\(\int \frac {-25+e^{-2+2 e^{\sqrt [4]{e}}-e^{x^2}-x} (-25-50 e^{x^2} x)}{1+e^{-4+4 e^{\sqrt [4]{e}}-2 e^{x^2}-2 x}+e^{-2+2 e^{\sqrt [4]{e}}-e^{x^2}-x} (-2-2 x)+2 x+x^2} \, dx\) [5080]

Optimal result

Integrand size = 99, antiderivative size = 32 \[ \int \frac {-25+e^{-2+2 e^{\sqrt [4]{e}}-e^{x^2}-x} \left (-25-50 e^{x^2} x\right )}{1+e^{-4+4 e^{\sqrt [4]{e}}-2 e^{x^2}-2 x}+e^{-2+2 e^{\sqrt [4]{e}}-e^{x^2}-x} (-2-2 x)+2 x+x^2} \, dx=\frac {25}{1-e^{-2+2 e^{\sqrt [4]{e}}-e^{x^2}-x}+x} \]

[Out]

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

Rubi [F]

\[ \int \frac {-25+e^{-2+2 e^{\sqrt [4]{e}}-e^{x^2}-x} \left (-25-50 e^{x^2} x\right )}{1+e^{-4+4 e^{\sqrt [4]{e}}-2 e^{x^2}-2 x}+e^{-2+2 e^{\sqrt [4]{e}}-e^{x^2}-x} (-2-2 x)+2 x+x^2} \, dx=\int \frac {-25+e^{-2+2 e^{\sqrt [4]{e}}-e^{x^2}-x} \left (-25-50 e^{x^2} x\right )}{1+e^{-4+4 e^{\sqrt [4]{e}}-2 e^{x^2}-2 x}+e^{-2+2 e^{\sqrt [4]{e}}-e^{x^2}-x} (-2-2 x)+2 x+x^2} \, dx \]

[In]

Int[(-25 + E^(-2 + 2*E^E^(1/4) - E^x^2 - x)*(-25 - 50*E^x^2*x))/(1 + E^(-4 + 4*E^E^(1/4) - 2*E^x^2 - 2*x) + E^
(-2 + 2*E^E^(1/4) - E^x^2 - x)*(-2 - 2*x) + 2*x + x^2),x]

[Out]

