\(\int \frac {e^{\frac {2 x^2}{1-8 x^2+16 x^4}} (-2 x+20 x^3-112 x^5+128 x^7)}{-1+12 x^2-48 x^4+64 x^6} \, dx\) [10174]

Optimal result

Integrand size = 60, antiderivative size = 28 \[ \int \frac {e^{\frac {2 x^2}{1-8 x^2+16 x^4}} \left (-2 x+20 x^3-112 x^5+128 x^7\right )}{-1+12 x^2-48 x^4+64 x^6} \, dx=e^{\frac {2 x^2}{\left (1-4 x^2\right )^2}} x^2-\frac {\log (4)}{e} \]

[Out]

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

Rubi [F]

\[ \int \frac {e^{\frac {2 x^2}{1-8 x^2+16 x^4}} \left (-2 x+20 x^3-112 x^5+128 x^7\right )}{-1+12 x^2-48 x^4+64 x^6} \, dx=\int \frac {e^{\frac {2 x^2}{1-8 x^2+16 x^4}} \left (-2 x+20 x^3-112 x^5+128 x^7\right )}{-1+12 x^2-48 x^4+64 x^6} \, dx \]

[In]

Int[(E^((2*x^2)/(1 - 8*x^2 + 16*x^4))*(-2*x + 20*x^3 - 112*x^5 + 128*x^7))/(-1 + 12*x^2 - 48*x^4 + 64*x^6),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 e^{\frac {2 x^2}{1-8 x^2+16 x^4}} x \left (1-10 x^2+56 x^4-64 x^6\right )}{\left (1-4 x^2\right )^3} \, dx \\ & = 2 \int \frac {e^{\frac {2 x^2}{1-8 x^2+16 x^4}} x \left (1-10 x^2+56 x^4-64 x^6\right )}{\left (1-4 x^2\right )^3} \, dx \\ & = \text {Subst}\left (\int \frac {e^{\frac {2 x}{1-8 x+16 x^2}} \left (1-10 x+56 x^2-64 x^3\right )}{(1-4 x)^3} \, dx,x,x^2\right ) \\ & = \text {Subst}\left (\int \frac {e^{\frac {2 x}{(1-4 x)^2}} \left (1-10 x+56 x^2-64 x^3\right )}{(1-4 x)^3} \, dx,x,x^2\right ) \\ & = \text {Subst}\left (\int \left (e^{\frac {2 x}{(1-4 x)^2}}-\frac {e^{\frac {2 x}{(1-4 x)^2}}}{(-1+4 x)^3}-\frac {3 e^{\frac {2 x}{(1-4 x)^2}}}{2 (-1+4 x)^2}-\frac {e^{\frac {2 x}{(1-4 x)^2}}}{2 (-1+4 x)}\right ) \, dx,x,x^2\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {e^{\frac {2 x}{(1-4 x)^2}}}{-1+4 x} \, dx,x,x^2\right )\right )-\frac {3}{2} \text {Subst}\left (\int \frac {e^{\frac {2 x}{(1-4 x)^2}}}{(-1+4 x)^2} \, dx,x,x^2\right )+\text {Subst}\left (\int e^{\frac {2 x}{(1-4 x)^2}} \, dx,x,x^2\right )-\text {Subst}\left (\int \frac {e^{\frac {2 x}{(1-4 x)^2}}}{(-1+4 x)^3} \, dx,x,x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {e^{\frac {2 x^2}{1-8 x^2+16 x^4}} \left (-2 x+20 x^3-112 x^5+128 x^7\right )}{-1+12 x^2-48 x^4+64 x^6} \, dx=e^{\frac {2 x^2}{\left (1-4 x^2\right )^2}} x^2 \]

[In]

Integrate[(E^((2*x^2)/(1 - 8*x^2 + 16*x^4))*(-2*x + 20*x^3 - 112*x^5 + 128*x^7))/(-1 + 12*x^2 - 48*x^4 + 64*x^
6),x]

[Out]

