\(\int \frac {e^{x+x^2} (1-2 x)+e^x (19880+282 x+x^2)}{1626347584+4 e^{2 x^2}[phone] x+483936 x^2+2272 x^3+4 x^4+e^{x^2} (161312+2272 x+8 x^2)} \, dx\) [10262]

Optimal result

Integrand size = 74, antiderivative size = 22 \[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}[phone] x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\frac {e^x}{4 \left (e^{x^2}+(-142-x)^2\right )} \]

[Out]

exp(x)/(4*(-142-x)^2+4*exp(x^2))

Rubi [F]

\[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}[phone] x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}[phone] x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx \]

[In]

Int[(E^(x + x^2)*(1 - 2*x) + E^x*(19880 + 282*x + x^2))/(1626347584 + 4*E^(2*x^2) + 45812608*x + 483936*x^2 +
2272*x^3 + 4*x^4 + E^x^2*(161312 + 2272*x + 8*x^2)),x]

[Out]

-71*Defer[Int][E^x/(20164 + E^x^2 + 284*x + x^2)^2, x] + (20163*Defer[Int][(E^x*x)/(20164 + E^x^2 + 284*x + x^
2)^2, x])/2 + 142*Defer[Int][(E^x*x^2)/(20164 + E^x^2 + 284*x + x^2)^2, x] + Defer[Int][(E^x*x^3)/(20164 + E^x
^2 + 284*x + x^2)^2, x]/2 + Defer[Int][E^x/(20164 + E^x^2 + 284*x + x^2), x]/4 - Defer[Int][(E^x*x)/(20164 + E
^x^2 + 284*x + x^2), x]/2

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^x \left (19880+e^{x^2} (1-2 x)+282 x+x^2\right )}{4 \left (e^{x^2}+(142+x)^2\right )^2} \, dx \\ & = \frac {1}{4} \int \frac {e^x \left (19880+e^{x^2} (1-2 x)+282 x+x^2\right )}{\left (e^{x^2}+(142+x)^2\right )^2} \, dx \\ & = \frac {1}{4} \int \left (-\frac {e^x (-1+2 x)}{20164+e^{x^2}+284 x+x^2}+\frac {2 e^x \left (-142+20163 x+284 x^2+x^3\right )}{\left (20164+e^{x^2}+284 x+x^2\right )^2}\right ) \, dx \\ & = -\left (\frac {1}{4} \int \frac {e^x (-1+2 x)}{20164+e^{x^2}+284 x+x^2} \, dx\right )+\frac {1}{2} \int \frac {e^x \left (-142+20163 x+284 x^2+x^3\right )}{\left (20164+e^{x^2}+284 x+x^2\right )^2} \, dx \\ & = -\left (\frac {1}{4} \int \left (-\frac {e^x}{20164+e^{x^2}+284 x+x^2}+\frac {2 e^x x}{20164+e^{x^2}+284 x+x^2}\right ) \, dx\right )+\frac {1}{2} \int \left (-\frac {142 e^x}{\left (20164+e^{x^2}+284 x+x^2\right )^2}+\frac {20163 e^x x}{\left (20164+e^{x^2}+284 x+x^2\right )^2}+\frac {284 e^x x^2}{\left (20164+e^{x^2}+284 x+x^2\right )^2}+\frac {e^x x^3}{\left (20164+e^{x^2}+284 x+x^2\right )^2}\right ) \, dx \\ & = \frac {1}{4} \int \frac {e^x}{20164+e^{x^2}+284 x+x^2} \, dx+\frac {1}{2} \int \frac {e^x x^3}{\left (20164+e^{x^2}+284 x+x^2\right )^2} \, dx-\frac {1}{2} \int \frac {e^x x}{20164+e^{x^2}+284 x+x^2} \, dx-71 \int \frac {e^x}{\left (20164+e^{x^2}+284 x+x^2\right )^2} \, dx+142 \int \frac {e^x x^2}{\left (20164+e^{x^2}+284 x+x^2\right )^2} \, dx+\frac {20163}{2} \int \frac {e^x x}{\left (20164+e^{x^2}+284 x+x^2\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 2.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}[phone] x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\frac {e^x}{4 \left (20164+e^{x^2}+284 x+x^2\right )} \]

[In]

Integrate[(E^(x + x^2)*(1 - 2*x) + E^x*(19880 + 282*x + x^2))/(1626347584 + 4*E^(2*x^2) + 45812608*x + 483936*
x^2 + 2272*x^3 + 4*x^4 + E^x^2*(161312 + 2272*x + 8*x^2)),x]

[Out]

