3.81.87 \(\int \frac {-2+x^2+e^{e^x} (1-e^x x)}{x^2} \, dx\)

Optimal. Leaf size=15 \[ x-\frac {-2+e^{e^x}+x}{x} \]

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Rubi [F]  time = 0.09, 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 {-2+x^2+e^{e^x} \left (1-e^x x\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 17, normalized size = 1.13 \begin {gather*} \frac {2}{x}-\frac {e^{e^x}}{x}+x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2/x - E^E^x/x + x

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fricas [A]  time = 0.60, size = 14, normalized size = 0.93 \begin {gather*} \frac {x^{2} - e^{\left (e^{x}\right )} + 2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2 - e^(e^x) + 2)/x

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giac [A]  time = 0.20, size = 26, normalized size = 1.73 \begin {gather*} \frac {{\left (x^{2} e^{x} - e^{\left (x + e^{x}\right )} + 2 \, e^{x}\right )} e^{\left (-x\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 15, normalized size = 1.00

method	result	size
norman	\(\frac {2+x^{2}-{\mathrm e}^{{\mathrm e}^{x}}}{x}\)	\(15\)
risch	\(x +\frac {2}{x}-\frac {{\mathrm e}^{{\mathrm e}^{x}}}{x}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.40, size = 15, normalized size = 1.00 \begin {gather*} x - \frac {e^{\left (e^{x}\right )}}{x} + \frac {2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - e^(e^x)/x + 2/x

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mupad [B]  time = 0.11, size = 12, normalized size = 0.80 \begin {gather*} x-\frac {{\mathrm {e}}^{{\mathrm {e}}^x}-2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - (exp(exp(x)) - 2)/x

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sympy [A]  time = 0.12, size = 10, normalized size = 0.67 \begin {gather*} x - \frac {e^{e^{x}}}{x} + \frac {2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - exp(exp(x))/x + 2/x

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