3.5.81 \(\int (1+e^{e^x+e^{\frac {1}{3} e^{e^x} (2 x+2 x^2)}+\frac {1}{3} e^{e^x} (2 x+2 x^2)} (2+4 x+e^x (2 x+2 x^2))) \, dx\) [481]

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

[Out]

x+3*exp(exp(2/3*exp(exp(x))*(1+x)*x))

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Rubi [F]
time = 2.45, 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 \left (1+\exp \left (e^x+e^{\frac {1}{3} e^{e^x} \left (2 x+2 x^2\right )}+\frac {1}{3} e^{e^x} \left (2 x+2 x^2\right )\right ) \left (2+4 x+e^x \left (2 x+2 x^2\right )\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1 + E^(E^x + E^((E^E^x*(2*x + 2*x^2))/3) + (E^E^x*(2*x + 2*x^2))/3)*(2 + 4*x + E^x*(2*x + 2*x^2)),x]

[Out]

x + 2*Defer[Int][E^(E^x + E^((E^E^x*(2*x + 2*x^2))/3) + (E^E^x*(2*x + 2*x^2))/3), x] + 4*Defer[Int][E^(E^x + E
^((E^E^x*(2*x + 2*x^2))/3) + (E^E^x*(2*x + 2*x^2))/3)*x, x] + 2*Defer[Int][E^(E^x + E^((E^E^x*(2*x + 2*x^2))/3
) + x + (E^E^x*(2*x + 2*x^2))/3)*x, x] + 2*Defer[Int][E^(E^x + E^((E^E^x*(2*x + 2*x^2))/3) + x + (E^E^x*(2*x +
2*x^2))/3)*x^2, x]

Rubi steps

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

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Mathematica [A]
time = 0.47, size = 21, normalized size = 1.00 \begin {gather*} 3 e^{e^{\frac {2}{3} e^{e^x} x (1+x)}}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + E^(E^x + E^((E^E^x*(2*x + 2*x^2))/3) + (E^E^x*(2*x + 2*x^2))/3)*(2 + 4*x + E^x*(2*x + 2*x^2)),x]

[Out]

3*E^E^((2*E^E^x*x*(1 + x))/3) + x

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Maple [A]
time = 0.14, size = 16, normalized size = 0.76

method	result	size
risch	\(x +3 \,{\mathrm e}^{{\mathrm e}^{\frac {2 \,{\mathrm e}^{{\mathrm e}^{x}} \left (x +1\right ) x}{3}}}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x+3*exp(exp(2/3*exp(exp(x))*(x+1)*x))

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Maxima [A]
time = 0.46, size = 21, normalized size = 1.00 \begin {gather*} x + 3 \, e^{\left (e^{\left (\frac {2}{3} \, x^{2} e^{\left (e^{x}\right )} + \frac {2}{3} \, x e^{\left (e^{x}\right )}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 3*e^(e^(2/3*x^2*e^(e^x) + 2/3*x*e^(e^x)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (15) = 30\).
time = 0.32, size = 61, normalized size = 2.90 \begin {gather*} {\left (x e^{\left (\frac {2}{3} \, {\left (x^{2} + x\right )} e^{\left (e^{x}\right )} + e^{x}\right )} + 3 \, e^{\left (\frac {2}{3} \, {\left (x^{2} + x\right )} e^{\left (e^{x}\right )} + e^{\left (\frac {2}{3} \, {\left (x^{2} + x\right )} e^{\left (e^{x}\right )}\right )} + e^{x}\right )}\right )} e^{\left (-\frac {2}{3} \, {\left (x^{2} + x\right )} e^{\left (e^{x}\right )} - e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*e^(2/3*(x^2 + x)*e^(e^x) + e^x) + 3*e^(2/3*(x^2 + x)*e^(e^x) + e^(2/3*(x^2 + x)*e^(e^x)) + e^x))*e^(-2/3*(x
^2 + x)*e^(e^x) - e^x)

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Sympy [A]
time = 1.78, size = 22, normalized size = 1.05 \begin {gather*} x + 3 e^{e^{\left (\frac {2 x^{2}}{3} + \frac {2 x}{3}\right ) e^{e^{x}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2+2*x)*exp(x)+4*x+2)*exp(exp(x))*exp(1/3*(2*x**2+2*x)*exp(exp(x)))*exp(exp(1/3*(2*x**2+2*x)*e
xp(exp(x))))+1,x)

[Out]

x + 3*exp(exp((2*x**2/3 + 2*x/3)*exp(exp(x))))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(2*((x^2 + x)*e^x + 2*x + 1)*e^(2/3*(x^2 + x)*e^(e^x) + e^(2/3*(x^2 + x)*e^(e^x)) + e^x) + 1, x)

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Mupad [B]
time = 0.59, size = 22, normalized size = 1.05 \begin {gather*} x+3\,{\mathrm {e}}^{{\mathrm {e}}^{\frac {2\,x\,{\mathrm {e}}^{{\mathrm {e}}^x}}{3}}\,{\mathrm {e}}^{\frac {2\,x^2\,{\mathrm {e}}^{{\mathrm {e}}^x}}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(exp(x))*exp((exp(exp(x))*(2*x + 2*x^2))/3)*exp(exp((exp(exp(x))*(2*x + 2*x^2))/3))*(4*x + exp(x)*(2*x
+ 2*x^2) + 2) + 1,x)

[Out]

x + 3*exp(exp((2*x*exp(exp(x)))/3)*exp((2*x^2*exp(exp(x)))/3))

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