3.89.88 $\int \frac{1}{3}(3+e^{-e^{3/x}x+x^2}(1+e^{3/x}(3-x)+2x^2)-\log (x))dx$

Optimal. Leaf size=34 $$x+\frac{1}{3}(-x+(2+e^{x(-e^{3/x}+x)})x-x\log (x))$$

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Rubi [B]  time = 0.10, antiderivative size = 77, normalized size of antiderivative = 2.26, number of steps used = 4, number of rules used = 3, integrand size = 47, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.064, Rules used = {12, 2288, 2295} $$\begin{array}{c}-\frac{e^{x^2-e^{3/x}x}(2x^2+e^{3/x}(3-x))}{3(-2x+e^{3/x}-\frac{3e^{3/x}}{x})}+\frac{4x}{3}-\frac{1}{3}x\log (x)\end{array}$$

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{1}{3}\int (3+e^{-e^{3/x}x+x^2}(1+e^{3/x}(3-x)+2x^2)-\log (x))dx\\ & =x+\frac{1}{3}\int e^{-e^{3/x}x+x^2}(1+e^{3/x}(3-x)+2x^2)dx-\frac{1}{3}\int \log (x)dx\\ & =\frac{4x}{3}-\frac{e^{-e^{3/x}x+x^2}(e^{3/x}(3-x)+2x^2)}{3(e^{3/x}-\frac{3e^{3/x}}{x}-2x)}-\frac{1}{3}x\log (x)\end{array}\end{array}$$

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Mathematica [A]  time = 0.11, size = 26, normalized size = 0.76 $$\begin{array}{c}\frac{1}{3}x(4+e^{x(-e^{3/x}+x)}-\log (x))\end{array}$$

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(4 + E^(x*(-E^(3/x) + x)) - Log[x]))/3

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fricas [A]  time = 0.51, size = 26, normalized size = 0.76 $$\begin{array}{c}\frac{1}{3}x{e}^{(x^2-x{e}^{\frac{3}{x}})}-\frac{1}{3}x\log (x)+\frac{4}{3}x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((3-x)*exp(3/x)+2*x^2+1)*exp(-x*exp(3/x)+x^2)+1-1/3*log(x),x, algorithm="fricas")

[Out]

1/3*x*e^(x^2 - x*e^(3/x)) - 1/3*x*log(x) + 4/3*x

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giac [A]  time = 0.25, size = 39, normalized size = 1.15 $$\begin{array}{c}\frac{1}{3}x{e}^{(\frac{x^3-x^2{e}^{\frac{3}{x}}+3}{x}-\frac{3}{x})}-\frac{1}{3}x\log (x)+\frac{4}{3}x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((3-x)*exp(3/x)+2*x^2+1)*exp(-x*exp(3/x)+x^2)+1-1/3*log(x),x, algorithm="giac")

[Out]

1/3*x*e^((x^3 - x^2*e^(3/x) + 3)/x - 3/x) - 1/3*x*log(x) + 4/3*x

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maple [A]  time = 0.07, size = 26, normalized size = 0.76

method	result	size
risch	$\frac{4x}{3}+\frac{x{\mathrm{e}}^{(x-{\mathrm{e}}^{\frac{3}{x}})x}}{3}-\frac{x\ln (x)}{3}$	$26$
default	$\frac{4x}{3}+\frac{x{\mathrm{e}}^{-x{\mathrm{e}}^{\frac{3}{x}}+x^2}}{3}-\frac{x\ln (x)}{3}$	$27$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((3-x)*exp(3/x)+2*x^2+1)*exp(-x*exp(3/x)+x^2)+1-1/3*ln(x),x,method=_RETURNVERBOSE)

[Out]

4/3*x+1/3*x*exp((x-exp(3/x))*x)-1/3*x*ln(x)

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maxima [A]  time = 0.43, size = 26, normalized size = 0.76 $$\begin{array}{c}\frac{1}{3}x{e}^{(x^2-x{e}^{\frac{3}{x}})}-\frac{1}{3}x\log (x)+\frac{4}{3}x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((3-x)*exp(3/x)+2*x^2+1)*exp(-x*exp(3/x)+x^2)+1-1/3*log(x),x, algorithm="maxima")

[Out]

1/3*x*e^(x^2 - x*e^(3/x)) - 1/3*x*log(x) + 4/3*x

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mupad [B]  time = 5.61, size = 23, normalized size = 0.68 $$\begin{array}{c}\frac{x({\mathrm{e}}^{x^2-x{\mathrm{e}}^{3/x}}-\ln (x)+4)}{3}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x^2 - x*exp(3/x))*(2*x^2 - exp(3/x)*(x - 3) + 1))/3 - log(x)/3 + 1,x)

[Out]

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

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sympy [A]  time = 1.95, size = 26, normalized size = 0.76 $$\begin{array}{c}\frac{x{e}^{x^2-x{e}^{\frac{3}{x}}}}{3}-\frac{x\log (x)}{3}+\frac{4x}{3}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((3-x)*exp(3/x)+2*x**2+1)*exp(-x*exp(3/x)+x**2)+1-1/3*ln(x),x)

[Out]

x*exp(x**2 - x*exp(3/x))/3 - x*log(x)/3 + 4*x/3

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