3.51.25 \(\int e^{-x+e^{-x} (e^{2 x}+e^x (-3-x)+x-e^x \log (x)+e^x \log (4 x))} (e^x (1-x)+x+e^{2 x} x-x^2) \, dx\)

Optimal. Leaf size=20 \[ 4 e^{-3+e^x-x+e^{-x} x} x \]

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Rubi [F]  time = 0.98, 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+e^{-x} \left (e^{2 x}+e^x (-3-x)+x-e^x \log (x)+e^x \log (4 x)\right )\right ) \left (e^x (1-x)+x+e^{2 x} x-x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(-x + (E^(2*x) + E^x*(-3 - x) + x - E^x*Log[x] + E^x*Log[4*x])/E^x)*(E^x*(1 - x) + x + E^(2*x)*x - x^2),
x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int 4 e^{-3+e^x-2 x+e^{-x} x} \left (e^x+x-e^x x+e^{2 x} x-x^2\right ) \, dx\\ &=4 \int e^{-3+e^x-2 x+e^{-x} x} \left (e^x+x-e^x x+e^{2 x} x-x^2\right ) \, dx\\ &=4 \int \left (e^{-3+e^x-x+e^{-x} x}+e^{-3+e^x+e^{-x} x} x+e^{-3+e^x-2 x+e^{-x} x} x-e^{-3+e^x-x+e^{-x} x} x-e^{-3+e^x-2 x+e^{-x} x} x^2\right ) \, dx\\ &=4 \int e^{-3+e^x-x+e^{-x} x} \, dx+4 \int e^{-3+e^x+e^{-x} x} x \, dx+4 \int e^{-3+e^x-2 x+e^{-x} x} x \, dx-4 \int e^{-3+e^x-x+e^{-x} x} x \, dx-4 \int e^{-3+e^x-2 x+e^{-x} x} x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.61, size = 20, normalized size = 1.00 \begin {gather*} 4 e^{-3+e^x-x+e^{-x} x} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-x + (E^(2*x) + E^x*(-3 - x) + x - E^x*Log[x] + E^x*Log[4*x])/E^x)*(E^x*(1 - x) + x + E^(2*x)*x -
x^2),x]

[Out]

4*E^(-3 + E^x - x + x/E^x)*x

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fricas [A]  time = 0.73, size = 33, normalized size = 1.65 \begin {gather*} x e^{\left (-{\left ({\left (2 \, x - 2 \, \log \relax (2) + 3\right )} e^{x} - x - e^{\left (2 \, x\right )}\right )} e^{\left (-x\right )} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(x)^2+(-x+1)*exp(x)-x^2+x)*exp((exp(x)*log(4*x)-exp(x)*log(x)+exp(x)^2+(-3-x)*exp(x)+x)/exp(x)
)/exp(x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 2.18, size = 20, normalized size = 1.00 \begin {gather*} 4 \, x e^{\left ({\left (x + e^{\left (2 \, x\right )}\right )} e^{\left (-x\right )} - x - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(x)^2+(-x+1)*exp(x)-x^2+x)*exp((exp(x)*log(4*x)-exp(x)*log(x)+exp(x)^2+(-3-x)*exp(x)+x)/exp(x)
)/exp(x),x, algorithm="giac")

[Out]

4*x*e^((x + e^(2*x))*e^(-x) - x - 3)

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maple [A]  time = 0.23, size = 30, normalized size = 1.50

method	result	size
risch	\(x \,{\mathrm e}^{\left (2 \,{\mathrm e}^{x} \ln \relax (2)-{\mathrm e}^{x} x -3 \,{\mathrm e}^{x}+{\mathrm e}^{2 x}+x \right ) {\mathrm e}^{-x}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*exp(x)^2+(1-x)*exp(x)-x^2+x)*exp((exp(x)*ln(4*x)-exp(x)*ln(x)+exp(x)^2+(-3-x)*exp(x)+x)/exp(x))/exp(x),
x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.47, size = 17, normalized size = 0.85 \begin {gather*} 4 \, x e^{\left (x e^{\left (-x\right )} - x + e^{x} - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(x)^2+(-x+1)*exp(x)-x^2+x)*exp((exp(x)*log(4*x)-exp(x)*log(x)+exp(x)^2+(-3-x)*exp(x)+x)/exp(x)
)/exp(x),x, algorithm="maxima")

[Out]

4*x*e^(x*e^(-x) - x + e^x - 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int {\mathrm {e}}^{-x}\,{\mathrm {e}}^{{\mathrm {e}}^{-x}\,\left (x+{\mathrm {e}}^{2\,x}-{\mathrm {e}}^x\,\left (x+3\right )-{\mathrm {e}}^x\,\ln \relax (x)+\ln \left (4\,x\right )\,{\mathrm {e}}^x\right )}\,\left (x+x\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^x\,\left (x-1\right )-x^2\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-x)*exp(exp(-x)*(x + exp(2*x) - exp(x)*(x + 3) - exp(x)*log(x) + log(4*x)*exp(x)))*(x + x*exp(2*x) - e
xp(x)*(x - 1) - x^2),x)

[Out]

int(exp(-x)*exp(exp(-x)*(x + exp(2*x) - exp(x)*(x + 3) - exp(x)*log(x) + log(4*x)*exp(x)))*(x + x*exp(2*x) - e
xp(x)*(x - 1) - x^2), x)

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sympy [B]  time = 0.43, size = 37, normalized size = 1.85 \begin {gather*} x e^{\left (x + \left (- x - 3\right ) e^{x} + \left (\log {\relax (x )} + \log {\relax (4 )}\right ) e^{x} + e^{2 x} - e^{x} \log {\relax (x )}\right ) e^{- x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(x)**2+(-x+1)*exp(x)-x**2+x)*exp((exp(x)*ln(4*x)-exp(x)*ln(x)+exp(x)**2+(-3-x)*exp(x)+x)/exp(x
))/exp(x),x)

[Out]

x*exp((x + (-x - 3)*exp(x) + (log(x) + log(4))*exp(x) + exp(2*x) - exp(x)*log(x))*exp(-x))

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