3.33.25 $\int \frac{e^{-2-2x^2}(-6-8x^2+e^{6-x^2}(-30-20x^2))}{x^4+25e^{12-2x^2}{x}^4+10e^{6-x^2}{x}^4}dx$

Optimal. Leaf size=30 $$\frac{2e^{-2-2x^2}}{x^2(x+5e^{6-x^2}x)}$$

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Rubi [F]  time = 1.36, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.000, Rules used = {} $$\begin{array}{c}\int \frac{e^{-2-2x^2}(-6-8x^2+e^{6-x^2}(-30-20x^2))}{x^4+25e^{12-2x^2}{x}^4+10e^{6-x^2}{x}^4}dx\end{array}$$

Verification is not applicable to the result.

[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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-6-8x^2+e^{6-x^2}(-30-20x^2)}{e^2{(5e^6+e^{x^2})}^2{x}^4}dx\\ & =\frac{\int \frac{-6-8x^2+e^{6-x^2}(-30-20x^2)}{{(5e^6+e^{x^2})}^2{x}^4}dx}{e^2}\\ & =\frac{\int (-\frac{4}{{(5e^6+e^{x^2})}^2{x}^2}-\frac{2e^{-6-x^2}(3+2x^2)}{5x^4}+\frac{2(3+2x^2)}{5e^6(5e^6+e^{x^2})x^4})dx}{e^2}\\ & =\frac{2\int \frac{3+2x^2}{(5e^6+e^{x^2})x^4}dx}{5e^8}-\frac{2\int \frac{e^{-6-x^2}(3+2x^2)}{x^4}dx}{5e^2}-\frac{4\int \frac{1}{{(5e^6+e^{x^2})}^2{x}^2}dx}{e^2}\\ & =\frac{2e^{-8-x^2}}{5x^3}+\frac{2\int (\frac{3}{(5e^6+e^{x^2})x^4}+\frac{2}{(5e^6+e^{x^2})x^2})dx}{5e^8}-\frac{4\int \frac{1}{{(5e^6+e^{x^2})}^2{x}^2}dx}{e^2}\\ & =\frac{2e^{-8-x^2}}{5x^3}+\frac{4\int \frac{1}{(5e^6+e^{x^2})x^2}dx}{5e^8}+\frac{6\int \frac{1}{(5e^6+e^{x^2})x^4}dx}{5e^8}-\frac{4\int \frac{1}{{(5e^6+e^{x^2})}^2{x}^2}dx}{e^2}\end{array}\end{array}$$

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Mathematica [A]  time = 0.22, size = 27, normalized size = 0.90 $$\begin{array}{c}\frac{2e^{-2-x^2}}{(5e^6+e^{x^2})x^3}\end{array}$$

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.54, size = 32, normalized size = 1.07 $$\begin{array}{c}\frac{2e^{(-2x^2+12)}}{x^3{e}^{14}+5x^3{e}^{(-x^2+20)}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [B]  time = 0.20, size = 61, normalized size = 2.03 $$\begin{array}{c}\frac{2(x^2{e}^{(-x^2+6)}+5x^2{e}^{(-2x^2+12)})}{10x^5{e}^{14}+x^5{e}^{(x^2+8)}+25x^5{e}^{(-x^2+20)}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.18, size = 34, normalized size = 1.13

method	result	size
norman	$\frac{2{\mathrm{e}}^{-14}{\mathrm{e}}^{-2x^2+12}}{x^3(1+5{\mathrm{e}}^{-x^2+6})}$	$34$
risch	$-\frac{2{\mathrm{e}}^{-14}}{25x^3}+\frac{2{\mathrm{e}}^{-x^2-8}}{5x^3}+\frac{2{\mathrm{e}}^{-14}}{25x^3(1+5{\mathrm{e}}^{-x^2+6})}$	$43$

Verification of antiderivative is not currently implemented for this CAS.

[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(7)^2*exp(-x^2+6)^2/x^3/(1+5*exp(-x^2+6))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}-2\int \frac{(4x^2+5(2x^2+3)e^{(-x^2+6)}+3)e^{(-2x^2-2)}}{10x^4{e}^{(-x^2+6)}+25x^4{e}^{(-2x^2+12)}+x^4}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 2.02, size = 32, normalized size = 1.07 $$\begin{array}{c}\frac{2{\mathrm{e}}^{12}{\mathrm{e}}^{-2x^2}}{x^3{\mathrm{e}}^{14}+5x^3{\mathrm{e}}^{20}{\mathrm{e}}^{-x^2}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.37, size = 51, normalized size = 1.70 $$\begin{array}{c}\frac{2}{125x^3{e}^{14}{e}^{6-x^2}+25x^3{e}^{14}}+\frac{2e^{6-x^2}}{5x^3{e}^{14}}-\frac{2}{25x^3{e}^{14}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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
)

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