3.73.23 \(\int e^{e^{\frac {1}{2} (e^{e^4 x}+2 x)}+\frac {1}{2} (e^{e^4 x}+2 x)} (26+13 e^{4+e^4 x}) \, dx\)

Optimal. Leaf size=19 \[ 26 e^{e^{\frac {e^{e^4 x}}{2}+x}} \]

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Rubi [F]  time = 0.46, 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 \exp \left (e^{\frac {1}{2} \left (e^{e^4 x}+2 x\right )}+\frac {1}{2} \left (e^{e^4 x}+2 x\right )\right ) \left (26+13 e^{4+e^4 x}\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(E^((E^(E^4*x) + 2*x)/2) + (E^(E^4*x) + 2*x)/2)*(26 + 13*E^(4 + E^4*x)),x]

[Out]

26*Defer[Int][E^((E^(E^4*x) + 2*E^(E^(E^4*x)/2 + x) + 2*x)/2), x] + 13*Defer[Int][E^((8 + E^(E^4*x) + 2*E^(E^(
E^4*x)/2 + x) + 2*(1 + E^4)*x)/2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int 13 e^{\frac {1}{2} \left (e^{e^4 x}+2 e^{\frac {e^{e^4 x}}{2}+x}+2 x\right )} \left (2+e^{4+e^4 x}\right ) \, dx\\ &=13 \int e^{\frac {1}{2} \left (e^{e^4 x}+2 e^{\frac {e^{e^4 x}}{2}+x}+2 x\right )} \left (2+e^{4+e^4 x}\right ) \, dx\\ &=13 \int \left (2 e^{\frac {1}{2} \left (e^{e^4 x}+2 e^{\frac {e^{e^4 x}}{2}+x}+2 x\right )}+\exp \left (4+e^4 x+\frac {1}{2} \left (e^{e^4 x}+2 e^{\frac {e^{e^4 x}}{2}+x}+2 x\right )\right )\right ) \, dx\\ &=13 \int \exp \left (4+e^4 x+\frac {1}{2} \left (e^{e^4 x}+2 e^{\frac {e^{e^4 x}}{2}+x}+2 x\right )\right ) \, dx+26 \int e^{\frac {1}{2} \left (e^{e^4 x}+2 e^{\frac {e^{e^4 x}}{2}+x}+2 x\right )} \, dx\\ &=13 \int \exp \left (\frac {1}{2} \left (8+e^{e^4 x}+2 e^{\frac {e^{e^4 x}}{2}+x}+2 \left (1+e^4\right ) x\right )\right ) \, dx+26 \int e^{\frac {1}{2} \left (e^{e^4 x}+2 e^{\frac {e^{e^4 x}}{2}+x}+2 x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 19, normalized size = 1.00 \begin {gather*} 26 e^{e^{\frac {e^{e^4 x}}{2}+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(E^((E^(E^4*x) + 2*x)/2) + (E^(E^4*x) + 2*x)/2)*(26 + 13*E^(4 + E^4*x)),x]

[Out]

26*E^E^(E^(E^4*x)/2 + x)

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fricas [B]  time = 0.70, size = 60, normalized size = 3.16 \begin {gather*} 26 \, e^{\left (\frac {1}{2} \, {\left (2 \, x e^{4} + e^{\left (x e^{4} + 4\right )} + 2 \, e^{\left (\frac {1}{2} \, {\left (2 \, x e^{4} + e^{\left (x e^{4} + 4\right )}\right )} e^{\left (-4\right )} + 4\right )}\right )} e^{\left (-4\right )} - \frac {1}{2} \, {\left (2 \, x e^{4} + e^{\left (x e^{4} + 4\right )}\right )} e^{\left (-4\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((13*exp(4)*exp(x*exp(4))+26)*exp(1/2*exp(x*exp(4))+x)*exp(exp(1/2*exp(x*exp(4))+x)),x, algorithm="fr
icas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int 13 \, {\left (e^{\left (x e^{4} + 4\right )} + 2\right )} e^{\left (x + \frac {1}{2} \, e^{\left (x e^{4}\right )} + e^{\left (x + \frac {1}{2} \, e^{\left (x e^{4}\right )}\right )}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((13*exp(4)*exp(x*exp(4))+26)*exp(1/2*exp(x*exp(4))+x)*exp(exp(1/2*exp(x*exp(4))+x)),x, algorithm="gi
ac")

[Out]

integrate(13*(e^(x*e^4 + 4) + 2)*e^(x + 1/2*e^(x*e^4) + e^(x + 1/2*e^(x*e^4))), x)

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maple [A]  time = 0.11, size = 14, normalized size = 0.74

method	result	size
norman	\(26 \,{\mathrm e}^{{\mathrm e}^{\frac {{\mathrm e}^{x \,{\mathrm e}^{4}}}{2}+x}}\)	\(14\)
risch	\(26 \,{\mathrm e}^{{\mathrm e}^{\frac {{\mathrm e}^{x \,{\mathrm e}^{4}}}{2}+x}}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((13*exp(4)*exp(x*exp(4))+26)*exp(1/2*exp(x*exp(4))+x)*exp(exp(1/2*exp(x*exp(4))+x)),x,method=_RETURNVERBOS
E)

[Out]

26*exp(exp(1/2*exp(x*exp(4))+x))

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maxima [A]  time = 0.48, size = 13, normalized size = 0.68 \begin {gather*} 26 \, e^{\left (e^{\left (x + \frac {1}{2} \, e^{\left (x e^{4}\right )}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((13*exp(4)*exp(x*exp(4))+26)*exp(1/2*exp(x*exp(4))+x)*exp(exp(1/2*exp(x*exp(4))+x)),x, algorithm="ma
xima")

[Out]

26*e^(e^(x + 1/2*e^(x*e^4)))

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mupad [B]  time = 0.20, size = 14, normalized size = 0.74 \begin {gather*} 26\,{\mathrm {e}}^{{\mathrm {e}}^{\frac {{\mathrm {e}}^{x\,{\mathrm {e}}^4}}{2}}\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x + exp(x*exp(4))/2)*exp(exp(x + exp(x*exp(4))/2))*(13*exp(4)*exp(x*exp(4)) + 26),x)

[Out]

26*exp(exp(exp(x*exp(4))/2)*exp(x))

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sympy [A]  time = 0.40, size = 14, normalized size = 0.74 \begin {gather*} 26 e^{e^{x + \frac {e^{x e^{4}}}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((13*exp(4)*exp(x*exp(4))+26)*exp(1/2*exp(x*exp(4))+x)*exp(exp(1/2*exp(x*exp(4))+x)),x)

[Out]

26*exp(exp(x + exp(x*exp(4))/2))

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