-25*Defer[Int][E^(E^x^2 + 2*(1 + E^E^(1/4)) + x)/((1 + x)*(-E^(2*E^E^(1/4)) + E^(2 + E^x^2 + x) + E^(2 + E^x^2
+ x)*x)^2), x] - 25*Defer[Int][E^(2 + E^x^2 + x)/((1 + x)*(-E^(2*E^E^(1/4)) + E^(2 + E^x^2 + x) + E^(2 + E^x^
2 + x)*x)), x] - 25*Defer[Int][E^(E^x^2 + 2*(1 + E^E^(1/4)) + x)/(E^(2*E^E^(1/4)) - E^(2 + E^x^2 + x)*(1 + x))
^2, x] - 50*Defer[Int][(E^(E^x^2 + 2*(1 + E^E^(1/4)) + x + x^2)*x)/(E^(2*E^E^(1/4)) - E^(2 + E^x^2 + x)*(1 + x
))^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{2 \left (2+e^{x^2}+x\right )} \left (-25+e^{-2+2 e^{\sqrt [4]{e}}-e^{x^2}-x} \left (-25-50 e^{x^2} x\right )\right )}{\left (e^{2 e^{\sqrt [4]{e}}}-e^{2+e^{x^2}+x}-e^{2+e^{x^2}+x} x\right )^2} \, dx \\ & = \int \left (-\frac {50 \exp \left (-e^{x^2}-2 \left (1-e^{\sqrt [4]{e}}\right )-x+x^2+2 \left (2+e^{x^2}+x\right )\right ) x}{\left (e^{2 e^{\sqrt [4]{e}}}-e^{2+e^{x^2}+x}-e^{2+e^{x^2}+x} x\right )^2}-\frac {25 e^{-2-e^{x^2}-x+2 \left (2+e^{x^2}+x\right )} \left (e^{2 e^{\sqrt [4]{e}}}+e^{2+e^{x^2}+x}\right )}{\left (-e^{2 e^{\sqrt [4]{e}}}+e^{2+e^{x^2}+x}+e^{2+e^{x^2}+x} x\right )^2}\right ) \, dx \\ & = -\left (25 \int \frac {e^{-2-e^{x^2}-x+2 \left (2+e^{x^2}+x\right )} \left (e^{2 e^{\sqrt [4]{e}}}+e^{2+e^{x^2}+x}\right )}{\left (-e^{2 e^{\sqrt [4]{e}}}+e^{2+e^{x^2}+x}+e^{2+e^{x^2}+x} x\right )^2} \, dx\right )-50 \int \frac {\exp \left (-e^{x^2}-2 \left (1-e^{\sqrt [4]{e}}\right )-x+x^2+2 \left (2+e^{x^2}+x\right )\right ) x}{\left (e^{2 e^{\sqrt [4]{e}}}-e^{2+e^{x^2}+x}-e^{2+e^{x^2}+x} x\right )^2} \, dx \\ & = -\left (25 \int \frac {e^{2+e^{x^2}+x} \left (e^{2 e^{\sqrt [4]{e}}}+e^{2+e^{x^2}+x}\right )}{\left (e^{2 e^{\sqrt [4]{e}}}-e^{2+e^{x^2}+x} (1+x)\right )^2} \, dx\right )-50 \int \frac {e^{e^{x^2}+2 \left (1+e^{\sqrt [4]{e}}\right )+x+x^2} x}{\left (e^{2 e^{\sqrt [4]{e}}}-e^{2+e^{x^2}+x} (1+x)\right )^2} \, dx \\ & = -\left (25 \int \left (\frac {e^{2+2 e^{\sqrt [4]{e}}+e^{x^2}+x} (2+x)}{(1+x) \left (e^{2 e^{\sqrt [4]{e}}}-e^{2+e^{x^2}+x}-e^{2+e^{x^2}+x} x\right )^2}+\frac {e^{2+e^{x^2}+x}}{(1+x) \left (-e^{2 e^{\sqrt [4]{e}}}+e^{2+e^{x^2}+x}+e^{2+e^{x^2}+x} x\right )}\right ) \, dx\right )-50 \int \frac {e^{e^{x^2}+2 \left (1+e^{\sqrt [4]{e}}\right )+x+x^2} x}{\left (e^{2 e^{\sqrt [4]{e}}}-e^{2+e^{x^2}+x} (1+x)\right )^2} \, dx \\ & = -\left (25 \int \frac {e^{2+2 e^{\sqrt [4]{e}}+e^{x^2}+x} (2+x)}{(1+x) \left (e^{2 e^{\sqrt [4]{e}}}-e^{2+e^{x^2}+x}-e^{2+e^{x^2}+x} x\right )^2} \, dx\right )-25 \int \frac {e^{2+e^{x^2}+x}}{(1+x) \left (-e^{2 e^{\sqrt [4]{e}}}+e^{2+e^{x^2}+x}+e^{2+e^{x^2}+x} x\right )} \, dx-50 \int \frac {e^{e^{x^2}+2 \left (1+e^{\sqrt [4]{e}}\right )+x+x^2} x}{\left (e^{2 e^{\sqrt [4]{e}}}-e^{2+e^{x^2}+x} (1+x)\right )^2} \, dx \\ & = -\left (25 \int \frac {e^{e^{x^2}+2 \left (1+e^{\sqrt [4]{e}}\right )+x} (2+x)}{(1+x) \left (e^{2 e^{\sqrt [4]{e}}}-e^{2+e^{x^2}+x}-e^{2+e^{x^2}+x} x\right )^2} \, dx\right )-25 \int \frac {e^{2+e^{x^2}+x}}{(1+x) \left (-e^{2 e^{\sqrt [4]{e}}}+e^{2+e^{x^2}+x}+e^{2+e^{x^2}+x} x\right )} \, dx-50 \int \frac {e^{e^{x^2}+2 \left (1+e^{\sqrt [4]{e}}\right )+x+x^2} x}{\left (e^{2 e^{\sqrt [4]{e}}}-e^{2+e^{x^2}+x} (1+x)\right )^2} \, dx \\ & = -\left (25 \int \frac {e^{2+e^{x^2}+x}}{(1+x) \left (-e^{2 e^{\sqrt [4]{e}}}+e^{2+e^{x^2}+x}+e^{2+e^{x^2}+x} x\right )} \, dx\right )-25 \int \frac {e^{e^{x^2}+2 \left (1+e^{\sqrt [4]{e}}\right )+x} (2+x)}{(1+x) \left (e^{2 e^{\sqrt [4]{e}}}-e^{2+e^{x^2}+x} (1+x)\right )^2} \, dx-50 \int \frac {e^{e^{x^2}+2 \left (1+e^{\sqrt [4]{e}}\right )+x+x^2} x}{\left (e^{2 e^{\sqrt [4]{e}}}-e^{2+e^{x^2}+x} (1+x)\right )^2} \, dx \\ & = -\left (25 \int \frac {e^{2+e^{x^2}+x}}{(1+x) \left (-e^{2 e^{\sqrt [4]{e}}}+e^{2+e^{x^2}+x}+e^{2+e^{x^2}+x} x\right )} \, dx\right )-25 \int \left (\frac {e^{e^{x^2}+2 \left (1+e^{\sqrt [4]{e}}\right )+x}}{\left (e^{2 e^{\sqrt [4]{e}}}-e^{2+e^{x^2}+x}-e^{2+e^{x^2}+x} x\right )^2}+\frac {e^{e^{x^2}+2 \left (1+e^{\sqrt [4]{e}}\right )+x}}{(1+x) \left (-e^{2 e^{\sqrt [4]{e}}}+e^{2+e^{x^2}+x}+e^{2+e^{x^2}+x} x\right )^2}\right ) \, dx-50 \int \frac {e^{e^{x^2}+2 \left (1+e^{\sqrt [4]{e}}\right )+x+x^2} x}{\left (e^{2 e^{\sqrt [4]{e}}}-e^{2+e^{x^2}+x} (1+x)\right )^2} \, dx \\ & = -\left (25 \int \frac {e^{e^{x^2}+2 \left (1+e^{\sqrt [4]{e}}\right )+x}}{\left (e^{2 e^{\sqrt [4]{e}}}-e^{2+e^{x^2}+x}-e^{2+e^{x^2}+x} x\right )^2} \, dx\right )-25 \int \frac {e^{e^{x^2}+2 \left (1+e^{\sqrt [4]{e}}\right )+x}}{(1+x) \left (-e^{2 e^{\sqrt [4]{e}}}+e^{2+e^{x^2}+x}+e^{2+e^{x^2}+x} x\right )^2} \, dx-25 \int \frac {e^{2+e^{x^2}+x}}{(1+x) \left (-e^{2 e^{\sqrt [4]{e}}}+e^{2+e^{x^2}+x}+e^{2+e^{x^2}+x} x\right )} \, dx-50 \int \frac {e^{e^{x^2}+2 \left (1+e^{\sqrt [4]{e}}\right )+x+x^2} x}{\left (e^{2 e^{\sqrt [4]{e}}}-e^{2+e^{x^2}+x} (1+x)\right )^2} \, dx \\ & = -\left (25 \int \frac {e^{e^{x^2}+2 \left (1+e^{\sqrt [4]{e}}\right )+x}}{(1+x) \left (-e^{2 e^{\sqrt [4]{e}}}+e^{2+e^{x^2}+x}+e^{2+e^{x^2}+x} x\right )^2} \, dx\right )-25 \int \frac {e^{2+e^{x^2}+x}}{(1+x) \left (-e^{2 e^{\sqrt [4]{e}}}+e^{2+e^{x^2}+x}+e^{2+e^{x^2}+x} x\right )} \, dx-25 \int \frac {e^{e^{x^2}+2 \left (1+e^{\sqrt [4]{e}}\right )+x}}{\left (e^{2 e^{\sqrt [4]{e}}}-e^{2+e^{x^2}+x} (1+x)\right )^2} \, dx-50 \int \frac {e^{e^{x^2}+2 \left (1+e^{\sqrt [4]{e}}\right )+x+x^2} x}{\left (e^{2 e^{\sqrt [4]{e}}}-e^{2+e^{x^2}+x} (1+x)\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {-25+e^{-2+2 e^{\sqrt [4]{e}}-e^{x^2}-x} \left (-25-50 e^{x^2} x\right )}{1+e^{-4+4 e^{\sqrt [4]{e}}-2 e^{x^2}-2 x}+e^{-2+2 e^{\sqrt [4]{e}}-e^{x^2}-x} (-2-2 x)+2 x+x^2} \, dx=\frac {25 e^{2+e^{x^2}+x}}{-e^{2 e^{\sqrt [4]{e}}}+e^{2+e^{x^2}+x} (1+x)} \]

