3.73.36 $\int e^{\frac{-2-x-x^2+\log (x)}{x}}(3+2x-x^2-\log (x))dx$

Optimal. Leaf size=22 $$e^{\frac{-2-x-x^2+\log (x)}{x}}{x}^2$$

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Rubi [B]  time = 0.04, antiderivative size = 63, normalized size of antiderivative = 2.86, number of steps used = 1, number of rules used = 1, integrand size = 33, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.030, Rules used = {2288} $$\begin{array}{c}-\frac{e^{-\frac{x^2+x+2}{x}}{x}^{\frac{1}{x}}(-x^2-\log (x)+3)}{\frac{2x-\frac{1}{x}+1}{x}-\frac{x^2+x-\log (x)+2}{x^2}}\end{array}$$

Antiderivative was successfully verified.

[In]

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

[Out]

-((x^x^(-1)*(3 - x^2 - Log[x]))/(E^((2 + x + x^2)/x)*((1 - x^(-1) + 2*x)/x - (2 + x + x^2 - Log[x])/x^2)))

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =-\frac{e^{-\frac{2+x+x^2}{x}}{x}^{\frac{1}{x}}(3-x^2-\log (x))}{\frac{1-\frac{1}{x}+2x}{x}-\frac{2+x+x^2-\log (x)}{x^2}}\end{array}\end{array}$$

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Mathematica [A]  time = 0.36, size = 20, normalized size = 0.91 $$\begin{array}{c}{e}^{-1-\frac{2}{x}-x}{x}^{2+\frac{1}{x}}\end{array}$$

Antiderivative was successfully verified.

[In]

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

[Out]

E^(-1 - 2/x - x)*x^(2 + x^(-1))

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fricas [A]  time = 1.10, size = 20, normalized size = 0.91 $$\begin{array}{c}{x}^2{e}^{(-\frac{x^2+x-\log (x)+2}{x})}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*e^(-(x^2 + x - log(x) + 2)/x)

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giac [A]  time = 0.22, size = 20, normalized size = 0.91 $$\begin{array}{c}{x}^2{e}^{(-\frac{x^2+x-\log (x)+2}{x})}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*e^(-(x^2 + x - log(x) + 2)/x)

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maple [A]  time = 0.02, size = 22, normalized size = 1.00

method	result	size
norman	${\mathrm{e}}^{\frac{\ln (x)-x^2-x-2}{x}}{x}^2$	$22$
risch	${\mathrm{e}}^{\frac{\ln (x)-x^2-x-2}{x}}{x}^2$	$22$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp((ln(x)-x^2-x-2)/x)*x^2

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maxima [A]  time = 0.43, size = 21, normalized size = 0.95 $$\begin{array}{c}{x}^2{e}^{(-x+\frac{\log (x)}{x}-\frac{2}{x}-1)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*e^(-x + log(x)/x - 2/x - 1)

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mupad [B]  time = 4.35, size = 20, normalized size = 0.91 $$\begin{array}{c}{x}^{\frac{1}{x}+2}{\mathrm{e}}^{-x}{\mathrm{e}}^{-1}{\mathrm{e}}^{-\frac{2}{x}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(1/x + 2)*exp(-x)*exp(-1)*exp(-2/x)

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sympy [A]  time = 0.32, size = 15, normalized size = 0.68 $$\begin{array}{c}{x}^2{e}^{\frac{-x^2-x+\log (x)-2}{x}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2*exp((-x**2 - x + log(x) - 2)/x)

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