E^((2*x^2)/(1 - 4*x^2)^2)*x^2

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89

method	result	size
risch	\(x^{2} {\mathrm e}^{\frac {2 x^{2}}{\left (-1+2 x \right )^{2} \left (1+2 x \right )^{2}}}\)	\(25\)
gosper	\(x^{2} {\mathrm e}^{\frac {2 x^{2}}{16 x^{4}-8 x^{2}+1}}\)	\(26\)
parallelrisch	\(x^{2} {\mathrm e}^{\frac {2 x^{2}}{16 x^{4}-8 x^{2}+1}}\)	\(26\)
norman	\(\frac {x^{2} {\mathrm e}^{\frac {2 x^{2}}{16 x^{4}-8 x^{2}+1}}-8 x^{4} {\mathrm e}^{\frac {2 x^{2}}{16 x^{4}-8 x^{2}+1}}+16 x^{6} {\mathrm e}^{\frac {2 x^{2}}{16 x^{4}-8 x^{2}+1}}}{\left (4 x^{2}-1\right )^{2}}\)	\(89\)

[In]

int((128*x^7-112*x^5+20*x^3-2*x)*exp(x^2/(16*x^4-8*x^2+1))^2/(64*x^6-48*x^4+12*x^2-1),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {e^{\frac {2 x^2}{1-8 x^2+16 x^4}} \left (-2 x+20 x^3-112 x^5+128 x^7\right )}{-1+12 x^2-48 x^4+64 x^6} \, dx=x^{2} e^{\left (\frac {2 \, x^{2}}{16 \, x^{4} - 8 \, x^{2} + 1}\right )} \]

[In]

integrate((128*x^7-112*x^5+20*x^3-2*x)*exp(x^2/(16*x^4-8*x^2+1))^2/(64*x^6-48*x^4+12*x^2-1),x, algorithm="fric
as")

[Out]

x^2*e^(2*x^2/(16*x^4 - 8*x^2 + 1))

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {e^{\frac {2 x^2}{1-8 x^2+16 x^4}} \left (-2 x+20 x^3-112 x^5+128 x^7\right )}{-1+12 x^2-48 x^4+64 x^6} \, dx=x^{2} e^{\frac {2 x^{2}}{16 x^{4} - 8 x^{2} + 1}} \]

[In]

integrate((128*x**7-112*x**5+20*x**3-2*x)*exp(x**2/(16*x**4-8*x**2+1))**2/(64*x**6-48*x**4+12*x**2-1),x)

[Out]

x**2*exp(2*x**2/(16*x**4 - 8*x**2 + 1))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86 \[ \int \frac {e^{\frac {2 x^2}{1-8 x^2+16 x^4}} \left (-2 x+20 x^3-112 x^5+128 x^7\right )}{-1+12 x^2-48 x^4+64 x^6} \, dx=x^{2} e^{\left (\frac {1}{8 \, {\left (4 \, x^{2} + 4 \, x + 1\right )}} + \frac {1}{8 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac {1}{8 \, {\left (2 \, x + 1\right )}} + \frac {1}{8 \, {\left (2 \, x - 1\right )}}\right )} \]

[In]

integrate((128*x^7-112*x^5+20*x^3-2*x)*exp(x^2/(16*x^4-8*x^2+1))^2/(64*x^6-48*x^4+12*x^2-1),x, algorithm="maxi
ma")

[Out]

x^2*e^(1/8/(4*x^2 + 4*x + 1) + 1/8/(4*x^2 - 4*x + 1) - 1/8/(2*x + 1) + 1/8/(2*x - 1))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {e^{\frac {2 x^2}{1-8 x^2+16 x^4}} \left (-2 x+20 x^3-112 x^5+128 x^7\right )}{-1+12 x^2-48 x^4+64 x^6} \, dx=x^{2} e^{\left (\frac {2 \, x^{2}}{16 \, x^{4} - 8 \, x^{2} + 1}\right )} \]

[In]

integrate((128*x^7-112*x^5+20*x^3-2*x)*exp(x^2/(16*x^4-8*x^2+1))^2/(64*x^6-48*x^4+12*x^2-1),x, algorithm="giac
")

[Out]

x^2*e^(2*x^2/(16*x^4 - 8*x^2 + 1))

Mupad [B] (verification not implemented)

Time = 15.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {e^{\frac {2 x^2}{1-8 x^2+16 x^4}} \left (-2 x+20 x^3-112 x^5+128 x^7\right )}{-1+12 x^2-48 x^4+64 x^6} \, dx=x^2\,{\mathrm {e}}^{\frac {2\,x^2}{16\,x^4-8\,x^2+1}} \]

[In]

int(-(exp((2*x^2)/(16*x^4 - 8*x^2 + 1))*(2*x - 20*x^3 + 112*x^5 - 128*x^7))/(12*x^2 - 48*x^4 + 64*x^6 - 1),x)

[Out]

x^2*exp((2*x^2)/(16*x^4 - 8*x^2 + 1))