[In]

Integrate[(-25 + E^(-2 + 2*E^E^(1/4) - E^x^2 - x)*(-25 - 50*E^x^2*x))/(1 + E^(-4 + 4*E^E^(1/4) - 2*E^x^2 - 2*x
) + E^(-2 + 2*E^E^(1/4) - E^x^2 - x)*(-2 - 2*x) + 2*x + x^2),x]

[Out]

(25*E^(2 + E^x^2 + x))/(-E^(2*E^E^(1/4)) + E^(2 + E^x^2 + x)*(1 + x))

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84

method	result	size
norman	\(\frac {25}{x +1-{\mathrm e}^{2 \,{\mathrm e}^{{\mathrm e}^{\frac {1}{4}}}-{\mathrm e}^{x^{2}}-x -2}}\)	\(27\)
risch	\(\frac {25}{x +1-{\mathrm e}^{2 \,{\mathrm e}^{{\mathrm e}^{\frac {1}{4}}}-{\mathrm e}^{x^{2}}-x -2}}\)	\(27\)
parallelrisch	\(\frac {25}{x +1-{\mathrm e}^{2 \,{\mathrm e}^{{\mathrm e}^{\frac {1}{4}}}-{\mathrm e}^{x^{2}}-x -2}}\)	\(27\)

[In]

int(((-50*exp(x^2)*x-25)*exp(2*exp(exp(1/4))-exp(x^2)-x-2)-25)/(exp(2*exp(exp(1/4))-exp(x^2)-x-2)^2+(-2-2*x)*e
xp(2*exp(exp(1/4))-exp(x^2)-x-2)+x^2+2*x+1),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {-25+e^{-2+2 e^{\sqrt [4]{e}}-e^{x^2}-x} \left (-25-50 e^{x^2} x\right )}{1+e^{-4+4 e^{\sqrt [4]{e}}-2 e^{x^2}-2 x}+e^{-2+2 e^{\sqrt [4]{e}}-e^{x^2}-x} (-2-2 x)+2 x+x^2} \, dx=\frac {25}{x - e^{\left (-x - e^{\left (x^{2}\right )} + 2 \, e^{\left (e^{\frac {1}{4}}\right )} - 2\right )} + 1} \]

