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

Optimal. Leaf size=15 \[ e^{e^{\left (-1-x^2\right )^2}}+x \]

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Rubi [F]  time = 0.41, 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+e^{1+e^{1+2 x^2+x^4}+2 x^2+x^4} \left (4 x+4 x^3\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

x + 2*Defer[Subst][Defer[Int][E^(1 + E^(1 + 2*x + x^2) + 2*x + x^2), x], x, x^2] + 2*Defer[Subst][Defer[Int][E
^(1 + E^(1 + 2*x + x^2) + 2*x + x^2)*x, x], x, x^2]

Rubi steps

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

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Mathematica [A]  time = 0.16, size = 13, normalized size = 0.87 \begin {gather*} e^{e^{\left (1+x^2\right )^2}}+x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^E^(1 + x^2)^2 + x

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fricas [B]  time = 0.60, size = 50, normalized size = 3.33 \begin {gather*} {\left (x e^{\left (x^{4} + 2 \, x^{2} + 1\right )} + e^{\left (x^{4} + 2 \, x^{2} + e^{\left (x^{4} + 2 \, x^{2} + 1\right )} + 1\right )}\right )} e^{\left (-x^{4} - 2 \, x^{2} - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int 4 \, {\left (x^{3} + x\right )} e^{\left (x^{4} + 2 \, x^{2} + e^{\left (x^{4} + 2 \, x^{2} + 1\right )} + 1\right )} + 1\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(4*(x^3 + x)*e^(x^4 + 2*x^2 + e^(x^4 + 2*x^2 + 1) + 1) + 1, x)

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maple [A]  time = 0.08, size = 12, normalized size = 0.80

method	result	size
risch	\(x +{\mathrm e}^{{\mathrm e}^{\left (x^{2}+1\right )^{2}}}\)	\(12\)
default	\(x +{\mathrm e}^{{\mathrm e}^{x^{4}+2 x^{2}+1}}\)	\(15\)
norman	\(x +{\mathrm e}^{{\mathrm e}^{x^{4}+2 x^{2}+1}}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x+exp(exp((x^2+1)^2))

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maxima [A]  time = 0.45, size = 14, normalized size = 0.93 \begin {gather*} x + e^{\left (e^{\left (x^{4} + 2 \, x^{2} + 1\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + e^(e^(x^4 + 2*x^2 + 1))

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mupad [B]  time = 3.15, size = 14, normalized size = 0.93 \begin {gather*} x+{\mathrm {e}}^{{\mathrm {e}}^{x^4+2\,x^2+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + exp(exp(2*x^2 + x^4 + 1))

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sympy [A]  time = 0.28, size = 14, normalized size = 0.93 \begin {gather*} x + e^{e^{x^{4} + 2 x^{2} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + exp(exp(x**4 + 2*x**2 + 1))

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