3.32.40 \(\int e^{-x+2 e^{-x} (-4+20 x+4 x \log (-\frac {x}{4}))} (14-10 x+(2-2 x) \log (-\frac {x}{4})) \, dx\)

Optimal. Leaf size=25 \[ \frac {1}{4} e^{8 e^{-x} \left (-1+x \left (5+\log \left (-\frac {x}{4}\right )\right )\right )} \]

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Rubi [F]  time = 2.11, 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 \exp \left (-x+2 e^{-x} \left (-4+20 x+4 x \log \left (-\frac {x}{4}\right )\right )\right ) \left (14-10 x+(2-2 x) \log \left (-\frac {x}{4}\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(-x + (2*(-4 + 20*x + 4*x*Log[-1/4*x]))/E^x)*(14 - 10*x + (2 - 2*x)*Log[-1/4*x]),x]

[Out]

14*Defer[Int][E^(-((8 - 40*x + E^x*x - 8*x*Log[-1/4*x])/E^x)), x] - 10*Defer[Int][x/E^((8 - 40*x + E^x*x - 8*x
*Log[-1/4*x])/E^x), x] + 2*Defer[Int][Log[-1/4*x]/E^((8 - 40*x + E^x*x - 8*x*Log[-1/4*x])/E^x), x] - 2*Defer[I
nt][(x*Log[-1/4*x])/E^((8 - 40*x + E^x*x - 8*x*Log[-1/4*x])/E^x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \exp \left (-e^{-x} \left (8-40 x+e^x x-8 x \log \left (-\frac {x}{4}\right )\right )\right ) \left (14-10 x+(2-2 x) \log \left (-\frac {x}{4}\right )\right ) \, dx\\ &=\int \left (14 \exp \left (-e^{-x} \left (8-40 x+e^x x-8 x \log \left (-\frac {x}{4}\right )\right )\right )-10 \exp \left (-e^{-x} \left (8-40 x+e^x x-8 x \log \left (-\frac {x}{4}\right )\right )\right ) x-2 \exp \left (-e^{-x} \left (8-40 x+e^x x-8 x \log \left (-\frac {x}{4}\right )\right )\right ) (-1+x) \log \left (-\frac {x}{4}\right )\right ) \, dx\\ &=-\left (2 \int \exp \left (-e^{-x} \left (8-40 x+e^x x-8 x \log \left (-\frac {x}{4}\right )\right )\right ) (-1+x) \log \left (-\frac {x}{4}\right ) \, dx\right )-10 \int \exp \left (-e^{-x} \left (8-40 x+e^x x-8 x \log \left (-\frac {x}{4}\right )\right )\right ) x \, dx+14 \int \exp \left (-e^{-x} \left (8-40 x+e^x x-8 x \log \left (-\frac {x}{4}\right )\right )\right ) \, dx\\ &=-\left (2 \int \left (-\exp \left (-e^{-x} \left (8-40 x+e^x x-8 x \log \left (-\frac {x}{4}\right )\right )\right ) \log \left (-\frac {x}{4}\right )+\exp \left (-e^{-x} \left (8-40 x+e^x x-8 x \log \left (-\frac {x}{4}\right )\right )\right ) x \log \left (-\frac {x}{4}\right )\right ) \, dx\right )-10 \int \exp \left (-e^{-x} \left (8-40 x+e^x x-8 x \log \left (-\frac {x}{4}\right )\right )\right ) x \, dx+14 \int \exp \left (-e^{-x} \left (8-40 x+e^x x-8 x \log \left (-\frac {x}{4}\right )\right )\right ) \, dx\\ &=2 \int \exp \left (-e^{-x} \left (8-40 x+e^x x-8 x \log \left (-\frac {x}{4}\right )\right )\right ) \log \left (-\frac {x}{4}\right ) \, dx-2 \int \exp \left (-e^{-x} \left (8-40 x+e^x x-8 x \log \left (-\frac {x}{4}\right )\right )\right ) x \log \left (-\frac {x}{4}\right ) \, dx-10 \int \exp \left (-e^{-x} \left (8-40 x+e^x x-8 x \log \left (-\frac {x}{4}\right )\right )\right ) x \, dx+14 \int \exp \left (-e^{-x} \left (8-40 x+e^x x-8 x \log \left (-\frac {x}{4}\right )\right )\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.56, size = 39, normalized size = 1.56 \begin {gather*} 4^{-1-8 e^{-x} x} e^{-8 e^{-x} (1-5 x)} (-x)^{8 e^{-x} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-x + (2*(-4 + 20*x + 4*x*Log[-1/4*x]))/E^x)*(14 - 10*x + (2 - 2*x)*Log[-1/4*x]),x]