E^x/(4*(20164 + E^x^2 + 284*x + x^2))

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86

method	result	size
norman	\(\frac {{\mathrm e}^{x}}{4 x^{2}+4 \,{\mathrm e}^{x^{2}}+1136 x +80656}\)	\(19\)
risch	\(\frac {{\mathrm e}^{x}}{4 x^{2}+4 \,{\mathrm e}^{x^{2}}+1136 x +80656}\)	\(19\)
parallelrisch	\(\frac {{\mathrm e}^{x}}{4 x^{2}+4 \,{\mathrm e}^{x^{2}}+1136 x +80656}\)	\(19\)

[In]

int(((1-2*x)*exp(x)*exp(x^2)+(x^2+282*x+19880)*exp(x))/(4*exp(x^2)^2+(8*x^2+2272*x+161312)*exp(x^2)+4*x^4+2272
*x^3+483936*x^2+45812608*x+1626347584),x,method=_RETURNVERBOSE)

[Out]

1/4*exp(x)/(x^2+exp(x^2)+284*x+20164)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}[phone] x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\frac {e^{\left (x^{2} + x\right )}}{4 \, {\left ({\left (x^{2} + 284 \, x + 20164\right )} e^{\left (x^{2}\right )} + e^{\left (2 \, x^{2}\right )}\right )}} \]

[In]

integrate(((1-2*x)*exp(x)*exp(x^2)+(x^2+282*x+19880)*exp(x))/(4*exp(x^2)^2+(8*x^2+2272*x+161312)*exp(x^2)+4*x^
4+2272*x^3+483936*x^2+45812608*x+1626347584),x, algorithm="fricas")

[Out]

1/4*e^(x^2 + x)/((x^2 + 284*x + 20164)*e^(x^2) + e^(2*x^2))

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}[phone] x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\frac {e^{x}}{4 x^{2} + 1136 x + 4 e^{x^{2}} + 80656} \]

[In]

integrate(((1-2*x)*exp(x)*exp(x**2)+(x**2+282*x+19880)*exp(x))/(4*exp(x**2)**2+(8*x**2+2272*x+161312)*exp(x**2
)+4*x**4+2272*x**3+483936*x**2+45812608*x+1626347584),x)

[Out]

exp(x)/(4*x**2 + 1136*x + 4*exp(x**2) + 80656)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}[phone] x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\frac {e^{x}}{4 \, {\left (x^{2} + 284 \, x + e^{\left (x^{2}\right )} + 20164\right )}} \]

[In]

integrate(((1-2*x)*exp(x)*exp(x^2)+(x^2+282*x+19880)*exp(x))/(4*exp(x^2)^2+(8*x^2+2272*x+161312)*exp(x^2)+4*x^
4+2272*x^3+483936*x^2+45812608*x+1626347584),x, algorithm="maxima")

[Out]

1/4*e^x/(x^2 + 284*x + e^(x^2) + 20164)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}[phone] x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\frac {e^{x}}{4 \, {\left (x^{2} + 284 \, x + e^{\left (x^{2}\right )} + 20164\right )}} \]

[In]

integrate(((1-2*x)*exp(x)*exp(x^2)+(x^2+282*x+19880)*exp(x))/(4*exp(x^2)^2+(8*x^2+2272*x+161312)*exp(x^2)+4*x^
4+2272*x^3+483936*x^2+45812608*x+1626347584),x, algorithm="giac")

[Out]

1/4*e^x/(x^2 + 284*x + e^(x^2) + 20164)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}[phone] x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\int \frac {{\mathrm {e}}^x\,\left (x^2+282\,x+19880\right )-{\mathrm {e}}^{x^2+x}\,\left (2\,x-1\right )}{45812608\,x+4\,{\mathrm {e}}^{2\,x^2}+{\mathrm {e}}^{x^2}\,\left (8\,x^2+2272\,x+161312\right )+483936\,x^2+2272\,x^3+4\,x^4+1626347584} \,d x \]

[In]

int((exp(x)*(282*x + x^2 + 19880) - exp(x^2)*exp(x)*(2*x - 1))/(45812608*x + 4*exp(2*x^2) + exp(x^2)*(2272*x +
8*x^2 + 161312) + 483936*x^2 + 2272*x^3 + 4*x^4 + 1626347584),x)

[Out]

int((exp(x)*(282*x + x^2 + 19880) - exp(x + x^2)*(2*x - 1))/(45812608*x + 4*exp(2*x^2) + exp(x^2)*(2272*x + 8*
x^2 + 161312) + 483936*x^2 + 2272*x^3 + 4*x^4 + 1626347584), x)