3.96.79 \(\int \frac {e^{\frac {2 e^{8 x^4}-4 e^{4 x^4} x^4+2 x^8}{x^4}} (8 x^8-64 e^{4 x^4} x^8+e^{8 x^4} (-8+64 x^4))}{x^5} \, dx\)

Optimal. Leaf size=22 \[ e^{2 \left (-\frac {e^{4 x^4}}{x^2}+x^2\right )^2} \]

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Rubi [A]  time = 0.56, antiderivative size = 30, normalized size of antiderivative = 1.36, number of steps used = 1, number of rules used = 1, integrand size = 70, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {6706} \begin {gather*} e^{\frac {2 \left (x^8-2 e^{4 x^4} x^4+e^{8 x^4}\right )}{x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((2*E^(8*x^4) - 4*E^(4*x^4)*x^4 + 2*x^8)/x^4)*(8*x^8 - 64*E^(4*x^4)*x^8 + E^(8*x^4)*(-8 + 64*x^4)))/x^5
,x]

[Out]

E^((2*(E^(8*x^4) - 2*E^(4*x^4)*x^4 + x^8))/x^4)

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

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

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Mathematica [A]  time = 0.55, size = 22, normalized size = 1.00 \begin {gather*} e^{\frac {2 \left (e^{4 x^4}-x^4\right )^2}{x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((2*E^(8*x^4) - 4*E^(4*x^4)*x^4 + 2*x^8)/x^4)*(8*x^8 - 64*E^(4*x^4)*x^8 + E^(8*x^4)*(-8 + 64*x^4)
))/x^5,x]

[Out]

E^((2*(E^(4*x^4) - x^4)^2)/x^4)

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fricas [A]  time = 0.49, size = 27, normalized size = 1.23 \begin {gather*} e^{\left (\frac {2 \, {\left (x^{8} - 2 \, x^{4} e^{\left (4 \, x^{4}\right )} + e^{\left (8 \, x^{4}\right )}\right )}}{x^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((64*x^4-8)*exp(4*x^4)^2-64*x^8*exp(4*x^4)+8*x^8)*exp((2*exp(4*x^4)^2-4*x^4*exp(4*x^4)+2*x^8)/x^4)/x
^5,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.21, size = 26, normalized size = 1.18 \begin {gather*} e^{\left (2 \, x^{4} + \frac {2 \, e^{\left (8 \, x^{4}\right )}}{x^{4}} - 4 \, e^{\left (4 \, x^{4}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((64*x^4-8)*exp(4*x^4)^2-64*x^8*exp(4*x^4)+8*x^8)*exp((2*exp(4*x^4)^2-4*x^4*exp(4*x^4)+2*x^8)/x^4)/x
^5,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 28, normalized size = 1.27

method	result	size
risch	\({\mathrm e}^{\frac {2 \,{\mathrm e}^{8 x^{4}}-4 x^{4} {\mathrm e}^{4 x^{4}}+2 x^{8}}{x^{4}}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((64*x^4-8)*exp(4*x^4)^2-64*x^8*exp(4*x^4)+8*x^8)*exp((2*exp(4*x^4)^2-4*x^4*exp(4*x^4)+2*x^8)/x^4)/x^5,x,m
ethod=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.44, size = 26, normalized size = 1.18 \begin {gather*} e^{\left (2 \, x^{4} + \frac {2 \, e^{\left (8 \, x^{4}\right )}}{x^{4}} - 4 \, e^{\left (4 \, x^{4}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((64*x^4-8)*exp(4*x^4)^2-64*x^8*exp(4*x^4)+8*x^8)*exp((2*exp(4*x^4)^2-4*x^4*exp(4*x^4)+2*x^8)/x^4)/x
^5,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 6.75, size = 28, normalized size = 1.27 \begin {gather*} {\mathrm {e}}^{\frac {2\,{\mathrm {e}}^{8\,x^4}}{x^4}}\,{\mathrm {e}}^{-4\,{\mathrm {e}}^{4\,x^4}}\,{\mathrm {e}}^{2\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((2*exp(8*x^4) - 4*x^4*exp(4*x^4) + 2*x^8)/x^4)*(exp(8*x^4)*(64*x^4 - 8) - 64*x^8*exp(4*x^4) + 8*x^8))
/x^5,x)

[Out]

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

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sympy [A]  time = 0.36, size = 29, normalized size = 1.32 \begin {gather*} e^{\frac {2 x^{8} - 4 x^{4} e^{4 x^{4}} + 2 e^{8 x^{4}}}{x^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((64*x**4-8)*exp(4*x**4)**2-64*x**8*exp(4*x**4)+8*x**8)*exp((2*exp(4*x**4)**2-4*x**4*exp(4*x**4)+2*x
**8)/x**4)/x**5,x)

[Out]

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

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