\(\int \frac {e^{\frac {192+96 x^2+12 x^4+e^{2 x^2} (12 x^2-4 e^4 x^2)+e^4 (-64-32 x^2-4 x^4)+e^{x^2} (96 x+24 x^3+e^4 (-32 x-8 x^3))}{x^4}} (-768-192 x^2+e^4 (256+64 x^2)+e^{2 x^2} (-24 x^2+48 x^4+e^4 (8 x^2-16 x^4))+e^{x^2} (-288 x+168 x^3+48 x^5+e^4 (96 x-56 x^3-16 x^5)))}{x^5} \, dx\) [6465]

Optimal result

Integrand size = 178, antiderivative size = 30 \[ \int \frac {e^{\frac {192+96 x^2+12 x^4+e^{2 x^2} \left (12 x^2-4 e^4 x^2\right )+e^4 \left (-64-32 x^2-4 x^4\right )+e^{x^2} \left (96 x+24 x^3+e^4 \left (-32 x-8 x^3\right )\right )}{x^4}} \left (-768-192 x^2+e^4 \left (256+64 x^2\right )+e^{2 x^2} \left (-24 x^2+48 x^4+e^4 \left (8 x^2-16 x^4\right )\right )+e^{x^2} \left (-288 x+168 x^3+48 x^5+e^4 \left (96 x-56 x^3-16 x^5\right )\right )\right )}{x^5} \, dx=5+e^{\frac {4 \left (3-e^4\right ) \left (e^{x^2}+\frac {4}{x}+x\right )^2}{x^2}} \]

[Out]

exp(4*(exp(x^2)+x+4/x)^2/x^2*(3-exp(4)))+5

Rubi [A] (verified)

Time = 3.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {6820, 12, 6838} \[ \int \frac {e^{\frac {192+96 x^2+12 x^4+e^{2 x^2} \left (12 x^2-4 e^4 x^2\right )+e^4 \left (-64-32 x^2-4 x^4\right )+e^{x^2} \left (96 x+24 x^3+e^4 \left (-32 x-8 x^3\right )\right )}{x^4}} \left (-768-192 x^2+e^4 \left (256+64 x^2\right )+e^{2 x^2} \left (-24 x^2+48 x^4+e^4 \left (8 x^2-16 x^4\right )\right )+e^{x^2} \left (-288 x+168 x^3+48 x^5+e^4 \left (96 x-56 x^3-16 x^5\right )\right )\right )}{x^5} \, dx=e^{\frac {4 \left (3-e^4\right ) \left (x^2+e^{x^2} x+4\right )^2}{x^4}} \]

[In]

Int[(E^((192 + 96*x^2 + 12*x^4 + E^(2*x^2)*(12*x^2 - 4*E^4*x^2) + E^4*(-64 - 32*x^2 - 4*x^4) + E^x^2*(96*x + 2
4*x^3 + E^4*(-32*x - 8*x^3)))/x^4)*(-768 - 192*x^2 + E^4*(256 + 64*x^2) + E^(2*x^2)*(-24*x^2 + 48*x^4 + E^4*(8
*x^2 - 16*x^4)) + E^x^2*(-288*x + 168*x^3 + 48*x^5 + E^4*(96*x - 56*x^3 - 16*x^5))))/x^5,x]

[Out]

E^((4*(3 - E^4)*(4 + E^x^2*x + x^2)^2)/x^4)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6838

