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

Optimal. Leaf size=25 \[ e^2+e^{\frac {2 x-e^{-2 x} x^4}{x^2}}+x \]

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Rubi [F]  time = 1.22, 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 \frac {e^{-2 x} \left (e^{2 x} x^2+e^{\frac {e^{-2 x} \left (2 e^{2 x}-x^3\right )}{x}} \left (-2 e^{2 x}-2 x^3+2 x^4\right )\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.34, size = 20, normalized size = 0.80 \begin {gather*} e^{\frac {2}{x}-e^{-2 x} x^2}+x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(2/x - x^2/E^(2*x)) + x

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fricas [A]  time = 0.48, size = 22, normalized size = 0.88 \begin {gather*} x + e^{\left (-\frac {{\left (x^{3} - 2 \, e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + e^(-(x^3 - 2*e^(2*x))*e^(-2*x)/x)

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giac [A]  time = 0.34, size = 18, normalized size = 0.72 \begin {gather*} x + e^{\left (-x^{2} e^{\left (-2 \, x\right )} + \frac {2}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + e^(-x^2*e^(-2*x) + 2/x)

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maple [A]  time = 0.10, size = 23, normalized size = 0.92

method	result	size
risch	\(x +{\mathrm e}^{-\frac {\left (-2 \,{\mathrm e}^{2 x}+x^{3}\right ) {\mathrm e}^{-2 x}}{x}}\)	\(23\)
norman	\(\frac {\left ({\mathrm e}^{2 x} x^{2}+x \,{\mathrm e}^{2 x} {\mathrm e}^{\frac {\left (2 \,{\mathrm e}^{2 x}-x^{3}\right ) {\mathrm e}^{-2 x}}{x}}\right ) {\mathrm e}^{-2 x}}{x}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*exp(x)^2+2*x^4-2*x^3)*exp((2*exp(x)^2-x^3)/x/exp(x)^2)+exp(x)^2*x^2)/exp(x)^2/x^2,x,method=_RETURNVER
BOSE)

[Out]

x+exp(-(-2*exp(2*x)+x^3)/x*exp(-2*x))

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maxima [A]  time = 0.62, size = 18, normalized size = 0.72 \begin {gather*} x + e^{\left (-x^{2} e^{\left (-2 \, x\right )} + \frac {2}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + e^(-x^2*e^(-2*x) + 2/x)

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mupad [B]  time = 1.54, size = 19, normalized size = 0.76 \begin {gather*} x+{\mathrm {e}}^{2/x}\,{\mathrm {e}}^{-x^2\,{\mathrm {e}}^{-2\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + exp(2/x)*exp(-x^2*exp(-2*x))

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sympy [A]  time = 0.22, size = 19, normalized size = 0.76 \begin {gather*} x + e^{\frac {\left (- x^{3} + 2 e^{2 x}\right ) e^{- 2 x}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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