\(\int \frac {e^{\frac {8-6 x+5 x^2-x^3-x^4+e^{4 e^{x^2} x} (4-5 x+x^2+x^3)}{e^{4 e^{x^2} x} (-4+x)+4 x-x^2}} (-32+16 x+14 x^2-8 x^3-11 x^4+2 x^5+e^{8 e^{x^2} x} (16-8 x-11 x^2+2 x^3)+e^{4 e^{x^2} x} (-32 x+16 x^2+22 x^3-4 x^4+e^{x^2} (128-64 x+264 x^2-128 x^3+16 x^4)))}{16 x^2-8 x^3+x^4+e^{8 e^{x^2} x} (16-8 x+x^2)+e^{4 e^{x^2} x} (-32 x+16 x^2-2 x^3)} \, dx\) [1125]

Optimal result

Integrand size = 238, antiderivative size = 32 \[ \int \frac {e^{\frac {8-6 x+5 x^2-x^3-x^4+e^{4 e^{x^2} x} \left (4-5 x+x^2+x^3\right )}{e^{4 e^{x^2} x} (-4+x)+4 x-x^2}} \left (-32+16 x+14 x^2-8 x^3-11 x^4+2 x^5+e^{8 e^{x^2} x} \left (16-8 x-11 x^2+2 x^3\right )+e^{4 e^{x^2} x} \left (-32 x+16 x^2+22 x^3-4 x^4+e^{x^2} \left (128-64 x+264 x^2-128 x^3+16 x^4\right )\right )\right )}{16 x^2-8 x^3+x^4+e^{8 e^{x^2} x} \left (16-8 x+x^2\right )+e^{4 e^{x^2} x} \left (-32 x+16 x^2-2 x^3\right )} \, dx=e^{-1-\frac {2}{e^{4 e^{x^2} x}-x}+x+\frac {x^3}{-4+x}} \]

[Out]

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

Rubi [F(-1)]

Timed out. \[ \int \frac {e^{\frac {8-6 x+5 x^2-x^3-x^4+e^{4 e^{x^2} x} \left (4-5 x+x^2+x^3\right )}{e^{4 e^{x^2} x} (-4+x)+4 x-x^2}} \left (-32+16 x+14 x^2-8 x^3-11 x^4+2 x^5+e^{8 e^{x^2} x} \left (16-8 x-11 x^2+2 x^3\right )+e^{4 e^{x^2} x} \left (-32 x+16 x^2+22 x^3-4 x^4+e^{x^2} \left (128-64 x+264 x^2-128 x^3+16 x^4\right )\right )\right )}{16 x^2-8 x^3+x^4+e^{8 e^{x^2} x} \left (16-8 x+x^2\right )+e^{4 e^{x^2} x} \left (-32 x+16 x^2-2 x^3\right )} \, dx=\text {\$Aborted} \]

[In]

Int[(E^((8 - 6*x + 5*x^2 - x^3 - x^4 + E^(4*E^x^2*x)*(4 - 5*x + x^2 + x^3))/(E^(4*E^x^2*x)*(-4 + x) + 4*x - x^
2))*(-32 + 16*x + 14*x^2 - 8*x^3 - 11*x^4 + 2*x^5 + E^(8*E^x^2*x)*(16 - 8*x - 11*x^2 + 2*x^3) + E^(4*E^x^2*x)*
(-32*x + 16*x^2 + 22*x^3 - 4*x^4 + E^x^2*(128 - 64*x + 264*x^2 - 128*x^3 + 16*x^4))))/(16*x^2 - 8*x^3 + x^4 +
E^(8*E^x^2*x)*(16 - 8*x + x^2) + E^(4*E^x^2*x)*(-32*x + 16*x^2 - 2*x^3)),x]

[Out]

$Aborted

Rubi steps Aborted

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {e^{\frac {8-6 x+5 x^2-x^3-x^4+e^{4 e^{x^2} x} \left (4-5 x+x^2+x^3\right )}{e^{4 e^{x^2} x} (-4+x)+4 x-x^2}} \left (-32+16 x+14 x^2-8 x^3-11 x^4+2 x^5+e^{8 e^{x^2} x} \left (16-8 x-11 x^2+2 x^3\right )+e^{4 e^{x^2} x} \left (-32 x+16 x^2+22 x^3-4 x^4+e^{x^2} \left (128-64 x+264 x^2-128 x^3+16 x^4\right )\right )\right )}{16 x^2-8 x^3+x^4+e^{8 e^{x^2} x} \left (16-8 x+x^2\right )+e^{4 e^{x^2} x} \left (-32 x+16 x^2-2 x^3\right )} \, dx=e^{15-\frac {2}{e^{4 e^{x^2} x}-x}+\frac {64}{-4+x}+5 x+x^2} \]