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{align*} \text {integral}& = \int \frac {8 e^{-\frac {4 \left (-3+e^4\right ) \left (4+e^{x^2} x+x^2\right )^2}{x^4}} \left (3-e^4\right ) \left (-8 \left (4+x^2\right )+e^{2 x^2} x^2 \left (-1+2 x^2\right )+e^{x^2} x \left (-12+7 x^2+2 x^4\right )\right )}{x^5} \, dx \\ & = \left (8 \left (3-e^4\right )\right ) \int \frac {e^{-\frac {4 \left (-3+e^4\right ) \left (4+e^{x^2} x+x^2\right )^2}{x^4}} \left (-8 \left (4+x^2\right )+e^{2 x^2} x^2 \left (-1+2 x^2\right )+e^{x^2} x \left (-12+7 x^2+2 x^4\right )\right )}{x^5} \, dx \\ & = e^{\frac {4 \left (3-e^4\right ) \left (4+e^{x^2} x+x^2\right )^2}{x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {e^{\frac {192+96 x^2+12 x^4+e^{2 x^2} \left (12 x^2-4 e^4 x^2\right )+e^4 \left (-64-32 x^2-4 x^4\right )+e^{x^2} \left (96 x+24 x^3+e^4 \left (-32 x-8 x^3\right )\right )}{x^4}} \left (-768-192 x^2+e^4 \left (256+64 x^2\right )+e^{2 x^2} \left (-24 x^2+48 x^4+e^4 \left (8 x^2-16 x^4\right )\right )+e^{x^2} \left (-288 x+168 x^3+48 x^5+e^4 \left (96 x-56 x^3-16 x^5\right )\right )\right )}{x^5} \, dx=e^{-\frac {4 \left (-3+e^4\right ) \left (4+e^{x^2} x+x^2\right )^2}{x^4}} \]

[In]

Integrate[(E^((192 + 96*x^2 + 12*x^4 + E^(2*x^2)*(12*x^2 - 4*E^4*x^2) + E^4*(-64 - 32*x^2 - 4*x^4) + E^x^2*(96
*x + 24*x^3 + E^4*(-32*x - 8*x^3)))/x^4)*(-768 - 192*x^2 + E^4*(256 + 64*x^2) + E^(2*x^2)*(-24*x^2 + 48*x^4 +
E^4*(8*x^2 - 16*x^4)) + E^x^2*(-288*x + 168*x^3 + 48*x^5 + E^4*(96*x - 56*x^3 - 16*x^5))))/x^5,x]

[Out]

E^((-4*(-3 + E^4)*(4 + E^x^2*x + x^2)^2)/x^4)

Maple [A] (verified)

Time = 10.84 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57

method	result	size
risch	\({\mathrm e}^{-\frac {4 \left ({\mathrm e}^{4}-3\right ) \left (2 x^{3} {\mathrm e}^{x^{2}}+x^{4}+{\mathrm e}^{2 x^{2}} x^{2}+8 \,{\mathrm e}^{x^{2}} x +8 x^{2}+16\right )}{x^{4}}}\)	\(47\)
parallelrisch	\({\mathrm e}^{\frac {\left (-4 x^{2} {\mathrm e}^{4}+12 x^{2}\right ) {\mathrm e}^{2 x^{2}}+\left (\left (-8 x^{3}-32 x \right ) {\mathrm e}^{4}+24 x^{3}+96 x \right ) {\mathrm e}^{x^{2}}+\left (-4 x^{4}-32 x^{2}-64\right ) {\mathrm e}^{4}+12 x^{4}+96 x^{2}+192}{x^{4}}}\)	\(79\)

[In]

int((((-16*x^4+8*x^2)*exp(4)+48*x^4-24*x^2)*exp(x^2)^2+((-16*x^5-56*x^3+96*x)*exp(4)+48*x^5+168*x^3-288*x)*exp
(x^2)+(64*x^2+256)*exp(4)-192*x^2-768)*exp(((-4*x^2*exp(4)+12*x^2)*exp(x^2)^2+((-8*x^3-32*x)*exp(4)+24*x^3+96*
x)*exp(x^2)+(-4*x^4-32*x^2-64)*exp(4)+12*x^4+96*x^2+192)/x^4)/x^5,x,method=_RETURNVERBOSE)

[Out]

exp(-4*(exp(4)-3)*(2*x^3*exp(x^2)+x^4+exp(2*x^2)*x^2+8*exp(x^2)*x+8*x^2+16)/x^4)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (25) = 50\).

