\(\int \frac {e^{2+\frac {2 x^2}{e^2}}+e^2 (-6+42 x+14 x^2)+e^{\frac {x^2}{e^2}} (e^2 (-1-14 x)-10 x^2+10 x^3)}{5 e^{2+\frac {2 x^2}{e^2}}+e^{2+\frac {x^2}{e^2}} (-30-20 x)+e^2 (45+60 x+20 x^2)} \, dx\) [2657]

Optimal result

Integrand size = 106, antiderivative size = 33 \[ \int \frac {e^{2+\frac {2 x^2}{e^2}}+e^2 \left (-6+42 x+14 x^2\right )+e^{\frac {x^2}{e^2}} \left (e^2 (-1-14 x)-10 x^2+10 x^3\right )}{5 e^{2+\frac {2 x^2}{e^2}}+e^{2+\frac {x^2}{e^2}} (-30-20 x)+e^2 \left (45+60 x+20 x^2\right )} \, dx=\frac {x}{5}-\frac {x-x^2}{3-e^{\frac {x^2}{e^2}}+2 x} \]

[Out]

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

Rubi [F]

\[ \int \frac {e^{2+\frac {2 x^2}{e^2}}+e^2 \left (-6+42 x+14 x^2\right )+e^{\frac {x^2}{e^2}} \left (e^2 (-1-14 x)-10 x^2+10 x^3\right )}{5 e^{2+\frac {2 x^2}{e^2}}+e^{2+\frac {x^2}{e^2}} (-30-20 x)+e^2 \left (45+60 x+20 x^2\right )} \, dx=\int \frac {e^{2+\frac {2 x^2}{e^2}}+e^2 \left (-6+42 x+14 x^2\right )+e^{\frac {x^2}{e^2}} \left (e^2 (-1-14 x)-10 x^2+10 x^3\right )}{5 e^{2+\frac {2 x^2}{e^2}}+e^{2+\frac {x^2}{e^2}} (-30-20 x)+e^2 \left (45+60 x+20 x^2\right )} \, dx \]

[In]

Int[(E^(2 + (2*x^2)/E^2) + E^2*(-6 + 42*x + 14*x^2) + E^(x^2/E^2)*(E^2*(-1 - 14*x) - 10*x^2 + 10*x^3))/(5*E^(2
+ (2*x^2)/E^2) + E^(2 + x^2/E^2)*(-30 - 20*x) + E^2*(45 + 60*x + 20*x^2)),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{2+\frac {2 x^2}{e^2}}+e^2 \left (-6+42 x+14 x^2\right )+e^{\frac {x^2}{e^2}} \left (e^2 (-1-14 x)-10 x^2+10 x^3\right )}{5 e^2 \left (3-e^{\frac {x^2}{e^2}}+2 x\right )^2} \, dx \\ & = \frac {\int \frac {e^{2+\frac {2 x^2}{e^2}}+e^2 \left (-6+42 x+14 x^2\right )+e^{\frac {x^2}{e^2}} \left (e^2 (-1-14 x)-10 x^2+10 x^3\right )}{\left (3-e^{\frac {x^2}{e^2}}+2 x\right )^2} \, dx}{5 e^2} \\ & = \frac {\int \left (e^2+\frac {10 (-1+x) x \left (-e^2+3 x+2 x^2\right )}{\left (-3+e^{\frac {x^2}{e^2}}-2 x\right )^2}+\frac {5 \left (e^2-2 e^2 x-2 x^2+2 x^3\right )}{-3+e^{\frac {x^2}{e^2}}-2 x}\right ) \, dx}{5 e^2} \\ & = \frac {x}{5}+\frac {\int \frac {e^2-2 e^2 x-2 x^2+2 x^3}{-3+e^{\frac {x^2}{e^2}}-2 x} \, dx}{e^2}+\frac {2 \int \frac {(-1+x) x \left (-e^2+3 x+2 x^2\right )}{\left (-3+e^{\frac {x^2}{e^2}}-2 x\right )^2} \, dx}{e^2} \\ & = \frac {x}{5}+\frac {\int \left (\frac {e^2}{-3+e^{\frac {x^2}{e^2}}-2 x}+\frac {2 x^3}{-3+e^{\frac {x^2}{e^2}}-2 x}+\frac {2 e^2 x}{3-e^{\frac {x^2}{e^2}}+2 x}+\frac {2 x^2}{3-e^{\frac {x^2}{e^2}}+2 x}\right ) \, dx}{e^2}+\frac {2 \int \left (\frac {e^2 x}{\left (-3+e^{\frac {x^2}{e^2}}-2 x\right )^2}-\frac {\left (3+e^2\right ) x^2}{\left (-3+e^{\frac {x^2}{e^2}}-2 x\right )^2}+\frac {x^3}{\left (-3+e^{\frac {x^2}{e^2}}-2 x\right )^2}+\frac {2 x^4}{\left (-3+e^{\frac {x^2}{e^2}}-2 x\right )^2}\right ) \, dx}{e^2} \\ & = \frac {x}{5}+2 \int \frac {x}{\left (-3+e^{\frac {x^2}{e^2}}-2 x\right )^2} \, dx+2 \int \frac {x}{3-e^{\frac {x^2}{e^2}}+2 x} \, dx+\frac {2 \int \frac {x^3}{\left (-3+e^{\frac {x^2}{e^2}}-2 x\right )^2} \, dx}{e^2}+\frac {2 \int \frac {x^3}{-3+e^{\frac {x^2}{e^2}}-2 x} \, dx}{e^2}+\frac {2 \int \frac {x^2}{3-e^{\frac {x^2}{e^2}}+2 x} \, dx}{e^2}+\frac {4 \int \frac {x^4}{\left (-3+e^{\frac {x^2}{e^2}}-2 x\right )^2} \, dx}{e^2}+\frac {\left (2 \left (-3-e^2\right )\right ) \int \frac {x^2}{\left (-3+e^{\frac {x^2}{e^2}}-2 x\right )^2} \, dx}{e^2}+\int \frac {1}{-3+e^{\frac {x^2}{e^2}}-2 x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 4.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {e^{2+\frac {2 x^2}{e^2}}+e^2 \left (-6+42 x+14 x^2\right )+e^{\frac {x^2}{e^2}} \left (e^2 (-1-14 x)-10 x^2+10 x^3\right )}{5 e^{2+\frac {2 x^2}{e^2}}+e^{2+\frac {x^2}{e^2}} (-30-20 x)+e^2 \left (45+60 x+20 x^2\right )} \, dx=\frac {1}{5} \left (x-\frac {5 (-1+x) x}{-3+e^{\frac {x^2}{e^2}}-2 x}\right ) \]