[In]

Integrate[(E^((8 - 6*x + 5*x^2 - x^3 - x^4 + E^(4*E^x^2*x)*(4 - 5*x + x^2 + x^3))/(E^(4*E^x^2*x)*(-4 + x) + 4*
x - x^2))*(-32 + 16*x + 14*x^2 - 8*x^3 - 11*x^4 + 2*x^5 + E^(8*E^x^2*x)*(16 - 8*x - 11*x^2 + 2*x^3) + E^(4*E^x
^2*x)*(-32*x + 16*x^2 + 22*x^3 - 4*x^4 + E^x^2*(128 - 64*x + 264*x^2 - 128*x^3 + 16*x^4))))/(16*x^2 - 8*x^3 +
x^4 + E^(8*E^x^2*x)*(16 - 8*x + x^2) + E^(4*E^x^2*x)*(-32*x + 16*x^2 - 2*x^3)),x]

[Out]

E^(15 - 2/(E^(4*E^x^2*x) - x) + 64/(-4 + x) + 5*x + x^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(73\) vs. \(2(29)=58\).

Time = 73.95 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.31

method	result	size
parallelrisch	\({\mathrm e}^{\frac {\left (x^{3}+x^{2}-5 x +4\right ) {\mathrm e}^{4 \,{\mathrm e}^{x^{2}} x}-x^{4}-x^{3}+5 x^{2}-6 x +8}{{\mathrm e}^{4 \,{\mathrm e}^{x^{2}} x} x -x^{2}-4 \,{\mathrm e}^{4 \,{\mathrm e}^{x^{2}} x}+4 x}}\)	\(74\)
risch	\({\mathrm e}^{\frac {-{\mathrm e}^{4 \,{\mathrm e}^{x^{2}} x} x^{3}+x^{4}-{\mathrm e}^{4 \,{\mathrm e}^{x^{2}} x} x^{2}+x^{3}+5 \,{\mathrm e}^{4 \,{\mathrm e}^{x^{2}} x} x -5 x^{2}-4 \,{\mathrm e}^{4 \,{\mathrm e}^{x^{2}} x}+6 x -8}{\left (x -4\right ) \left (-{\mathrm e}^{4 \,{\mathrm e}^{x^{2}} x}+x \right )}}\)	\(85\)

[In]

int(((2*x^3-11*x^2-8*x+16)*exp(4*exp(x^2)*x)^2+((16*x^4-128*x^3+264*x^2-64*x+128)*exp(x^2)-4*x^4+22*x^3+16*x^2
-32*x)*exp(4*exp(x^2)*x)+2*x^5-11*x^4-8*x^3+14*x^2+16*x-32)*exp(((x^3+x^2-5*x+4)*exp(4*exp(x^2)*x)-x^4-x^3+5*x
^2-6*x+8)/((x-4)*exp(4*exp(x^2)*x)-x^2+4*x))/((x^2-8*x+16)*exp(4*exp(x^2)*x)^2+(-2*x^3+16*x^2-32*x)*exp(4*exp(
x^2)*x)+x^4-8*x^3+16*x^2),x,method=_RETURNVERBOSE)

[Out]

exp(((x^3+x^2-5*x+4)*exp(4*exp(x^2)*x)-x^4-x^3+5*x^2-6*x+8)/(exp(4*exp(x^2)*x)*x-x^2-4*exp(4*exp(x^2)*x)+4*x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (29) = 58\).

Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.91 \[ \int \frac {e^{\frac {8-6 x+5 x^2-x^3-x^4+e^{4 e^{x^2} x} \left (4-5 x+x^2+x^3\right )}{e^{4 e^{x^2} x} (-4+x)+4 x-x^2}} \left (-32+16 x+14 x^2-8 x^3-11 x^4+2 x^5+e^{8 e^{x^2} x} \left (16-8 x-11 x^2+2 x^3\right )+e^{4 e^{x^2} x} \left (-32 x+16 x^2+22 x^3-4 x^4+e^{x^2} \left (128-64 x+264 x^2-128 x^3+16 x^4\right )\right )\right )}{16 x^2-8 x^3+x^4+e^{8 e^{x^2} x} \left (16-8 x+x^2\right )+e^{4 e^{x^2} x} \left (-32 x+16 x^2-2 x^3\right )} \, dx=e^{\left (\frac {x^{4} + x^{3} - 5 \, x^{2} - {\left (x^{3} + x^{2} - 5 \, x + 4\right )} e^{\left (4 \, x e^{\left (x^{2}\right )}\right )} + 6 \, x - 8}{x^{2} - {\left (x - 4\right )} e^{\left (4 \, x e^{\left (x^{2}\right )}\right )} - 4 \, x}\right )} \]