Time = 0.26 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.60 \[ \int \frac {e^{\frac {192+96 x^2+12 x^4+e^{2 x^2} \left (12 x^2-4 e^4 x^2\right )+e^4 \left (-64-32 x^2-4 x^4\right )+e^{x^2} \left (96 x+24 x^3+e^4 \left (-32 x-8 x^3\right )\right )}{x^4}} \left (-768-192 x^2+e^4 \left (256+64 x^2\right )+e^{2 x^2} \left (-24 x^2+48 x^4+e^4 \left (8 x^2-16 x^4\right )\right )+e^{x^2} \left (-288 x+168 x^3+48 x^5+e^4 \left (96 x-56 x^3-16 x^5\right )\right )\right )}{x^5} \, dx=e^{\left (\frac {4 \, {\left (3 \, x^{4} + 24 \, x^{2} - {\left (x^{4} + 8 \, x^{2} + 16\right )} e^{4} - {\left (x^{2} e^{4} - 3 \, x^{2}\right )} e^{\left (2 \, x^{2}\right )} + 2 \, {\left (3 \, x^{3} - {\left (x^{3} + 4 \, x\right )} e^{4} + 12 \, x\right )} e^{\left (x^{2}\right )} + 48\right )}}{x^{4}}\right )} \]

[In]

integrate((((-16*x^4+8*x^2)*exp(4)+48*x^4-24*x^2)*exp(x^2)^2+((-16*x^5-56*x^3+96*x)*exp(4)+48*x^5+168*x^3-288*
x)*exp(x^2)+(64*x^2+256)*exp(4)-192*x^2-768)*exp(((-4*x^2*exp(4)+12*x^2)*exp(x^2)^2+((-8*x^3-32*x)*exp(4)+24*x
^3+96*x)*exp(x^2)+(-4*x^4-32*x^2-64)*exp(4)+12*x^4+96*x^2+192)/x^4)/x^5,x, algorithm="fricas")

[Out]

e^(4*(3*x^4 + 24*x^2 - (x^4 + 8*x^2 + 16)*e^4 - (x^2*e^4 - 3*x^2)*e^(2*x^2) + 2*(3*x^3 - (x^3 + 4*x)*e^4 + 12*
x)*e^(x^2) + 48)/x^4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (24) = 48\).

Time = 0.53 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.67 \[ \int \frac {e^{\frac {192+96 x^2+12 x^4+e^{2 x^2} \left (12 x^2-4 e^4 x^2\right )+e^4 \left (-64-32 x^2-4 x^4\right )+e^{x^2} \left (96 x+24 x^3+e^4 \left (-32 x-8 x^3\right )\right )}{x^4}} \left (-768-192 x^2+e^4 \left (256+64 x^2\right )+e^{2 x^2} \left (-24 x^2+48 x^4+e^4 \left (8 x^2-16 x^4\right )\right )+e^{x^2} \left (-288 x+168 x^3+48 x^5+e^4 \left (96 x-56 x^3-16 x^5\right )\right )\right )}{x^5} \, dx=e^{\frac {12 x^{4} + 96 x^{2} + \left (- 4 x^{2} e^{4} + 12 x^{2}\right ) e^{2 x^{2}} + \left (24 x^{3} + 96 x + \left (- 8 x^{3} - 32 x\right ) e^{4}\right ) e^{x^{2}} + \left (- 4 x^{4} - 32 x^{2} - 64\right ) e^{4} + 192}{x^{4}}} \]

[In]

integrate((((-16*x**4+8*x**2)*exp(4)+48*x**4-24*x**2)*exp(x**2)**2+((-16*x**5-56*x**3+96*x)*exp(4)+48*x**5+168
*x**3-288*x)*exp(x**2)+(64*x**2+256)*exp(4)-192*x**2-768)*exp(((-4*x**2*exp(4)+12*x**2)*exp(x**2)**2+((-8*x**3
-32*x)*exp(4)+24*x**3+96*x)*exp(x**2)+(-4*x**4-32*x**2-64)*exp(4)+12*x**4+96*x**2+192)/x**4)/x**5,x)

[Out]

exp((12*x**4 + 96*x**2 + (-4*x**2*exp(4) + 12*x**2)*exp(2*x**2) + (24*x**3 + 96*x + (-8*x**3 - 32*x)*exp(4))*e
xp(x**2) + (-4*x**4 - 32*x**2 - 64)*exp(4) + 192)/x**4)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (25) = 50\).

