*x + (1 - E^4)*Log[20] + E^x*(-4*E^4*x^2 + 4*E^4*
x*Log[20])),x]

[Out]

E^(2/E^4 - 4*E^x)*x^(1 - E^(-4))*(x - Log[20])

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fricas [A]  time = 0.70, size = 34, normalized size = 1.31 \begin {gather*} {\left (x^{2} e^{4} - x e^{4} \log \left (20\right )\right )} e^{\left (-{\left (4 \, e^{4} + 4 \, e^{\left (x + 4\right )} + \log \relax (x) - 2\right )} e^{\left (-4\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*exp(4)*log(20)-4*x^2*exp(4))*exp(x)+(1-exp(4))*log(20)+2*x*exp(4)-x)*exp((-log(x)-4*exp(4)*exp
(x)+2)/exp(4))/exp(4),x, algorithm="fricas")

[Out]

(x^2*e^4 - x*e^4*log(20))*e^(-(4*e^4 + 4*e^(x + 4) + log(x) - 2)*e^(-4))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (2 \, x e^{4} - 4 \, {\left (x^{2} e^{4} - x e^{4} \log \left (20\right )\right )} e^{x} - {\left (e^{4} - 1\right )} \log \left (20\right ) - x\right )} e^{\left (-{\left (4 \, e^{\left (x + 4\right )} + \log \relax (x) - 2\right )} e^{\left (-4\right )} - 4\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*exp(4)*log(20)-4*x^2*exp(4))*exp(x)+(1-exp(4))*log(20)+2*x*exp(4)-x)*exp((-log(x)-4*exp(4)*exp
(x)+2)/exp(4))/exp(4),x, algorithm="giac")

[Out]

integrate((2*x*e^4 - 4*(x^2*e^4 - x*e^4*log(20))*e^x - (e^4 - 1)*log(20) - x)*e^(-(4*e^(x + 4) + log(x) - 2)*e
^(-4) - 4), x)

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maple [A]  time = 0.22, size = 44, normalized size = 1.69

method	result	size
risch	\(\left (-x \,{\mathrm e}^{4} \ln \relax (5)-2 x \,{\mathrm e}^{4} \ln \relax (2)+x^{2} {\mathrm e}^{4}\right ) {\mathrm e}^{-{\mathrm e}^{-4} \ln \relax (x )-4 \,{\mathrm e}^{-4} {\mathrm e}^{4+x}+2 \,{\mathrm e}^{-4}-4}\)	\(44\)
norman	\(x^{2} {\mathrm e}^{\left (-\ln \relax (x )-4 \,{\mathrm e}^{4} {\mathrm e}^{x}+2\right ) {\mathrm e}^{-4}}-x \ln \left (20\right ) {\mathrm e}^{\left (-\ln \relax (x )-4 \,{\mathrm e}^{4} {\mathrm e}^{x}+2\right ) {\mathrm e}^{-4}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x*exp(4)*ln(20)-4*x^2*exp(4))*exp(x)+(1-exp(4))*ln(20)+2*x*exp(4)-x)*exp((-ln(x)-4*exp(4)*exp(x)+2)/ex
p(4))/exp(4),x,method=_RETURNVERBOSE)

[Out]

(-x*exp(4)*ln(5)-2*x*exp(4)*ln(2)+x^2*exp(4))*exp(-exp(-4)*ln(x)-4*exp(-4)*exp(4+x)+2*exp(-4)-4)

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maxima [A]  time = 1.04, size = 38, normalized size = 1.46 \begin {gather*} {\left (x^{2} e^{\left (2 \, e^{\left (-4\right )}\right )} - x {\left (\log \relax (5) + 2 \, \log \relax (2)\right )} e^{\left (2 \, e^{\left (-4\right )}\right )}\right )} e^{\left (-e^{\left (-4\right )} \log \relax (x) - 4 \, e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*exp(4)*log(20)-4*x^2*exp(4))*exp(x)+(1-exp(4))*log(20)+2*x*exp(4)-x)*exp((-log(x)-4*exp(4)*exp
(x)+2)/exp(4))/exp(4),x, algorithm="maxima")

[Out]

(x^2*e^(2*e^(-4)) - x*(log(5) + 2*log(2))*e^(2*e^(-4)))*e^(-e^(-4)*log(x) - 4*e^x)

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mupad [B]  time = 1.30, size = 24, normalized size = 0.92 \begin {gather*} \frac {x\,{\mathrm {e}}^{2\,{\mathrm {e}}^{-4}-4\,{\mathrm {e}}^x}\,\left (x-\ln \left (20\right )\right )}{x^{{\mathrm {e}}^{-4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-exp(-4)*(log(x) + 4*exp(4)*exp(x) - 2))*exp(-4)*(x - 2*x*exp(4) + log(20)*(exp(4) - 1) + exp(x)*(4*x
^2*exp(4) - 4*x*exp(4)*log(20))),x)

[Out]

(x*exp(2*exp(-4) - 4*exp(x))*(x - log(20)))/x^exp(-4)

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sympy [A]  time = 0.42, size = 26, normalized size = 1.00 \begin {gather*} \left (x^{2} - x \log {\left (20 \right )}\right ) e^{\frac {- 4 e^{4} e^{x} - \log {\relax (x )} + 2}{e^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*exp(4)*ln(20)-4*x**2*exp(4))*exp(x)+(1-exp(4))*ln(20)+2*x*exp(4)-x)*exp((-ln(x)-4*exp(4)*exp(x
)+2)/exp(4))/exp(4),x)

[Out]

(x**2 - x*log(20))*exp((-4*exp(4)*exp(x) - log(x) + 2)*exp(-4))

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