[Out]

(4^(-1 - (8*x)/E^x)*(-x)^((8*x)/E^x))/E^((8*(1 - 5*x))/E^x)

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fricas [A]  time = 0.68, size = 26, normalized size = 1.04 \begin {gather*} \frac {1}{4} \, e^{\left (8 \, x e^{\left (-x\right )} \log \left (-\frac {1}{4} \, x\right ) + 8 \, {\left (5 \, x - 1\right )} e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+2)*log(-1/4*x)-10*x+14)*exp((4*x*log(-1/4*x)+20*x-4)/exp(x))^2/exp(x),x, algorithm="fricas")

[Out]

1/4*e^(8*x*e^(-x)*log(-1/4*x) + 8*(5*x - 1)*e^(-x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -2 \, {\left ({\left (x - 1\right )} \log \left (-\frac {1}{4} \, x\right ) + 5 \, x - 7\right )} e^{\left (8 \, {\left (x \log \left (-\frac {1}{4} \, x\right ) + 5 \, x - 1\right )} e^{\left (-x\right )} - x\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+2)*log(-1/4*x)-10*x+14)*exp((4*x*log(-1/4*x)+20*x-4)/exp(x))^2/exp(x),x, algorithm="giac")

[Out]

integrate(-2*((x - 1)*log(-1/4*x) + 5*x - 7)*e^(8*(x*log(-1/4*x) + 5*x - 1)*e^(-x) - x), x)

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maple [A]  time = 0.03, size = 21, normalized size = 0.84

method	result	size
risch	\(\frac {{\mathrm e}^{8 \left (x \ln \left (-\frac {x}{4}\right )+5 x -1\right ) {\mathrm e}^{-x}}}{4}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x+2)*ln(-1/4*x)-10*x+14)*exp((4*x*ln(-1/4*x)+20*x-4)/exp(x))^2/exp(x),x,method=_RETURNVERBOSE)

[Out]

1/4*exp(8*(x*ln(-1/4*x)+5*x-1)*exp(-x))

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maxima [A]  time = 0.85, size = 37, normalized size = 1.48 \begin {gather*} \frac {1}{4} \, e^{\left (-16 \, x e^{\left (-x\right )} \log \relax (2) + 8 \, x e^{\left (-x\right )} \log \left (-x\right ) + 40 \, x e^{\left (-x\right )} - 8 \, e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+2)*log(-1/4*x)-10*x+14)*exp((4*x*log(-1/4*x)+20*x-4)/exp(x))^2/exp(x),x, algorithm="maxima")

[Out]

1/4*e^(-16*x*e^(-x)*log(2) + 8*x*e^(-x)*log(-x) + 40*x*e^(-x) - 8*e^(-x))

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mupad [B]  time = 1.92, size = 28, normalized size = 1.12 \begin {gather*} \frac {{\mathrm {e}}^{-8\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{40\,x\,{\mathrm {e}}^{-x}}\,{\left (-\frac {x}{4}\right )}^{8\,x\,{\mathrm {e}}^{-x}}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-x)*exp(2*exp(-x)*(20*x + 4*x*log(-x/4) - 4))*(10*x + log(-x/4)*(2*x - 2) - 14),x)

[Out]

(exp(-8*exp(-x))*exp(40*x*exp(-x))*(-x/4)^(8*x*exp(-x)))/4

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sympy [A]  time = 0.51, size = 20, normalized size = 0.80 \begin {gather*} \frac {e^{2 \left (4 x \log {\left (- \frac {x}{4} \right )} + 20 x - 4\right ) e^{- x}}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+2)*ln(-1/4*x)-10*x+14)*exp((4*x*ln(-1/4*x)+20*x-4)/exp(x))**2/exp(x),x)

[Out]

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

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