[In]

integrate(((2*x^3-11*x^2-8*x+16)*exp(4*exp(x^2)*x)^2+((16*x^4-128*x^3+264*x^2-64*x+128)*exp(x^2)-4*x^4+22*x^3+
16*x^2-32*x)*exp(4*exp(x^2)*x)+2*x^5-11*x^4-8*x^3+14*x^2+16*x-32)*exp(((x^3+x^2-5*x+4)*exp(4*exp(x^2)*x)-x^4-x
^3+5*x^2-6*x+8)/((x-4)*exp(4*exp(x^2)*x)-x^2+4*x))/((x^2-8*x+16)*exp(4*exp(x^2)*x)^2+(-2*x^3+16*x^2-32*x)*exp(
4*exp(x^2)*x)+x^4-8*x^3+16*x^2),x, algorithm="fricas")

[Out]

e^((x^4 + x^3 - 5*x^2 - (x^3 + x^2 - 5*x + 4)*e^(4*x*e^(x^2)) + 6*x - 8)/(x^2 - (x - 4)*e^(4*x*e^(x^2)) - 4*x)
)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (24) = 48\).

Time = 2.21 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.88 \[ \int \frac {e^{\frac {8-6 x+5 x^2-x^3-x^4+e^{4 e^{x^2} x} \left (4-5 x+x^2+x^3\right )}{e^{4 e^{x^2} x} (-4+x)+4 x-x^2}} \left (-32+16 x+14 x^2-8 x^3-11 x^4+2 x^5+e^{8 e^{x^2} x} \left (16-8 x-11 x^2+2 x^3\right )+e^{4 e^{x^2} x} \left (-32 x+16 x^2+22 x^3-4 x^4+e^{x^2} \left (128-64 x+264 x^2-128 x^3+16 x^4\right )\right )\right )}{16 x^2-8 x^3+x^4+e^{8 e^{x^2} x} \left (16-8 x+x^2\right )+e^{4 e^{x^2} x} \left (-32 x+16 x^2-2 x^3\right )} \, dx=e^{\frac {- x^{4} - x^{3} + 5 x^{2} - 6 x + \left (x^{3} + x^{2} - 5 x + 4\right ) e^{4 x e^{x^{2}}} + 8}{- x^{2} + 4 x + \left (x - 4\right ) e^{4 x e^{x^{2}}}}} \]

[In]

integrate(((2*x**3-11*x**2-8*x+16)*exp(4*exp(x**2)*x)**2+((16*x**4-128*x**3+264*x**2-64*x+128)*exp(x**2)-4*x**
4+22*x**3+16*x**2-32*x)*exp(4*exp(x**2)*x)+2*x**5-11*x**4-8*x**3+14*x**2+16*x-32)*exp(((x**3+x**2-5*x+4)*exp(4
*exp(x**2)*x)-x**4-x**3+5*x**2-6*x+8)/((x-4)*exp(4*exp(x**2)*x)-x**2+4*x))/((x**2-8*x+16)*exp(4*exp(x**2)*x)**
2+(-2*x**3+16*x**2-32*x)*exp(4*exp(x**2)*x)+x**4-8*x**3+16*x**2),x)

[Out]

exp((-x**4 - x**3 + 5*x**2 - 6*x + (x**3 + x**2 - 5*x + 4)*exp(4*x*exp(x**2)) + 8)/(-x**2 + 4*x + (x - 4)*exp(
4*x*exp(x**2))))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (29) = 58\).