[In]

Integrate[(E^(2 + (2*x^2)/E^2) + E^2*(-6 + 42*x + 14*x^2) + E^(x^2/E^2)*(E^2*(-1 - 14*x) - 10*x^2 + 10*x^3))/(
5*E^(2 + (2*x^2)/E^2) + E^(2 + x^2/E^2)*(-30 - 20*x) + E^2*(45 + 60*x + 20*x^2)),x]

[Out]

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

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79

method	result	size
risch	\(\frac {x}{5}+\frac {x \left (-1+x \right )}{2 x -{\mathrm e}^{{\mathrm e}^{-2} x^{2}}+3}\)	\(26\)
norman	\(\frac {\left (-\frac {2 x \,{\mathrm e}}{5}+\frac {7 x^{2} {\mathrm e}}{5}-\frac {x \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{-2} x^{2}}}{5}\right ) {\mathrm e}^{-1}}{2 x -{\mathrm e}^{{\mathrm e}^{-2} x^{2}}+3}\)	\(51\)
parallelrisch	\(\frac {\left (7 x^{2} {\mathrm e}^{2}-{\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{-2} x^{2}} x -2 \,{\mathrm e}^{2} x \right ) {\mathrm e}^{-2}}{10 x -5 \,{\mathrm e}^{{\mathrm e}^{-2} x^{2}}+15}\)	\(58\)

[In]

int((exp(1)^2*exp(x^2/exp(1)^2)^2+((-14*x-1)*exp(1)^2+10*x^3-10*x^2)*exp(x^2/exp(1)^2)+(14*x^2+42*x-6)*exp(1)^
2)/(5*exp(1)^2*exp(x^2/exp(1)^2)^2+(-20*x-30)*exp(1)^2*exp(x^2/exp(1)^2)+(20*x^2+60*x+45)*exp(1)^2),x,method=_
RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.67 \[ \int \frac {e^{2+\frac {2 x^2}{e^2}}+e^2 \left (-6+42 x+14 x^2\right )+e^{\frac {x^2}{e^2}} \left (e^2 (-1-14 x)-10 x^2+10 x^3\right )}{5 e^{2+\frac {2 x^2}{e^2}}+e^{2+\frac {x^2}{e^2}} (-30-20 x)+e^2 \left (45+60 x+20 x^2\right )} \, dx=\frac {{\left (7 \, x^{2} - 2 \, x\right )} e^{2} - x e^{\left ({\left (x^{2} + 2 \, e^{2}\right )} e^{\left (-2\right )}\right )}}{5 \, {\left ({\left (2 \, x + 3\right )} e^{2} - e^{\left ({\left (x^{2} + 2 \, e^{2}\right )} e^{\left (-2\right )}\right )}\right )}} \]

[In]

integrate((exp(1)^2*exp(x^2/exp(1)^2)^2+((-14*x-1)*exp(1)^2+10*x^3-10*x^2)*exp(x^2/exp(1)^2)+(14*x^2+42*x-6)*e
xp(1)^2)/(5*exp(1)^2*exp(x^2/exp(1)^2)^2+(-20*x-30)*exp(1)^2*exp(x^2/exp(1)^2)+(20*x^2+60*x+45)*exp(1)^2),x, a
lgorithm="fricas")

[Out]

1/5*((7*x^2 - 2*x)*e^2 - x*e^((x^2 + 2*e^2)*e^(-2)))/((2*x + 3)*e^2 - e^((x^2 + 2*e^2)*e^(-2)))

