\(\int \frac {e^{-2-2 x^2} (-6-8 x^2+e^{6-x^2} (-30-20 x^2))}{x^4+25 e^{12-2 x^2} x^4+10 e^{6-x^2} x^4} \, dx\) [3225]

Optimal result

Integrand size = 68, antiderivative size = 30 \[ \int \frac {e^{-2-2 x^2} \left (-6-8 x^2+e^{6-x^2} \left (-30-20 x^2\right )\right )}{x^4+25 e^{12-2 x^2} x^4+10 e^{6-x^2} x^4} \, dx=\frac {2 e^{-2-2 x^2}}{x^2 \left (x+5 e^{6-x^2} x\right )} \]

[Out]

2/x^2/exp(x^2+1)^2/(5*x*exp(-x^2+6)+x)

Rubi [F]

\[ \int \frac {e^{-2-2 x^2} \left (-6-8 x^2+e^{6-x^2} \left (-30-20 x^2\right )\right )}{x^4+25 e^{12-2 x^2} x^4+10 e^{6-x^2} x^4} \, dx=\int \frac {e^{-2-2 x^2} \left (-6-8 x^2+e^{6-x^2} \left (-30-20 x^2\right )\right )}{x^4+25 e^{12-2 x^2} x^4+10 e^{6-x^2} x^4} \, dx \]

[In]

Int[(E^(-2 - 2*x^2)*(-6 - 8*x^2 + E^(6 - x^2)*(-30 - 20*x^2)))/(x^4 + 25*E^(12 - 2*x^2)*x^4 + 10*E^(6 - x^2)*x
^4),x]

[Out]

(2*E^(-8 - x^2))/(5*x^3) + (6*Defer[Int][1/((5*E^6 + E^x^2)*x^4), x])/(5*E^8) - (4*Defer[Int][1/((5*E^6 + E^x^
2)^2*x^2), x])/E^2 + (4*Defer[Int][1/((5*E^6 + E^x^2)*x^2), x])/(5*E^8)

Rubi steps \begin{align*} \text {integral}& = \int \frac {-6-8 x^2+e^{6-x^2} \left (-30-20 x^2\right )}{e^2 \left (5 e^6+e^{x^2}\right )^2 x^4} \, dx \\ & = \frac {\int \frac {-6-8 x^2+e^{6-x^2} \left (-30-20 x^2\right )}{\left (5 e^6+e^{x^2}\right )^2 x^4} \, dx}{e^2} \\ & = \frac {\int \left (-\frac {4}{\left (5 e^6+e^{x^2}\right )^2 x^2}-\frac {2 e^{-6-x^2} \left (3+2 x^2\right )}{5 x^4}+\frac {2 \left (3+2 x^2\right )}{5 e^6 \left (5 e^6+e^{x^2}\right ) x^4}\right ) \, dx}{e^2} \\ & = \frac {2 \int \frac {3+2 x^2}{\left (5 e^6+e^{x^2}\right ) x^4} \, dx}{5 e^8}-\frac {2 \int \frac {e^{-6-x^2} \left (3+2 x^2\right )}{x^4} \, dx}{5 e^2}-\frac {4 \int \frac {1}{\left (5 e^6+e^{x^2}\right )^2 x^2} \, dx}{e^2} \\ & = \frac {2 e^{-8-x^2}}{5 x^3}+\frac {2 \int \left (\frac {3}{\left (5 e^6+e^{x^2}\right ) x^4}+\frac {2}{\left (5 e^6+e^{x^2}\right ) x^2}\right ) \, dx}{5 e^8}-\frac {4 \int \frac {1}{\left (5 e^6+e^{x^2}\right )^2 x^2} \, dx}{e^2} \\ & = \frac {2 e^{-8-x^2}}{5 x^3}+\frac {4 \int \frac {1}{\left (5 e^6+e^{x^2}\right ) x^2} \, dx}{5 e^8}+\frac {6 \int \frac {1}{\left (5 e^6+e^{x^2}\right ) x^4} \, dx}{5 e^8}-\frac {4 \int \frac {1}{\left (5 e^6+e^{x^2}\right )^2 x^2} \, dx}{e^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.36 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-2-2 x^2} \left (-6-8 x^2+e^{6-x^2} \left (-30-20 x^2\right )\right )}{x^4+25 e^{12-2 x^2} x^4+10 e^{6-x^2} x^4} \, dx=\frac {2 e^{-2-x^2}}{\left (5 e^6+e^{x^2}\right ) x^3} \]

[In]

Integrate[(E^(-2 - 2*x^2)*(-6 - 8*x^2 + E^(6 - x^2)*(-30 - 20*x^2)))/(x^4 + 25*E^(12 - 2*x^2)*x^4 + 10*E^(6 -
x^2)*x^4),x]

[Out]

(2*E^(-2 - x^2))/((5*E^6 + E^x^2)*x^3)

Maple [A] (verified)

Time = 2.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93

method	result	size
risch	\(\frac {2 \,{\mathrm e}^{-2 x^{2}-2}}{x^{3} \left (1+5 \,{\mathrm e}^{-x^{2}+6}\right )}\)	\(28\)
parallelrisch	\(\frac {2 \,{\mathrm e}^{-2 x^{2}-2}}{x^{3} \left (1+5 \,{\mathrm e}^{-x^{2}+6}\right )}\)	\(28\)
norman	\(\frac {2 \,{\mathrm e}^{-14} {\mathrm e}^{-2 x^{2}+12}}{x^{3} \left (1+5 \,{\mathrm e}^{-x^{2}+6}\right )}\)	\(34\)

