3.52.12 \(\int e^{-e^{x^2}} (3 x^2-2 e^{x^2} x^4+e^{e^{x^2}} (2+e^{6+x} (1+x))) \, dx\)

Optimal. Leaf size=22 \[ x \left (2+e^{6+x}+e^{-e^{x^2}} x^2\right ) \]

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Rubi [F]  time = 0.30, 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 e^{-e^{x^2}} \left (3 x^2-2 e^{x^2} x^4+e^{e^{x^2}} \left (2+e^{6+x} (1+x)\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3*x^2 - 2*E^x^2*x^4 + E^E^x^2*(2 + E^(6 + x)*(1 + x)))/E^E^x^2,x]

[Out]

2*x + E^(6 + x)*x + 3*Defer[Int][x^2/E^E^x^2, x] - 2*Defer[Int][E^(-E^x^2 + x^2)*x^4, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2+e^{6+x}+e^{6+x} x+3 e^{-e^{x^2}} x^2-2 e^{-e^{x^2}+x^2} x^4\right ) \, dx\\ &=2 x-2 \int e^{-e^{x^2}+x^2} x^4 \, dx+3 \int e^{-e^{x^2}} x^2 \, dx+\int e^{6+x} \, dx+\int e^{6+x} x \, dx\\ &=e^{6+x}+2 x+e^{6+x} x-2 \int e^{-e^{x^2}+x^2} x^4 \, dx+3 \int e^{-e^{x^2}} x^2 \, dx-\int e^{6+x} \, dx\\ &=2 x+e^{6+x} x-2 \int e^{-e^{x^2}+x^2} x^4 \, dx+3 \int e^{-e^{x^2}} x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.50, size = 24, normalized size = 1.09 \begin {gather*} 2 x+e^{6+x} x+e^{-e^{x^2}} x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*x^2 - 2*E^x^2*x^4 + E^E^x^2*(2 + E^(6 + x)*(1 + x)))/E^E^x^2,x]

[Out]

2*x + E^(6 + x)*x + x^3/E^E^x^2

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fricas [A]  time = 0.49, size = 28, normalized size = 1.27 \begin {gather*} {\left (x^{3} + {\left (x e^{\left (x + 6\right )} + 2 \, x\right )} e^{\left (e^{\left (x^{2}\right )}\right )}\right )} e^{\left (-e^{\left (x^{2}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x+1)*exp(x+6)+2)*exp(exp(x^2))-2*x^4*exp(x^2)+3*x^2)/exp(exp(x^2)),x, algorithm="fricas")

[Out]

(x^3 + (x*e^(x + 6) + 2*x)*e^(e^(x^2)))*e^(-e^(x^2))

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giac [A]  time = 0.15, size = 21, normalized size = 0.95 \begin {gather*} x^{3} e^{\left (-e^{\left (x^{2}\right )}\right )} + x e^{\left (x + 6\right )} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x+1)*exp(x+6)+2)*exp(exp(x^2))-2*x^4*exp(x^2)+3*x^2)/exp(exp(x^2)),x, algorithm="giac")

[Out]

x^3*e^(-e^(x^2)) + x*e^(x + 6) + 2*x

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maple [A]  time = 0.04, size = 22, normalized size = 1.00

method	result	size
risch	\({\mathrm e}^{x +6} x +2 x +x^{3} {\mathrm e}^{-{\mathrm e}^{x^{2}}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x+1)*exp(x+6)+2)*exp(exp(x^2))-2*x^4*exp(x^2)+3*x^2)/exp(exp(x^2)),x,method=_RETURNVERBOSE)

[Out]

exp(x+6)*x+2*x+x^3*exp(-exp(x^2))

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maxima [A]  time = 0.41, size = 31, normalized size = 1.41 \begin {gather*} x^{3} e^{\left (-e^{\left (x^{2}\right )}\right )} + {\left (x e^{6} - e^{6}\right )} e^{x} + 2 \, x + e^{\left (x + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x+1)*exp(x+6)+2)*exp(exp(x^2))-2*x^4*exp(x^2)+3*x^2)/exp(exp(x^2)),x, algorithm="maxima")

[Out]

x^3*e^(-e^(x^2)) + (x*e^6 - e^6)*e^x + 2*x + e^(x + 6)

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mupad [B]  time = 3.24, size = 21, normalized size = 0.95 \begin {gather*} 2\,x+x^3\,{\mathrm {e}}^{-{\mathrm {e}}^{x^2}}+x\,{\mathrm {e}}^6\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-exp(x^2))*(exp(exp(x^2))*(exp(x + 6)*(x + 1) + 2) - 2*x^4*exp(x^2) + 3*x^2),x)

[Out]

2*x + x^3*exp(-exp(x^2)) + x*exp(6)*exp(x)

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sympy [A]  time = 4.76, size = 19, normalized size = 0.86 \begin {gather*} x^{3} e^{- e^{x^{2}}} + x e^{x + 6} + 2 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x+1)*exp(x+6)+2)*exp(exp(x**2))-2*x**4*exp(x**2)+3*x**2)/exp(exp(x**2)),x)

[Out]

x**3*exp(-exp(x**2)) + x*exp(x + 6) + 2*x

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