3.47.13 \(\int (-8+2 x+e^{2 e^{4 x+x^2}+4 x+x^2} (8+4 x)+e^{e^{4 x+x^2}} (2+e^{4 x+x^2} (-32-8 x+4 x^2))) \, dx\)

Optimal. Leaf size=14 \[ \left (-4+e^{e^{x (4+x)}}+x\right )^2 \]

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Rubi [F]  time = 0.22, 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 (-8+2 x+e^{2 e^{4 x+x^2}+4 x+x^2} (8+4 x)+e^{e^{4 x+x^2}} \left (2+e^{4 x+x^2} \left (-32-8 x+4 x^2\right )\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[-8 + 2*x + E^(2*E^(4*x + x^2) + 4*x + x^2)*(8 + 4*x) + E^E^(4*x + x^2)*(2 + E^(4*x + x^2)*(-32 - 8*x + 4*x
^2)),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-8 x+x^2+\int e^{2 e^{4 x+x^2}+4 x+x^2} (8+4 x) \, dx+\int e^{e^{4 x+x^2}} \left (2+e^{4 x+x^2} \left (-32-8 x+4 x^2\right )\right ) \, dx\\ &=-8 x+x^2-\frac {2 e^{e^{4 x+x^2}} \left (8+2 x-x^2\right )}{2+x}+\int \left (8 e^{2 e^{4 x+x^2}+4 x+x^2}+4 e^{2 e^{4 x+x^2}+4 x+x^2} x\right ) \, dx\\ &=-8 x+x^2-\frac {2 e^{e^{4 x+x^2}} \left (8+2 x-x^2\right )}{2+x}+4 \int e^{2 e^{4 x+x^2}+4 x+x^2} x \, dx+8 \int e^{2 e^{4 x+x^2}+4 x+x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 14, normalized size = 1.00 \begin {gather*} \left (-4+e^{e^{x (4+x)}}+x\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-8 + 2*x + E^(2*E^(4*x + x^2) + 4*x + x^2)*(8 + 4*x) + E^E^(4*x + x^2)*(2 + E^(4*x + x^2)*(-32 - 8*x
+ 4*x^2)),x]

[Out]

(-4 + E^E^(x*(4 + x)) + x)^2

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fricas [B]  time = 0.75, size = 32, normalized size = 2.29 \begin {gather*} x^{2} + 2 \, {\left (x - 4\right )} e^{\left (e^{\left (x^{2} + 4 \, x\right )}\right )} - 8 \, x + e^{\left (2 \, e^{\left (x^{2} + 4 \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x+8)*exp(x^2+4*x)*exp(exp(x^2+4*x))^2+((4*x^2-8*x-32)*exp(x^2+4*x)+2)*exp(exp(x^2+4*x))+2*x-8,x,
algorithm="fricas")

[Out]

x^2 + 2*(x - 4)*e^(e^(x^2 + 4*x)) - 8*x + e^(2*e^(x^2 + 4*x))

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giac [B]  time = 0.20, size = 67, normalized size = 4.79 \begin {gather*} x^{2} + 2 \, {\left (x e^{\left (x^{2} + 4 \, x + e^{\left (x^{2} + 4 \, x\right )}\right )} - 4 \, e^{\left (x^{2} + 4 \, x + e^{\left (x^{2} + 4 \, x\right )}\right )}\right )} e^{\left (-x^{2} - 4 \, x\right )} - 8 \, x + e^{\left (2 \, e^{\left (x^{2} + 4 \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x+8)*exp(x^2+4*x)*exp(exp(x^2+4*x))^2+((4*x^2-8*x-32)*exp(x^2+4*x)+2)*exp(exp(x^2+4*x))+2*x-8,x,
algorithm="giac")

[Out]

x^2 + 2*(x*e^(x^2 + 4*x + e^(x^2 + 4*x)) - 4*e^(x^2 + 4*x + e^(x^2 + 4*x)))*e^(-x^2 - 4*x) - 8*x + e^(2*e^(x^2
+ 4*x))

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maple [B]  time = 0.08, size = 30, normalized size = 2.14

method	result	size
risch	\({\mathrm e}^{2 \,{\mathrm e}^{\left (4+x \right ) x}}+\left (2 x -8\right ) {\mathrm e}^{{\mathrm e}^{\left (4+x \right ) x}}+x^{2}-8 x\)	\(30\)
default	\(-8 x +2 x \,{\mathrm e}^{{\mathrm e}^{x^{2}+4 x}}-8 \,{\mathrm e}^{{\mathrm e}^{x^{2}+4 x}}+{\mathrm e}^{2 \,{\mathrm e}^{x^{2}+4 x}}+x^{2}\)	\(42\)
norman	\(-8 x +2 x \,{\mathrm e}^{{\mathrm e}^{x^{2}+4 x}}-8 \,{\mathrm e}^{{\mathrm e}^{x^{2}+4 x}}+{\mathrm e}^{2 \,{\mathrm e}^{x^{2}+4 x}}+x^{2}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x+8)*exp(x^2+4*x)*exp(exp(x^2+4*x))^2+((4*x^2-8*x-32)*exp(x^2+4*x)+2)*exp(exp(x^2+4*x))+2*x-8,x,method=
_RETURNVERBOSE)

[Out]

exp(2*exp((4+x)*x))+(2*x-8)*exp(exp((4+x)*x))+x^2-8*x

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maxima [B]  time = 0.37, size = 32, normalized size = 2.29 \begin {gather*} x^{2} + 2 \, {\left (x - 4\right )} e^{\left (e^{\left (x^{2} + 4 \, x\right )}\right )} - 8 \, x + e^{\left (2 \, e^{\left (x^{2} + 4 \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x+8)*exp(x^2+4*x)*exp(exp(x^2+4*x))^2+((4*x^2-8*x-32)*exp(x^2+4*x)+2)*exp(exp(x^2+4*x))+2*x-8,x,
algorithm="maxima")

[Out]

x^2 + 2*(x - 4)*e^(e^(x^2 + 4*x)) - 8*x + e^(2*e^(x^2 + 4*x))

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mupad [B]  time = 3.53, size = 43, normalized size = 3.07 \begin {gather*} {\mathrm {e}}^{2\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{x^2}}-8\,{\mathrm {e}}^{{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{x^2}}-8\,x+2\,x\,{\mathrm {e}}^{{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{x^2}}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x - exp(exp(4*x + x^2))*(exp(4*x + x^2)*(8*x - 4*x^2 + 32) - 2) + exp(2*exp(4*x + x^2))*exp(4*x + x^2)*(
4*x + 8) - 8,x)

[Out]

exp(2*exp(4*x)*exp(x^2)) - 8*exp(exp(4*x)*exp(x^2)) - 8*x + 2*x*exp(exp(4*x)*exp(x^2)) + x^2

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sympy [B]  time = 8.35, size = 32, normalized size = 2.29 \begin {gather*} x^{2} - 8 x + \left (2 x - 8\right ) e^{e^{x^{2} + 4 x}} + e^{2 e^{x^{2} + 4 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x+8)*exp(x**2+4*x)*exp(exp(x**2+4*x))**2+((4*x**2-8*x-32)*exp(x**2+4*x)+2)*exp(exp(x**2+4*x))+2*x
-8,x)

[Out]

x**2 - 8*x + (2*x - 8)*exp(exp(x**2 + 4*x)) + exp(2*exp(x**2 + 4*x))

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