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

Optimal. Leaf size=25 \[ x^2 \left (e^{2+4 x+e^{-2 x} x^2}-\log (x)\right ) \]

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Rubi [F]  time = 1.23, 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^{-2 x} \left (-e^{2 x} x+e^{e^{-2 x} \left (x^2+e^{2 x} (2+4 x)\right )} \left (2 x^3-2 x^4+e^{2 x} \left (2 x+4 x^2\right )\right )-2 e^{2 x} x \log (x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int x \left (-1+2 e^{2+2 x+e^{-2 x} x^2} \left (-\left ((-1+x) x^2\right )+e^{2 x} (1+2 x)\right )-2 \log (x)\right ) \, dx\\ &=\int \left (2 e^{2+2 x+e^{-2 x} x^2} x \left (e^{2 x}+2 e^{2 x} x+x^2-x^3\right )-x (1+2 \log (x))\right ) \, dx\\ &=2 \int e^{2+2 x+e^{-2 x} x^2} x \left (e^{2 x}+2 e^{2 x} x+x^2-x^3\right ) \, dx-\int x (1+2 \log (x)) \, dx\\ &=-x^2 \log (x)+2 \int \left (-e^{2+2 x+e^{-2 x} x^2} (-1+x) x^3+e^{2+4 x+e^{-2 x} x^2} x (1+2 x)\right ) \, dx\\ &=-x^2 \log (x)-2 \int e^{2+2 x+e^{-2 x} x^2} (-1+x) x^3 \, dx+2 \int e^{2+4 x+e^{-2 x} x^2} x (1+2 x) \, dx\\ &=-x^2 \log (x)+2 \int \left (e^{2+4 x+e^{-2 x} x^2} x+2 e^{2+4 x+e^{-2 x} x^2} x^2\right ) \, dx-2 \int \left (-e^{2+2 x+e^{-2 x} x^2} x^3+e^{2+2 x+e^{-2 x} x^2} x^4\right ) \, dx\\ &=-x^2 \log (x)+2 \int e^{2+4 x+e^{-2 x} x^2} x \, dx+2 \int e^{2+2 x+e^{-2 x} x^2} x^3 \, dx-2 \int e^{2+2 x+e^{-2 x} x^2} x^4 \, dx+4 \int e^{2+4 x+e^{-2 x} x^2} x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.44, size = 25, normalized size = 1.00 \begin {gather*} x^2 \left (e^{2+4 x+e^{-2 x} x^2}-\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.62, size = 33, normalized size = 1.32 \begin {gather*} x^{2} e^{\left ({\left (x^{2} + 2 \, {\left (2 \, x + 1\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x\right )}\right )} - x^{2} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*e^((x^2 + 2*(2*x + 1)*e^(2*x))*e^(-2*x)) - x^2*log(x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -{\left (2 \, x e^{\left (2 \, x\right )} \log \relax (x) + 2 \, {\left (x^{4} - x^{3} - {\left (2 \, x^{2} + x\right )} e^{\left (2 \, x\right )}\right )} e^{\left ({\left (x^{2} + 2 \, {\left (2 \, x + 1\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x\right )}\right )} + x e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-(2*x*e^(2*x)*log(x) + 2*(x^4 - x^3 - (2*x^2 + x)*e^(2*x))*e^((x^2 + 2*(2*x + 1)*e^(2*x))*e^(-2*x))
+ x*e^(2*x))*e^(-2*x), x)

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maple [A]  time = 0.15, size = 36, normalized size = 1.44

method	result	size
risch	\(-x^{2} \ln \relax (x )+{\mathrm e}^{\left (4 x \,{\mathrm e}^{2 x}+x^{2}+2 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{-2 x}} x^{2}\)	\(36\)
default	\(-x^{2} \ln \relax (x )+{\mathrm e}^{\left (4 x \,{\mathrm e}^{2 x}+x^{2}+2 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{-2 x}} x^{2}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -x^{2} \log \relax (x) - \int -2 \, {\left ({\left (2 \, x^{2} e^{2} + x e^{2}\right )} e^{\left (4 \, x\right )} - {\left (x^{4} e^{2} - x^{3} e^{2}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (x^{2} e^{\left (-2 \, x\right )}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2*log(x) - integrate(-2*((2*x^2*e^2 + x*e^2)*e^(4*x) - (x^4*e^2 - x^3*e^2)*e^(2*x))*e^(x^2*e^(-2*x)), x)

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mupad [B]  time = 1.32, size = 27, normalized size = 1.08 \begin {gather*} x^2\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^2\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{-2\,x}}-x^2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.42, size = 29, normalized size = 1.16 \begin {gather*} x^{2} e^{\left (x^{2} + \left (4 x + 2\right ) e^{2 x}\right ) e^{- 2 x}} - x^{2} \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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