3.35.50 \(\int \frac {e^{e^{4+256 x^2+e^{e^4} x^2+256 x^3+96 x^4+16 x^5+x^6}} (2+e^{4+256 x^2+e^{e^4} x^2+256 x^3+96 x^4+16 x^5+x^6} (-1024 x^2-1024 x^3+224 x^5+68 x^6+6 x^7+e^{e^4} (-4 x^2+2 x^3)))}{x^2} \, dx\) [3450]

Optimal. Leaf size=28 \[ \frac {e^{e^{4+x^2 \left (e^{e^4}+(4+x)^4\right )}} (-2+x)}{x} \]

[Out]

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

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(119\) vs. \(2(28)=56\).
time = 0.56, antiderivative size = 119, normalized size of antiderivative = 4.25, number of steps used = 1, number of rules used = 1, integrand size = 124, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {2326} \begin {gather*} -\frac {\left (-3 x^7-34 x^6-112 x^5+512 x^3+512 x^2+e^{e^4} \left (2 x^2-x^3\right )\right ) \exp \left (\exp \left (x^6+16 x^5+96 x^4+256 x^3+e^{e^4} x^2+256 x^2+4\right )\right )}{x^2 \left (3 x^5+40 x^4+192 x^3+384 x^2+e^{e^4} x+256 x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^E^(4 + 256*x^2 + E^E^4*x^2 + 256*x^3 + 96*x^4 + 16*x^5 + x^6)*(2 + E^(4 + 256*x^2 + E^E^4*x^2 + 256*x^3
+ 96*x^4 + 16*x^5 + x^6)*(-1024*x^2 - 1024*x^3 + 224*x^5 + 68*x^6 + 6*x^7 + E^E^4*(-4*x^2 + 2*x^3))))/x^2,x]

[Out]

-((E^E^(4 + 256*x^2 + E^E^4*x^2 + 256*x^3 + 96*x^4 + 16*x^5 + x^6)*(512*x^2 + 512*x^3 - 112*x^5 - 34*x^6 - 3*x
^7 + E^E^4*(2*x^2 - x^3)))/(x^2*(256*x + E^E^4*x + 384*x^2 + 192*x^3 + 40*x^4 + 3*x^5)))

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {\exp \left (\exp \left (4+256 x^2+e^{e^4} x^2+256 x^3+96 x^4+16 x^5+x^6\right )\right ) \left (512 x^2+512 x^3-112 x^5-34 x^6-3 x^7+e^{e^4} \left (2 x^2-x^3\right )\right )}{x^2 \left (256 x+e^{e^4} x+384 x^2+192 x^3+40 x^4+3 x^5\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.12, size = 43, normalized size = 1.54 \begin {gather*} e^{e^{4+\left (256+e^{e^4}\right ) x^2+256 x^3+96 x^4+16 x^5+x^6}} \left (1-\frac {2}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^E^(4 + 256*x^2 + E^E^4*x^2 + 256*x^3 + 96*x^4 + 16*x^5 + x^6)*(2 + E^(4 + 256*x^2 + E^E^4*x^2 + 2
56*x^3 + 96*x^4 + 16*x^5 + x^6)*(-1024*x^2 - 1024*x^3 + 224*x^5 + 68*x^6 + 6*x^7 + E^E^4*(-4*x^2 + 2*x^3))))/x
^2,x]

[Out]

E^E^(4 + (256 + E^E^4)*x^2 + 256*x^3 + 96*x^4 + 16*x^5 + x^6)*(1 - 2/x)

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Maple [A]
time = 0.39, size = 42, normalized size = 1.50

method	result	size
risch	\(\frac {\left (x -2\right ) {\mathrm e}^{{\mathrm e}^{x^{2} {\mathrm e}^{{\mathrm e}^{4}}+x^{6}+16 x^{5}+96 x^{4}+256 x^{3}+256 x^{2}+4}}}{x}\)	\(42\)
norman	\(\frac {x \,{\mathrm e}^{{\mathrm e}^{x^{2} {\mathrm e}^{{\mathrm e}^{4}}+x^{6}+16 x^{5}+96 x^{4}+256 x^{3}+256 x^{2}+4}}-2 \,{\mathrm e}^{{\mathrm e}^{x^{2} {\mathrm e}^{{\mathrm e}^{4}}+x^{6}+16 x^{5}+96 x^{4}+256 x^{3}+256 x^{2}+4}}}{x}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^3-4*x^2)*exp(exp(4))+6*x^7+68*x^6+224*x^5-1024*x^3-1024*x^2)*exp(x^2*exp(exp(4))+x^6+16*x^5+96*x^4+
256*x^3+256*x^2+4)+2)*exp(exp(x^2*exp(exp(4))+x^6+16*x^5+96*x^4+256*x^3+256*x^2+4))/x^2,x,method=_RETURNVERBOS
E)

[Out]

(x-2)/x*exp(exp(x^2*exp(exp(4))+x^6+16*x^5+96*x^4+256*x^3+256*x^2+4))