Time = 0.57 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.17 \[ \int \frac {e^{\frac {192+96 x^2+12 x^4+e^{2 x^2} \left (12 x^2-4 e^4 x^2\right )+e^4 \left (-64-32 x^2-4 x^4\right )+e^{x^2} \left (96 x+24 x^3+e^4 \left (-32 x-8 x^3\right )\right )}{x^4}} \left (-768-192 x^2+e^4 \left (256+64 x^2\right )+e^{2 x^2} \left (-24 x^2+48 x^4+e^4 \left (8 x^2-16 x^4\right )\right )+e^{x^2} \left (-288 x+168 x^3+48 x^5+e^4 \left (96 x-56 x^3-16 x^5\right )\right )\right )}{x^5} \, dx=e^{\left (-\frac {8 \, e^{\left (x^{2} + 4\right )}}{x} + \frac {24 \, e^{\left (x^{2}\right )}}{x} - \frac {32 \, e^{4}}{x^{2}} + \frac {12 \, e^{\left (2 \, x^{2}\right )}}{x^{2}} - \frac {4 \, e^{\left (2 \, x^{2} + 4\right )}}{x^{2}} + \frac {96}{x^{2}} - \frac {32 \, e^{\left (x^{2} + 4\right )}}{x^{3}} + \frac {96 \, e^{\left (x^{2}\right )}}{x^{3}} - \frac {64 \, e^{4}}{x^{4}} + \frac {192}{x^{4}} - 4 \, e^{4} + 12\right )} \]

[In]

integrate((((-16*x^4+8*x^2)*exp(4)+48*x^4-24*x^2)*exp(x^2)^2+((-16*x^5-56*x^3+96*x)*exp(4)+48*x^5+168*x^3-288*
x)*exp(x^2)+(64*x^2+256)*exp(4)-192*x^2-768)*exp(((-4*x^2*exp(4)+12*x^2)*exp(x^2)^2+((-8*x^3-32*x)*exp(4)+24*x
^3+96*x)*exp(x^2)+(-4*x^4-32*x^2-64)*exp(4)+12*x^4+96*x^2+192)/x^4)/x^5,x, algorithm="maxima")

[Out]

e^(-8*e^(x^2 + 4)/x + 24*e^(x^2)/x - 32*e^4/x^2 + 12*e^(2*x^2)/x^2 - 4*e^(2*x^2 + 4)/x^2 + 96/x^2 - 32*e^(x^2
+ 4)/x^3 + 96*e^(x^2)/x^3 - 64*e^4/x^4 + 192/x^4 - 4*e^4 + 12)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (25) = 50\).

Time = 0.56 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.17 \[ \int \frac {e^{\frac {192+96 x^2+12 x^4+e^{2 x^2} \left (12 x^2-4 e^4 x^2\right )+e^4 \left (-64-32 x^2-4 x^4\right )+e^{x^2} \left (96 x+24 x^3+e^4 \left (-32 x-8 x^3\right )\right )}{x^4}} \left (-768-192 x^2+e^4 \left (256+64 x^2\right )+e^{2 x^2} \left (-24 x^2+48 x^4+e^4 \left (8 x^2-16 x^4\right )\right )+e^{x^2} \left (-288 x+168 x^3+48 x^5+e^4 \left (96 x-56 x^3-16 x^5\right )\right )\right )}{x^5} \, dx=e^{\left (-\frac {8 \, e^{\left (x^{2} + 4\right )}}{x} + \frac {24 \, e^{\left (x^{2}\right )}}{x} - \frac {32 \, e^{4}}{x^{2}} + \frac {12 \, e^{\left (2 \, x^{2}\right )}}{x^{2}} - \frac {4 \, e^{\left (2 \, x^{2} + 4\right )}}{x^{2}} + \frac {96}{x^{2}} - \frac {32 \, e^{\left (x^{2} + 4\right )}}{x^{3}} + \frac {96 \, e^{\left (x^{2}\right )}}{x^{3}} - \frac {64 \, e^{4}}{x^{4}} + \frac {192}{x^{4}} - 4 \, e^{4} + 12\right )} \]