[In]

integrate(((-50*exp(x^2)*x-25)*exp(2*exp(exp(1/4))-exp(x^2)-x-2)-25)/(exp(2*exp(exp(1/4))-exp(x^2)-x-2)^2+(-2-
2*x)*exp(2*exp(exp(1/4))-exp(x^2)-x-2)+x^2+2*x+1),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {-25+e^{-2+2 e^{\sqrt [4]{e}}-e^{x^2}-x} \left (-25-50 e^{x^2} x\right )}{1+e^{-4+4 e^{\sqrt [4]{e}}-2 e^{x^2}-2 x}+e^{-2+2 e^{\sqrt [4]{e}}-e^{x^2}-x} (-2-2 x)+2 x+x^2} \, dx=- \frac {25}{- x + e^{- x - e^{x^{2}} - 2 + 2 e^{e^{\frac {1}{4}}}} - 1} \]

[In]

integrate(((-50*exp(x**2)*x-25)*exp(2*exp(exp(1/4))-exp(x**2)-x-2)-25)/(exp(2*exp(exp(1/4))-exp(x**2)-x-2)**2+
(-2-2*x)*exp(2*exp(exp(1/4))-exp(x**2)-x-2)+x**2+2*x+1),x)

[Out]

-25/(-x + exp(-x - exp(x**2) - 2 + 2*exp(exp(1/4))) - 1)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {-25+e^{-2+2 e^{\sqrt [4]{e}}-e^{x^2}-x} \left (-25-50 e^{x^2} x\right )}{1+e^{-4+4 e^{\sqrt [4]{e}}-2 e^{x^2}-2 x}+e^{-2+2 e^{\sqrt [4]{e}}-e^{x^2}-x} (-2-2 x)+2 x+x^2} \, dx=\frac {25 \, e^{\left (x + e^{\left (x^{2}\right )} + 2\right )}}{{\left (x e^{2} + e^{2}\right )} e^{\left (x + e^{\left (x^{2}\right )}\right )} - e^{\left (2 \, e^{\left (e^{\frac {1}{4}}\right )}\right )}} \]

[In]

integrate(((-50*exp(x^2)*x-25)*exp(2*exp(exp(1/4))-exp(x^2)-x-2)-25)/(exp(2*exp(exp(1/4))-exp(x^2)-x-2)^2+(-2-
2*x)*exp(2*exp(exp(1/4))-exp(x^2)-x-2)+x^2+2*x+1),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {-25+e^{-2+2 e^{\sqrt [4]{e}}-e^{x^2}-x} \left (-25-50 e^{x^2} x\right )}{1+e^{-4+4 e^{\sqrt [4]{e}}-2 e^{x^2}-2 x}+e^{-2+2 e^{\sqrt [4]{e}}-e^{x^2}-x} (-2-2 x)+2 x+x^2} \, dx=\frac {25}{x - e^{\left (-x - e^{\left (x^{2}\right )} + 2 \, e^{\left (e^{\frac {1}{4}}\right )} - 2\right )} + 1} \]

[In]

integrate(((-50*exp(x^2)*x-25)*exp(2*exp(exp(1/4))-exp(x^2)-x-2)-25)/(exp(2*exp(exp(1/4))-exp(x^2)-x-2)^2+(-2-
2*x)*exp(2*exp(exp(1/4))-exp(x^2)-x-2)+x^2+2*x+1),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 15.80 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {-25+e^{-2+2 e^{\sqrt [4]{e}}-e^{x^2}-x} \left (-25-50 e^{x^2} x\right )}{1+e^{-4+4 e^{\sqrt [4]{e}}-2 e^{x^2}-2 x}+e^{-2+2 e^{\sqrt [4]{e}}-e^{x^2}-x} (-2-2 x)+2 x+x^2} \, dx=\frac {25}{x-{\mathrm {e}}^{-{\mathrm {e}}^{x^2}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-2}\,{\mathrm {e}}^{2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}}+1} \]

[In]

int(-(exp(2*exp(exp(1/4)) - exp(x^2) - x - 2)*(50*x*exp(x^2) + 25) + 25)/(2*x + exp(4*exp(exp(1/4)) - 2*exp(x^
2) - 2*x - 4) + x^2 - exp(2*exp(exp(1/4)) - exp(x^2) - x - 2)*(2*x + 2) + 1),x)

[Out]

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