3.85.94 $\int \frac{e^{\frac{1}{3}(3e^{e^x}+7x-e^{\frac{12+4x}{x}}x+x^2)}(e^{\frac{12+4x}{x}}(12-x)+7x+3e^{e^x+x}x+2x^2)}{3x}dx$

Optimal. Leaf size=27 $$e^{e^{e^x}+\frac{1}{3}x(7-e^{4+\frac{12}{x}}+x)}$$

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Rubi [A]  time = 1.11, antiderivative size = 33, normalized size of antiderivative = 1.22, number of steps used = 2, number of rules used = 2, integrand size = 77, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.026, Rules used = {12, 6706} $$\begin{array}{c}\exp \left(\frac{1}{3}(x^2-e^{\frac{4(x+3)}{x}}x+7x+3e^{e^x})\right)\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(E^((3*E^E^x + 7*x - E^((12 + 4*x)/x)*x + x^2)/3)*(E^((12 + 4*x)/x)*(12 - x) + 7*x + 3*E^(E^x + x)*x + 2*x
^2))/(3*x),x]

[Out]

E^((3*E^E^x + 7*x - E^((4*(3 + x))/x)*x + x^2)/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 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{1}{3}\int \frac{e^{\frac{1}{3}(3e^{e^x}+7x-e^{\frac{12+4x}{x}}x+x^2)}(e^{\frac{12+4x}{x}}(12-x)+7x+3e^{e^x+x}x+2x^2)}{x}dx\\ & =\exp \left(\frac{1}{3}(3e^{e^x}+7x-e^{\frac{4(3+x)}{x}}x+x^2)\right)\end{array}\end{array}$$

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Mathematica [A]  time = 1.66, size = 30, normalized size = 1.11 $$\begin{array}{c}{e}^{e^{e^x}-\frac{1}{3}{e}^{4+\frac{12}{x}}x+\frac{1}{3}x(7+x)}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^((3*E^E^x + 7*x - E^((12 + 4*x)/x)*x + x^2)/3)*(E^((12 + 4*x)/x)*(12 - x) + 7*x + 3*E^(E^x + x)*x
+ 2*x^2))/(3*x),x]

[Out]

E^(E^E^x - (E^(4 + 12/x)*x)/3 + (x*(7 + x))/3)

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fricas [A]  time = 0.50, size = 37, normalized size = 1.37 $$\begin{array}{c}{e}^{\left(\frac{1}{3}((x^2-x{e}^{\left(\frac{4(x+3)}{x}\right)}+7x)e^x+3e^{(x+e^x)})e^{(-x)}\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*x*exp(x)*exp(exp(x))+(12-x)*exp((4*x+12)/x)+2*x^2+7*x)*exp(exp(exp(x))-1/3*x*exp((4*x+12)/x)+
1/3*x^2+7/3*x)/x,x, algorithm="fricas")

[Out]

e^(1/3*((x^2 - x*e^(4*(x + 3)/x) + 7*x)*e^x + 3*e^(x + e^x))*e^(-x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\int \frac{(2x^2+3x{e}^{(x+e^x)}-(x-12)e^{\left(\frac{4(x+3)}{x}\right)}+7x)e^{(\frac{1}{3}{x}^2-\frac{1}{3}x{e}^{\left(\frac{4(x+3)}{x}\right)}+\frac{7}{3}x+e^{\left(e^x\right)})}}{3x}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*x*exp(x)*exp(exp(x))+(12-x)*exp((4*x+12)/x)+2*x^2+7*x)*exp(exp(exp(x))-1/3*x*exp((4*x+12)/x)+
1/3*x^2+7/3*x)/x,x, algorithm="giac")

[Out]

integrate(1/3*(2*x^2 + 3*x*e^(x + e^x) - (x - 12)*e^(4*(x + 3)/x) + 7*x)*e^(1/3*x^2 - 1/3*x*e^(4*(x + 3)/x) +
7/3*x + e^(e^x))/x, x)

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maple [A]  time = 0.28, size = 26, normalized size = 0.96

method	result	size
risch	${\mathrm{e}}^{{\mathrm{e}}^{{\mathrm{e}}^x}-\frac{x{\mathrm{e}}^{\frac{4x+12}{x}}}{3}+\frac{x^2}{3}+\frac{7x}{3}}$	$26$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(3*x*exp(x)*exp(exp(x))+(12-x)*exp((4*x+12)/x)+2*x^2+7*x)*exp(exp(exp(x))-1/3*x*exp((4*x+12)/x)+1/3*x^
2+7/3*x)/x,x,method=_RETURNVERBOSE)

[Out]

exp(exp(exp(x))-1/3*x*exp(4*(3+x)/x)+1/3*x^2+7/3*x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\frac{1}{3}\int \frac{(2x^2+3x{e}^{(x+e^x)}-(x-12)e^{\left(\frac{4(x+3)}{x}\right)}+7x)e^{(\frac{1}{3}{x}^2-\frac{1}{3}x{e}^{\left(\frac{4(x+3)}{x}\right)}+\frac{7}{3}x+e^{\left(e^x\right)})}}{x}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*x*exp(x)*exp(exp(x))+(12-x)*exp((4*x+12)/x)+2*x^2+7*x)*exp(exp(exp(x))-1/3*x*exp((4*x+12)/x)+
1/3*x^2+7/3*x)/x,x, algorithm="maxima")

[Out]

1/3*integrate((2*x^2 + 3*x*e^(x + e^x) - (x - 12)*e^(4*(x + 3)/x) + 7*x)*e^(1/3*x^2 - 1/3*x*e^(4*(x + 3)/x) +
7/3*x + e^(e^x))/x, x)

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mupad [B]  time = 5.70, size = 27, normalized size = 1.00 $$\begin{array}{c}{\mathrm{e}}^{\frac{7x}{3}}{\mathrm{e}}^{-\frac{x{\mathrm{e}}^4{\mathrm{e}}^{12/x}}{3}}{\mathrm{e}}^{\frac{x^2}{3}}{\mathrm{e}}^{{\mathrm{e}}^{{\mathrm{e}}^x}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((7*x)/3 + exp(exp(x)) - (x*exp((4*x + 12)/x))/3 + x^2/3)*(7*x - exp((4*x + 12)/x)*(x - 12) + 2*x^2 +
3*x*exp(exp(x))*exp(x)))/(3*x),x)

[Out]

exp((7*x)/3)*exp(-(x*exp(4)*exp(12/x))/3)*exp(x^2/3)*exp(exp(exp(x)))

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sympy [A]  time = 0.89, size = 27, normalized size = 1.00 $$\begin{array}{c}{e}^{\frac{x^2}{3}-\frac{x{e}^{\frac{4x+12}{x}}}{3}+\frac{7x}{3}+e^{e^x}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*x*exp(x)*exp(exp(x))+(12-x)*exp((4*x+12)/x)+2*x**2+7*x)*exp(exp(exp(x))-1/3*x*exp((4*x+12)/x)
+1/3*x**2+7/3*x)/x,x)

[Out]

exp(x**2/3 - x*exp((4*x + 12)/x)/3 + 7*x/3 + exp(exp(x)))

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