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.61 \[ \int \frac {e^{2+\frac {2 x^2}{e^2}}+e^2 \left (-6+42 x+14 x^2\right )+e^{\frac {x^2}{e^2}} \left (e^2 (-1-14 x)-10 x^2+10 x^3\right )}{5 e^{2+\frac {2 x^2}{e^2}}+e^{2+\frac {x^2}{e^2}} (-30-20 x)+e^2 \left (45+60 x+20 x^2\right )} \, dx=\frac {x}{5} + \frac {- x^{2} + x}{- 2 x + e^{\frac {x^{2}}{e^{2}}} - 3} \]

[In]

integrate((exp(1)**2*exp(x**2/exp(1)**2)**2+((-14*x-1)*exp(1)**2+10*x**3-10*x**2)*exp(x**2/exp(1)**2)+(14*x**2
+42*x-6)*exp(1)**2)/(5*exp(1)**2*exp(x**2/exp(1)**2)**2+(-20*x-30)*exp(1)**2*exp(x**2/exp(1)**2)+(20*x**2+60*x
+45)*exp(1)**2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {e^{2+\frac {2 x^2}{e^2}}+e^2 \left (-6+42 x+14 x^2\right )+e^{\frac {x^2}{e^2}} \left (e^2 (-1-14 x)-10 x^2+10 x^3\right )}{5 e^{2+\frac {2 x^2}{e^2}}+e^{2+\frac {x^2}{e^2}} (-30-20 x)+e^2 \left (45+60 x+20 x^2\right )} \, dx=\frac {7 \, x^{2} - x e^{\left (x^{2} e^{\left (-2\right )}\right )} - 2 \, x}{5 \, {\left (2 \, x - e^{\left (x^{2} e^{\left (-2\right )}\right )} + 3\right )}} \]

[In]

integrate((exp(1)^2*exp(x^2/exp(1)^2)^2+((-14*x-1)*exp(1)^2+10*x^3-10*x^2)*exp(x^2/exp(1)^2)+(14*x^2+42*x-6)*e
xp(1)^2)/(5*exp(1)^2*exp(x^2/exp(1)^2)^2+(-20*x-30)*exp(1)^2*exp(x^2/exp(1)^2)+(20*x^2+60*x+45)*exp(1)^2),x, a
lgorithm="maxima")

[Out]

1/5*(7*x^2 - x*e^(x^2*e^(-2)) - 2*x)/(2*x - e^(x^2*e^(-2)) + 3)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {e^{2+\frac {2 x^2}{e^2}}+e^2 \left (-6+42 x+14 x^2\right )+e^{\frac {x^2}{e^2}} \left (e^2 (-1-14 x)-10 x^2+10 x^3\right )}{5 e^{2+\frac {2 x^2}{e^2}}+e^{2+\frac {x^2}{e^2}} (-30-20 x)+e^2 \left (45+60 x+20 x^2\right )} \, dx=\frac {7 \, x^{2} - x e^{\left (x^{2} e^{\left (-2\right )}\right )} - 2 \, x}{5 \, {\left (2 \, x - e^{\left (x^{2} e^{\left (-2\right )}\right )} + 3\right )}} \]

[In]

integrate((exp(1)^2*exp(x^2/exp(1)^2)^2+((-14*x-1)*exp(1)^2+10*x^3-10*x^2)*exp(x^2/exp(1)^2)+(14*x^2+42*x-6)*e
xp(1)^2)/(5*exp(1)^2*exp(x^2/exp(1)^2)^2+(-20*x-30)*exp(1)^2*exp(x^2/exp(1)^2)+(20*x^2+60*x+45)*exp(1)^2),x, a
lgorithm="giac")

[Out]

1/5*(7*x^2 - x*e^(x^2*e^(-2)) - 2*x)/(2*x - e^(x^2*e^(-2)) + 3)

Mupad [B] (verification not implemented)

Time = 8.93 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {e^{2+\frac {2 x^2}{e^2}}+e^2 \left (-6+42 x+14 x^2\right )+e^{\frac {x^2}{e^2}} \left (e^2 (-1-14 x)-10 x^2+10 x^3\right )}{5 e^{2+\frac {2 x^2}{e^2}}+e^{2+\frac {x^2}{e^2}} (-30-20 x)+e^2 \left (45+60 x+20 x^2\right )} \, dx=-\frac {x\,\left ({\mathrm {e}}^{x^2\,{\mathrm {e}}^{-2}}-7\,x+2\right )}{5\,\left (2\,x-{\mathrm {e}}^{x^2\,{\mathrm {e}}^{-2}}+3\right )} \]

[In]

int((exp(2)*(42*x + 14*x^2 - 6) - exp(x^2*exp(-2))*(10*x^2 - 10*x^3 + exp(2)*(14*x + 1)) + exp(2*x^2*exp(-2))*
exp(2))/(exp(2)*(60*x + 20*x^2 + 45) + 5*exp(2*x^2*exp(-2))*exp(2) - exp(x^2*exp(-2))*exp(2)*(20*x + 30)),x)

[Out]

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