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

Optimal. Leaf size=27 \[ -6-e^{2 e^{2/x}}+e^{x+x^2}+2 x+\log (4) \]

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Rubi [F]  time = 0.13, 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 {4 e^{2 e^{2/x}+\frac {2}{x}}+2 x^2+e^{x+x^2} \left (x^2+2 x^3\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 24, normalized size = 0.89 \begin {gather*} -e^{2 e^{2/x}}+e^{x+x^2}+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-E^(2*E^(2/x)) + E^(x + x^2) + 2*x

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fricas [A]  time = 0.74, size = 46, normalized size = 1.70 \begin {gather*} {\left (2 \, x e^{\frac {2}{x}} + e^{\left (x^{2} + x + \frac {2}{x}\right )} - e^{\left (\frac {2 \, {\left (x e^{\frac {2}{x}} + 1\right )}}{x}\right )}\right )} e^{\left (-\frac {2}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.18, size = 46, normalized size = 1.70 \begin {gather*} {\left (2 \, x e^{\frac {2}{x}} + e^{\left (x^{2} + x + \frac {2}{x}\right )} - e^{\left (\frac {2 \, {\left (x e^{\frac {2}{x}} + 1\right )}}{x}\right )}\right )} e^{\left (-\frac {2}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 22, normalized size = 0.81

method	result	size
default	\(2 x -{\mathrm e}^{2 \,{\mathrm e}^{\frac {2}{x}}}+{\mathrm e}^{x^{2}+x}\)	\(22\)
risch	\(-{\mathrm e}^{2 \,{\mathrm e}^{\frac {2}{x}}}+2 x +{\mathrm e}^{\left (x +1\right ) x}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x-exp(exp(1/x)^2)^2+exp(x^2+x)

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maxima [C]  time = 0.39, size = 81, normalized size = 3.00 \begin {gather*} -\frac {1}{2} i \, \sqrt {\pi } \operatorname {erf}\left (i \, x + \frac {1}{2} i\right ) e^{\left (-\frac {1}{4}\right )} - \frac {1}{2} \, {\left (\frac {\sqrt {\pi } {\left (2 \, x + 1\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-{\left (2 \, x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (2 \, x + 1\right )}^{2}}} - 2 \, e^{\left (\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}\right )} e^{\left (-\frac {1}{4}\right )} + 2 \, x - e^{\left (2 \, e^{\frac {2}{x}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*I*sqrt(pi)*erf(I*x + 1/2*I)*e^(-1/4) - 1/2*(sqrt(pi)*(2*x + 1)*(erf(1/2*sqrt(-(2*x + 1)^2)) - 1)/sqrt(-(2
*x + 1)^2) - 2*e^(1/4*(2*x + 1)^2))*e^(-1/4) + 2*x - e^(2*e^(2/x))

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mupad [B]  time = 7.40, size = 21, normalized size = 0.78 \begin {gather*} 2\,x+{\mathrm {e}}^{x^2+x}-{\mathrm {e}}^{2\,{\mathrm {e}}^{2/x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x + exp(x + x^2) - exp(2*exp(2/x))

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sympy [A]  time = 0.43, size = 17, normalized size = 0.63 \begin {gather*} 2 x + e^{x^{2} + x} - e^{2 e^{\frac {2}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x + exp(x**2 + x) - exp(2*exp(2/x))

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