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

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

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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 {gather*} \exp \left (\frac {1}{3} \left (x^2-e^{\frac {4 (x+3)}{x}} x+7 x+3 e^{e^x}\right )\right ) \end {gather*}

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 {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {e^{\frac {1}{3} \left (3 e^{e^x}+7 x-e^{\frac {12+4 x}{x}} x+x^2\right )} \left (e^{\frac {12+4 x}{x}} (12-x)+7 x+3 e^{e^x+x} x+2 x^2\right )}{x} \, dx\\ &=\exp \left (\frac {1}{3} \left (3 e^{e^x}+7 x-e^{\frac {4 (3+x)}{x}} x+x^2\right )\right )\\ \end {aligned} \end {gather*}

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

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 {gather*} e^{\left (\frac {1}{3} \, {\left ({\left (x^{2} - x e^{\left (\frac {4 \, {\left (x + 3\right )}}{x}\right )} + 7 \, x\right )} e^{x} + 3 \, e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )}\right )} \end {gather*}

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 {gather*} \int \frac {{\left (2 \, x^{2} + 3 \, x e^{\left (x + e^{x}\right )} - {\left (x - 12\right )} e^{\left (\frac {4 \, {\left (x + 3\right )}}{x}\right )} + 7 \, x\right )} e^{\left (\frac {1}{3} \, x^{2} - \frac {1}{3} \, x e^{\left (\frac {4 \, {\left (x + 3\right )}}{x}\right )} + \frac {7}{3} \, x + e^{\left (e^{x}\right )}\right )}}{3 \, x}\,{d x} \end {gather*}

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	\({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}-\frac {x \,{\mathrm e}^{\frac {4 x +12}{x}}}{3}+\frac {x^{2}}{3}+\frac {7 x}{3}}\)	\(26\)

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 {gather*} \frac {1}{3} \, \int \frac {{\left (2 \, x^{2} + 3 \, x e^{\left (x + e^{x}\right )} - {\left (x - 12\right )} e^{\left (\frac {4 \, {\left (x + 3\right )}}{x}\right )} + 7 \, x\right )} e^{\left (\frac {1}{3} \, x^{2} - \frac {1}{3} \, x e^{\left (\frac {4 \, {\left (x + 3\right )}}{x}\right )} + \frac {7}{3} \, x + e^{\left (e^{x}\right )}\right )}}{x}\,{d x} \end {gather*}

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 {gather*} {\mathrm {e}}^{\frac {7\,x}{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 {gather*}

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 {gather*} e^{\frac {x^{2}}{3} - \frac {x e^{\frac {4 x + 12}{x}}}{3} + \frac {7 x}{3} + e^{e^{x}}} \end {gather*}

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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