[In]

int(((-20*x^2-30)*exp(-x^2+6)-8*x^2-6)/(25*x^4*exp(-x^2+6)^2+10*x^4*exp(-x^2+6)+x^4)/exp(x^2+1)^2,x,method=_RE
TURNVERBOSE)

[Out]

2*exp(-2*x^2-2)/x^3/(1+5*exp(-x^2+6))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-2-2 x^2} \left (-6-8 x^2+e^{6-x^2} \left (-30-20 x^2\right )\right )}{x^4+25 e^{12-2 x^2} x^4+10 e^{6-x^2} x^4} \, dx=\frac {2 \, e^{\left (-2 \, x^{2} + 12\right )}}{x^{3} e^{14} + 5 \, x^{3} e^{\left (-x^{2} + 20\right )}} \]

[In]

integrate(((-20*x^2-30)*exp(-x^2+6)-8*x^2-6)/(25*x^4*exp(-x^2+6)^2+10*x^4*exp(-x^2+6)+x^4)/exp(x^2+1)^2,x, alg
orithm="fricas")

[Out]

2*e^(-2*x^2 + 12)/(x^3*e^14 + 5*x^3*e^(-x^2 + 20))

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.70 \[ \int \frac {e^{-2-2 x^2} \left (-6-8 x^2+e^{6-x^2} \left (-30-20 x^2\right )\right )}{x^4+25 e^{12-2 x^2} x^4+10 e^{6-x^2} x^4} \, dx=\frac {2}{125 x^{3} e^{14} e^{6 - x^{2}} + 25 x^{3} e^{14}} + \frac {2 e^{6 - x^{2}}}{5 x^{3} e^{14}} - \frac {2}{25 x^{3} e^{14}} \]

[In]

integrate(((-20*x**2-30)*exp(-x**2+6)-8*x**2-6)/(25*x**4*exp(-x**2+6)**2+10*x**4*exp(-x**2+6)+x**4)/exp(x**2+1
)**2,x)

[Out]

2/(125*x**3*exp(14)*exp(6 - x**2) + 25*x**3*exp(14)) + 2*exp(-14)*exp(6 - x**2)/(5*x**3) - 2*exp(-14)/(25*x**3
)

Maxima [F]

\[ \int \frac {e^{-2-2 x^2} \left (-6-8 x^2+e^{6-x^2} \left (-30-20 x^2\right )\right )}{x^4+25 e^{12-2 x^2} x^4+10 e^{6-x^2} x^4} \, dx=\int { -\frac {2 \, {\left (4 \, x^{2} + 5 \, {\left (2 \, x^{2} + 3\right )} e^{\left (-x^{2} + 6\right )} + 3\right )} e^{\left (-2 \, x^{2} - 2\right )}}{10 \, x^{4} e^{\left (-x^{2} + 6\right )} + 25 \, x^{4} e^{\left (-2 \, x^{2} + 12\right )} + x^{4}} \,d x } \]

[In]

integrate(((-20*x^2-30)*exp(-x^2+6)-8*x^2-6)/(25*x^4*exp(-x^2+6)^2+10*x^4*exp(-x^2+6)+x^4)/exp(x^2+1)^2,x, alg
orithm="maxima")

[Out]

-2*integrate((4*x^2 + 5*(2*x^2 + 3)*e^(-x^2 + 6) + 3)*e^(-2*x^2 - 2)/(10*x^4*e^(-x^2 + 6) + 25*x^4*e^(-2*x^2 +
12) + x^4), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (28) = 56\).

Time = 0.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.03 \[ \int \frac {e^{-2-2 x^2} \left (-6-8 x^2+e^{6-x^2} \left (-30-20 x^2\right )\right )}{x^4+25 e^{12-2 x^2} x^4+10 e^{6-x^2} x^4} \, dx=\frac {2 \, {\left (x^{2} e^{\left (-x^{2} + 6\right )} + 5 \, x^{2} e^{\left (-2 \, x^{2} + 12\right )}\right )}}{10 \, x^{5} e^{14} + x^{5} e^{\left (x^{2} + 8\right )} + 25 \, x^{5} e^{\left (-x^{2} + 20\right )}} \]

[In]

integrate(((-20*x^2-30)*exp(-x^2+6)-8*x^2-6)/(25*x^4*exp(-x^2+6)^2+10*x^4*exp(-x^2+6)+x^4)/exp(x^2+1)^2,x, alg
orithm="giac")

[Out]

2*(x^2*e^(-x^2 + 6) + 5*x^2*e^(-2*x^2 + 12))/(10*x^5*e^14 + x^5*e^(x^2 + 8) + 25*x^5*e^(-x^2 + 20))

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-2-2 x^2} \left (-6-8 x^2+e^{6-x^2} \left (-30-20 x^2\right )\right )}{x^4+25 e^{12-2 x^2} x^4+10 e^{6-x^2} x^4} \, dx=\frac {2\,{\mathrm {e}}^{12}\,{\mathrm {e}}^{-2\,x^2}}{x^3\,{\mathrm {e}}^{14}+5\,x^3\,{\mathrm {e}}^{20}\,{\mathrm {e}}^{-x^2}} \]

[In]

int(-(exp(- 2*x^2 - 2)*(8*x^2 + exp(6 - x^2)*(20*x^2 + 30) + 6))/(10*x^4*exp(6 - x^2) + 25*x^4*exp(12 - 2*x^2)
+ x^4),x)

[Out]

(2*exp(12)*exp(-2*x^2))/(x^3*exp(14) + 5*x^3*exp(20)*exp(-x^2))