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Maxima [A]
time = 0.34, size = 41, normalized size = 1.46 \begin {gather*} \frac {{\left (x - 2\right )} e^{\left (e^{\left (x^{6} + 16 \, x^{5} + 96 \, x^{4} + 256 \, x^{3} + x^{2} e^{\left (e^{4}\right )} + 256 \, x^{2} + 4\right )}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3-4*x^2)*exp(exp(4))+6*x^7+68*x^6+224*x^5-1024*x^3-1024*x^2)*exp(x^2*exp(exp(4))+x^6+16*x^5+9
6*x^4+256*x^3+256*x^2+4)+2)*exp(exp(x^2*exp(exp(4))+x^6+16*x^5+96*x^4+256*x^3+256*x^2+4))/x^2,x, algorithm="ma
xima")

[Out]

(x - 2)*e^(e^(x^6 + 16*x^5 + 96*x^4 + 256*x^3 + x^2*e^(e^4) + 256*x^2 + 4))/x

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Fricas [A]
time = 0.36, size = 41, normalized size = 1.46 \begin {gather*} \frac {{\left (x - 2\right )} e^{\left (e^{\left (x^{6} + 16 \, x^{5} + 96 \, x^{4} + 256 \, x^{3} + x^{2} e^{\left (e^{4}\right )} + 256 \, x^{2} + 4\right )}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3-4*x^2)*exp(exp(4))+6*x^7+68*x^6+224*x^5-1024*x^3-1024*x^2)*exp(x^2*exp(exp(4))+x^6+16*x^5+9
6*x^4+256*x^3+256*x^2+4)+2)*exp(exp(x^2*exp(exp(4))+x^6+16*x^5+96*x^4+256*x^3+256*x^2+4))/x^2,x, algorithm="fr
icas")

[Out]

(x - 2)*e^(e^(x^6 + 16*x^5 + 96*x^4 + 256*x^3 + x^2*e^(e^4) + 256*x^2 + 4))/x

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Sympy [A]
time = 0.13, size = 41, normalized size = 1.46 \begin {gather*} \frac {\left (x - 2\right ) e^{e^{x^{6} + 16 x^{5} + 96 x^{4} + 256 x^{3} + 256 x^{2} + x^{2} e^{e^{4}} + 4}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**3-4*x**2)*exp(exp(4))+6*x**7+68*x**6+224*x**5-1024*x**3-1024*x**2)*exp(x**2*exp(exp(4))+x**6
+16*x**5+96*x**4+256*x**3+256*x**2+4)+2)*exp(exp(x**2*exp(exp(4))+x**6+16*x**5+96*x**4+256*x**3+256*x**2+4))/x
**2,x)

[Out]

(x - 2)*exp(exp(x**6 + 16*x**5 + 96*x**4 + 256*x**3 + 256*x**2 + x**2*exp(exp(4)) + 4))/x

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3-4*x^2)*exp(exp(4))+6*x^7+68*x^6+224*x^5-1024*x^3-1024*x^2)*exp(x^2*exp(exp(4))+x^6+16*x^5+9
6*x^4+256*x^3+256*x^2+4)+2)*exp(exp(x^2*exp(exp(4))+x^6+16*x^5+96*x^4+256*x^3+256*x^2+4))/x^2,x, algorithm="gi
ac")

[Out]

integrate(2*((3*x^7 + 34*x^6 + 112*x^5 - 512*x^3 - 512*x^2 + (x^3 - 2*x^2)*e^(e^4))*e^(x^6 + 16*x^5 + 96*x^4 +
256*x^3 + x^2*e^(e^4) + 256*x^2 + 4) + 1)*e^(e^(x^6 + 16*x^5 + 96*x^4 + 256*x^3 + x^2*e^(e^4) + 256*x^2 + 4))
/x^2, x)

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Mupad [B]
time = 2.36, size = 47, normalized size = 1.68 \begin {gather*} \frac {{\mathrm {e}}^{{\mathrm {e}}^{x^6}\,{\mathrm {e}}^4\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{{\mathrm {e}}^4}}\,{\mathrm {e}}^{16\,x^5}\,{\mathrm {e}}^{96\,x^4}\,{\mathrm {e}}^{256\,x^2}\,{\mathrm {e}}^{256\,x^3}}\,\left (x-2\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(x^2*exp(exp(4)) + 256*x^2 + 256*x^3 + 96*x^4 + 16*x^5 + x^6 + 4))*(exp(x^2*exp(exp(4)) + 256*x^2
+ 256*x^3 + 96*x^4 + 16*x^5 + x^6 + 4)*(1024*x^2 + 1024*x^3 - 224*x^5 - 68*x^6 - 6*x^7 + exp(exp(4))*(4*x^2 -
2*x^3)) - 2))/x^2,x)

[Out]

(exp(exp(x^6)*exp(4)*exp(x^2*exp(exp(4)))*exp(16*x^5)*exp(96*x^4)*exp(256*x^2)*exp(256*x^3))*(x - 2))/x

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