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

Optimal. Leaf size=21 \[ -1+e^{\frac {e^x}{x}}-i \pi +2 x+\log (3) \]

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Rubi [F]  time = 0.25, 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^{\frac {e^x}{x}+x} (-1+x)+2 x^2}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 13, normalized size = 0.62 \begin {gather*} e^{\frac {e^x}{x}}+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(E^x/x) + 2*x

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fricas [A]  time = 0.68, size = 22, normalized size = 1.05 \begin {gather*} {\left (2 \, x e^{x} + e^{\left (\frac {x^{2} + e^{x}}{x}\right )}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x*e^x + e^((x^2 + e^x)/x))*e^(-x)

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giac [A]  time = 0.20, size = 22, normalized size = 1.05 \begin {gather*} {\left (2 \, x e^{x} + e^{\left (\frac {x^{2} + e^{x}}{x}\right )}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x*e^x + e^((x^2 + e^x)/x))*e^(-x)

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maple [A]  time = 0.05, size = 12, normalized size = 0.57

method	result	size
risch	\(2 x +{\mathrm e}^{\frac {{\mathrm e}^{x}}{x}}\)	\(12\)
norman	\(\frac {x \,{\mathrm e}^{\frac {{\mathrm e}^{x}}{x}}+2 x^{2}}{x}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.41, size = 11, normalized size = 0.52 \begin {gather*} 2 \, x + e^{\left (\frac {e^{x}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x + e^(e^x/x)

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mupad [B]  time = 4.20, size = 11, normalized size = 0.52 \begin {gather*} 2\,x+{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x + exp(exp(x)/x)

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sympy [A]  time = 0.14, size = 8, normalized size = 0.38 \begin {gather*} 2 x + e^{\frac {e^{x}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x + exp(exp(x)/x)

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