Time = 1.01 (sec) , antiderivative size = 127, normalized size of antiderivative = 3.97 \[ \int \frac {e^{\frac {8-6 x+5 x^2-x^3-x^4+e^{4 e^{x^2} x} \left (4-5 x+x^2+x^3\right )}{e^{4 e^{x^2} x} (-4+x)+4 x-x^2}} \left (-32+16 x+14 x^2-8 x^3-11 x^4+2 x^5+e^{8 e^{x^2} x} \left (16-8 x-11 x^2+2 x^3\right )+e^{4 e^{x^2} x} \left (-32 x+16 x^2+22 x^3-4 x^4+e^{x^2} \left (128-64 x+264 x^2-128 x^3+16 x^4\right )\right )\right )}{16 x^2-8 x^3+x^4+e^{8 e^{x^2} x} \left (16-8 x+x^2\right )+e^{4 e^{x^2} x} \left (-32 x+16 x^2-2 x^3\right )} \, dx=e^{\left (x^{2} + 5 \, x + \frac {2 \, e^{\left (4 \, x e^{\left (x^{2}\right )}\right )}}{{\left (x + 4\right )} e^{\left (4 \, x e^{\left (x^{2}\right )}\right )} - 4 \, x - e^{\left (8 \, x e^{\left (x^{2}\right )}\right )}} + \frac {64 \, e^{\left (4 \, x e^{\left (x^{2}\right )}\right )}}{{\left (x - 4\right )} e^{\left (4 \, x e^{\left (x^{2}\right )}\right )} - 4 \, x + 16} - \frac {8}{{\left (x + 4\right )} e^{\left (4 \, x e^{\left (x^{2}\right )}\right )} - 4 \, x - e^{\left (8 \, x e^{\left (x^{2}\right )}\right )}} - \frac {256}{{\left (x - 4\right )} e^{\left (4 \, x e^{\left (x^{2}\right )}\right )} - 4 \, x + 16} + 15\right )} \]

[In]

integrate(((2*x^3-11*x^2-8*x+16)*exp(4*exp(x^2)*x)^2+((16*x^4-128*x^3+264*x^2-64*x+128)*exp(x^2)-4*x^4+22*x^3+
16*x^2-32*x)*exp(4*exp(x^2)*x)+2*x^5-11*x^4-8*x^3+14*x^2+16*x-32)*exp(((x^3+x^2-5*x+4)*exp(4*exp(x^2)*x)-x^4-x
^3+5*x^2-6*x+8)/((x-4)*exp(4*exp(x^2)*x)-x^2+4*x))/((x^2-8*x+16)*exp(4*exp(x^2)*x)^2+(-2*x^3+16*x^2-32*x)*exp(
4*exp(x^2)*x)+x^4-8*x^3+16*x^2),x, algorithm="maxima")

[Out]

e^(x^2 + 5*x + 2*e^(4*x*e^(x^2))/((x + 4)*e^(4*x*e^(x^2)) - 4*x - e^(8*x*e^(x^2))) + 64*e^(4*x*e^(x^2))/((x -
4)*e^(4*x*e^(x^2)) - 4*x + 16) - 8/((x + 4)*e^(4*x*e^(x^2)) - 4*x - e^(8*x*e^(x^2))) - 256/((x - 4)*e^(4*x*e^(
x^2)) - 4*x + 16) + 15)

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{\frac {8-6 x+5 x^2-x^3-x^4+e^{4 e^{x^2} x} \left (4-5 x+x^2+x^3\right )}{e^{4 e^{x^2} x} (-4+x)+4 x-x^2}} \left (-32+16 x+14 x^2-8 x^3-11 x^4+2 x^5+e^{8 e^{x^2} x} \left (16-8 x-11 x^2+2 x^3\right )+e^{4 e^{x^2} x} \left (-32 x+16 x^2+22 x^3-4 x^4+e^{x^2} \left (128-64 x+264 x^2-128 x^3+16 x^4\right )\right )\right )}{16 x^2-8 x^3+x^4+e^{8 e^{x^2} x} \left (16-8 x+x^2\right )+e^{4 e^{x^2} x} \left (-32 x+16 x^2-2 x^3\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(((2*x^3-11*x^2-8*x+16)*exp(4*exp(x^2)*x)^2+((16*x^4-128*x^3+264*x^2-64*x+128)*exp(x^2)-4*x^4+22*x^3+
16*x^2-32*x)*exp(4*exp(x^2)*x)+2*x^5-11*x^4-8*x^3+14*x^2+16*x-32)*exp(((x^3+x^2-5*x+4)*exp(4*exp(x^2)*x)-x^4-x
^3+5*x^2-6*x+8)/((x-4)*exp(4*exp(x^2)*x)-x^2+4*x))/((x^2-8*x+16)*exp(4*exp(x^2)*x)^2+(-2*x^3+16*x^2-32*x)*exp(
4*exp(x^2)*x)+x^4-8*x^3+16*x^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{4194304,[0,6,35]%%%}+%%%{-257949696,[0,6,34]%%%}+%%%{739
4557952,[0,

