3.6.93 \(\int \frac {e^{e^{\frac {-4 x^2+4 x^4+e^4 (1+x)}{4 x}}+\frac {-4 x^2+4 x^4+e^4 (1+x)}{4 x}} (-e^4-4 x^2+12 x^4)}{4 x^2} \, dx\)

Optimal. Leaf size=23 \[ e^{e^{\left (-1+\frac {e^4}{4 x^2}+x\right ) \left (x+x^2\right )}} \]

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Rubi [F]  time = 2.77, 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 \frac {\exp \left (e^{\frac {-4 x^2+4 x^4+e^4 (1+x)}{4 x}}+\frac {-4 x^2+4 x^4+e^4 (1+x)}{4 x}\right ) \left (-e^4-4 x^2+12 x^4\right )}{4 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(E^((-4*x^2 + 4*x^4 + E^4*(1 + x))/(4*x)) + (-4*x^2 + 4*x^4 + E^4*(1 + x))/(4*x))*(-E^4 - 4*x^2 + 12*x^
4))/(4*x^2),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {\exp \left (e^{\frac {-4 x^2+4 x^4+e^4 (1+x)}{4 x}}+\frac {-4 x^2+4 x^4+e^4 (1+x)}{4 x}\right ) \left (-e^4-4 x^2+12 x^4\right )}{x^2} \, dx\\ &=\frac {1}{4} \int \left (-4 \exp \left (e^{\frac {-4 x^2+4 x^4+e^4 (1+x)}{4 x}}+\frac {-4 x^2+4 x^4+e^4 (1+x)}{4 x}\right )-\frac {\exp \left (4+e^{\frac {-4 x^2+4 x^4+e^4 (1+x)}{4 x}}+\frac {-4 x^2+4 x^4+e^4 (1+x)}{4 x}\right )}{x^2}+12 \exp \left (e^{\frac {-4 x^2+4 x^4+e^4 (1+x)}{4 x}}+\frac {-4 x^2+4 x^4+e^4 (1+x)}{4 x}\right ) x^2\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {\exp \left (4+e^{\frac {-4 x^2+4 x^4+e^4 (1+x)}{4 x}}+\frac {-4 x^2+4 x^4+e^4 (1+x)}{4 x}\right )}{x^2} \, dx\right )+3 \int \exp \left (e^{\frac {-4 x^2+4 x^4+e^4 (1+x)}{4 x}}+\frac {-4 x^2+4 x^4+e^4 (1+x)}{4 x}\right ) x^2 \, dx-\int \exp \left (e^{\frac {-4 x^2+4 x^4+e^4 (1+x)}{4 x}}+\frac {-4 x^2+4 x^4+e^4 (1+x)}{4 x}\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.75, size = 28, normalized size = 1.22 \begin {gather*} e^{e^{\frac {e^4}{4}+\frac {e^4}{4 x}-x+x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^E^(E^4/4 + E^4/(4*x) - x + x^3)

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fricas [B]  time = 0.79, size = 72, normalized size = 3.13 \begin {gather*} e^{\left (\frac {4 \, x^{4} - 4 \, x^{2} + {\left (x + 1\right )} e^{4} + 4 \, x e^{\left (\frac {4 \, x^{4} - 4 \, x^{2} + {\left (x + 1\right )} e^{4}}{4 \, x}\right )}}{4 \, x} - \frac {4 \, x^{4} - 4 \, x^{2} + {\left (x + 1\right )} e^{4}}{4 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-exp(4)+12*x^4-4*x^2)*exp(1/4*((x+1)*exp(4)+4*x^4-4*x^2)/x)*exp(exp(1/4*((x+1)*exp(4)+4*x^4-4*x
^2)/x))/x^2,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (12 \, x^{4} - 4 \, x^{2} - e^{4}\right )} e^{\left (\frac {4 \, x^{4} - 4 \, x^{2} + {\left (x + 1\right )} e^{4}}{4 \, x} + e^{\left (\frac {4 \, x^{4} - 4 \, x^{2} + {\left (x + 1\right )} e^{4}}{4 \, x}\right )}\right )}}{4 \, x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-exp(4)+12*x^4-4*x^2)*exp(1/4*((x+1)*exp(4)+4*x^4-4*x^2)/x)*exp(exp(1/4*((x+1)*exp(4)+4*x^4-4*x
^2)/x))/x^2,x, algorithm="giac")

[Out]

integrate(1/4*(12*x^4 - 4*x^2 - e^4)*e^(1/4*(4*x^4 - 4*x^2 + (x + 1)*e^4)/x + e^(1/4*(4*x^4 - 4*x^2 + (x + 1)*
e^4)/x))/x^2, x)

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maple [A]  time = 0.30, size = 24, normalized size = 1.04

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{\frac {\left (x +1\right ) \left (4 x^{3}-4 x^{2}+{\mathrm e}^{4}\right )}{4 x}}}\)	\(24\)
norman	\({\mathrm e}^{{\mathrm e}^{\frac {\left (x +1\right ) {\mathrm e}^{4}+4 x^{4}-4 x^{2}}{4 x}}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(-exp(4)+12*x^4-4*x^2)*exp(1/4*((x+1)*exp(4)+4*x^4-4*x^2)/x)*exp(exp(1/4*((x+1)*exp(4)+4*x^4-4*x^2)/x)
)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 1.28, size = 20, normalized size = 0.87 \begin {gather*} e^{\left (e^{\left (x^{3} - x + \frac {e^{4}}{4 \, x} + \frac {1}{4} \, e^{4}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-exp(4)+12*x^4-4*x^2)*exp(1/4*((x+1)*exp(4)+4*x^4-4*x^2)/x)*exp(exp(1/4*((x+1)*exp(4)+4*x^4-4*x
^2)/x))/x^2,x, algorithm="maxima")

[Out]

e^(e^(x^3 - x + 1/4*e^4/x + 1/4*e^4))

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mupad [B]  time = 0.81, size = 23, normalized size = 1.00 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^{\frac {{\mathrm {e}}^4}{4\,x}}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^4}{4}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{x^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(exp(exp(4)/(4*x))*exp(exp(4)/4)*exp(-x)*exp(x^3))

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sympy [A]  time = 0.40, size = 19, normalized size = 0.83 \begin {gather*} e^{e^{\frac {x^{4} - x^{2} + \frac {\left (x + 1\right ) e^{4}}{4}}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-exp(4)+12*x**4-4*x**2)*exp(1/4*((x+1)*exp(4)+4*x**4-4*x**2)/x)*exp(exp(1/4*((x+1)*exp(4)+4*x**
4-4*x**2)/x))/x**2,x)

[Out]

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

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