[In]

integrate((((-16*x^4+8*x^2)*exp(4)+48*x^4-24*x^2)*exp(x^2)^2+((-16*x^5-56*x^3+96*x)*exp(4)+48*x^5+168*x^3-288*
x)*exp(x^2)+(64*x^2+256)*exp(4)-192*x^2-768)*exp(((-4*x^2*exp(4)+12*x^2)*exp(x^2)^2+((-8*x^3-32*x)*exp(4)+24*x
^3+96*x)*exp(x^2)+(-4*x^4-32*x^2-64)*exp(4)+12*x^4+96*x^2+192)/x^4)/x^5,x, algorithm="giac")

[Out]

e^(-8*e^(x^2 + 4)/x + 24*e^(x^2)/x - 32*e^4/x^2 + 12*e^(2*x^2)/x^2 - 4*e^(2*x^2 + 4)/x^2 + 96/x^2 - 32*e^(x^2
+ 4)/x^3 + 96*e^(x^2)/x^3 - 64*e^4/x^4 + 192/x^4 - 4*e^4 + 12)

Mupad [B] (verification not implemented)

Time = 12.81 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.53 \[ \int \frac {e^{\frac {192+96 x^2+12 x^4+e^{2 x^2} \left (12 x^2-4 e^4 x^2\right )+e^4 \left (-64-32 x^2-4 x^4\right )+e^{x^2} \left (96 x+24 x^3+e^4 \left (-32 x-8 x^3\right )\right )}{x^4}} \left (-768-192 x^2+e^4 \left (256+64 x^2\right )+e^{2 x^2} \left (-24 x^2+48 x^4+e^4 \left (8 x^2-16 x^4\right )\right )+e^{x^2} \left (-288 x+168 x^3+48 x^5+e^4 \left (96 x-56 x^3-16 x^5\right )\right )\right )}{x^5} \, dx={\mathrm {e}}^{-\frac {32\,{\mathrm {e}}^4}{x^2}}\,{\mathrm {e}}^{-\frac {64\,{\mathrm {e}}^4}{x^4}}\,{\mathrm {e}}^{-4\,{\mathrm {e}}^4}\,{\mathrm {e}}^{\frac {12\,{\mathrm {e}}^{2\,x^2}}{x^2}}\,{\mathrm {e}}^{-\frac {8\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^4}{x}}\,{\mathrm {e}}^{-\frac {32\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^4}{x^3}}\,{\mathrm {e}}^{12}\,{\mathrm {e}}^{\frac {96}{x^2}}\,{\mathrm {e}}^{\frac {192}{x^4}}\,{\mathrm {e}}^{-\frac {4\,{\mathrm {e}}^4\,{\mathrm {e}}^{2\,x^2}}{x^2}}\,{\mathrm {e}}^{\frac {24\,{\mathrm {e}}^{x^2}}{x}}\,{\mathrm {e}}^{\frac {96\,{\mathrm {e}}^{x^2}}{x^3}} \]

[In]

int(-(exp((exp(x^2)*(96*x - exp(4)*(32*x + 8*x^3) + 24*x^3) - exp(2*x^2)*(4*x^2*exp(4) - 12*x^2) - exp(4)*(32*
x^2 + 4*x^4 + 64) + 96*x^2 + 12*x^4 + 192)/x^4)*(exp(x^2)*(288*x + exp(4)*(56*x^3 - 96*x + 16*x^5) - 168*x^3 -
48*x^5) - exp(4)*(64*x^2 + 256) - exp(2*x^2)*(exp(4)*(8*x^2 - 16*x^4) - 24*x^2 + 48*x^4) + 192*x^2 + 768))/x^
5,x)

[Out]

exp(-(32*exp(4))/x^2)*exp(-(64*exp(4))/x^4)*exp(-4*exp(4))*exp((12*exp(2*x^2))/x^2)*exp(-(8*exp(x^2)*exp(4))/x
)*exp(-(32*exp(x^2)*exp(4))/x^3)*exp(12)*exp(96/x^2)*exp(192/x^4)*exp(-(4*exp(4)*exp(2*x^2))/x^2)*exp((24*exp(
x^2))/x)*exp((96*exp(x^2))/x^3)