Mupad [B] (verification not implemented)

Time = 9.23 (sec) , antiderivative size = 354, normalized size of antiderivative = 11.06 \[ \int \frac {e^{\frac {8-6 x+5 x^2-x^3-x^4+e^{4 e^{x^2} x} \left (4-5 x+x^2+x^3\right )}{e^{4 e^{x^2} x} (-4+x)+4 x-x^2}} \left (-32+16 x+14 x^2-8 x^3-11 x^4+2 x^5+e^{8 e^{x^2} x} \left (16-8 x-11 x^2+2 x^3\right )+e^{4 e^{x^2} x} \left (-32 x+16 x^2+22 x^3-4 x^4+e^{x^2} \left (128-64 x+264 x^2-128 x^3+16 x^4\right )\right )\right )}{16 x^2-8 x^3+x^4+e^{8 e^{x^2} x} \left (16-8 x+x^2\right )+e^{4 e^{x^2} x} \left (-32 x+16 x^2-2 x^3\right )} \, dx={\mathrm {e}}^{\frac {x^2\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}}}{4\,x-4\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}}+x\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}}-x^2}}\,{\mathrm {e}}^{\frac {x^3\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}}}{4\,x-4\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}}+x\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}}-x^2}}\,{\mathrm {e}}^{-\frac {6\,x}{4\,x-4\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}}+x\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}}-x^2}}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}}}{4\,x-4\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}}+x\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}}-x^2}}\,{\mathrm {e}}^{-\frac {x^3}{4\,x-4\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}}+x\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}}-x^2}}\,{\mathrm {e}}^{-\frac {x^4}{4\,x-4\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}}+x\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}}-x^2}}\,{\mathrm {e}}^{\frac {5\,x^2}{4\,x-4\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}}+x\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}}-x^2}}\,{\mathrm {e}}^{-\frac {5\,x\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}}}{4\,x-4\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}}+x\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}}-x^2}}\,{\mathrm {e}}^{\frac {8}{4\,x-4\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}}+x\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{x^2}}-x^2}} \]

[In]

int((exp(-(6*x - exp(4*x*exp(x^2))*(x^2 - 5*x + x^3 + 4) - 5*x^2 + x^3 + x^4 - 8)/(4*x + exp(4*x*exp(x^2))*(x
- 4) - x^2))*(16*x + exp(4*x*exp(x^2))*(exp(x^2)*(264*x^2 - 64*x - 128*x^3 + 16*x^4 + 128) - 32*x + 16*x^2 + 2
2*x^3 - 4*x^4) - exp(8*x*exp(x^2))*(8*x + 11*x^2 - 2*x^3 - 16) + 14*x^2 - 8*x^3 - 11*x^4 + 2*x^5 - 32))/(16*x^
2 - exp(4*x*exp(x^2))*(32*x - 16*x^2 + 2*x^3) - 8*x^3 + x^4 + exp(8*x*exp(x^2))*(x^2 - 8*x + 16)),x)

[Out]

exp((x^2*exp(4*x*exp(x^2)))/(4*x - 4*exp(4*x*exp(x^2)) + x*exp(4*x*exp(x^2)) - x^2))*exp((x^3*exp(4*x*exp(x^2)
))/(4*x - 4*exp(4*x*exp(x^2)) + x*exp(4*x*exp(x^2)) - x^2))*exp(-(6*x)/(4*x - 4*exp(4*x*exp(x^2)) + x*exp(4*x*
exp(x^2)) - x^2))*exp((4*exp(4*x*exp(x^2)))/(4*x - 4*exp(4*x*exp(x^2)) + x*exp(4*x*exp(x^2)) - x^2))*exp(-x^3/
(4*x - 4*exp(4*x*exp(x^2)) + x*exp(4*x*exp(x^2)) - x^2))*exp(-x^4/(4*x - 4*exp(4*x*exp(x^2)) + x*exp(4*x*exp(x
^2)) - x^2))*exp((5*x^2)/(4*x - 4*exp(4*x*exp(x^2)) + x*exp(4*x*exp(x^2)) - x^2))*exp(-(5*x*exp(4*x*exp(x^2)))
/(4*x - 4*exp(4*x*exp(x^2)) + x*exp(4*x*exp(x^2)) - x^2))*exp(8/(4*x - 4*exp(4*x*exp(x^2)) + x*exp(4*x*exp(x^2
)) - x^2))