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

Optimal. Leaf size=22 \[ \frac {e^x}{4 \left (e^{x^2}+(-142-x)^2\right )} \]

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Rubi [F]  time = 1.67, 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 {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.52, size = 22, normalized size = 1.00 \begin {gather*} \frac {e^x}{4 \left (20164+e^{x^2}+284 x+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[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))

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fricas [A]  time = 0.56, size = 30, normalized size = 1.36 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.17, size = 18, normalized size = 0.82 \begin {gather*} \frac {e^{x}}{4 \, {\left (x^{2} + 284 \, x + e^{\left (x^{2}\right )} + 20164\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.10, size = 19, normalized size = 0.86

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.40, size = 18, normalized size = 0.82 \begin {gather*} \frac {e^{x}}{4 \, {\left (x^{2} + 284 \, x + e^{\left (x^{2}\right )} + 20164\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.18, size = 19, normalized size = 0.86 \begin {gather*} \frac {e^{x}}{4 x^{2} + 1136 x + 4 e^{x^{2}} + 80656} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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