\(\int \frac {-3+36 x-12 e^{x^2} x-6 x^2}{e^{3 x^2}+e^{2 x^2} (-12+2 x)+e^{x^2} (36-12 x+x^2)} \, dx\) [5048]

Optimal result

Integrand size = 56, antiderivative size = 19 \[ \int \frac {-3+36 x-12 e^{x^2} x-6 x^2}{e^{3 x^2}+e^{2 x^2} (-12+2 x)+e^{x^2} \left (36-12 x+x^2\right )} \, dx=\frac {3 e^{-x^2}}{-6+e^{x^2}+x} \]

[Out]

3/(exp(x^2)+x-6)/exp(x^2)

Rubi [F]

\[ \int \frac {-3+36 x-12 e^{x^2} x-6 x^2}{e^{3 x^2}+e^{2 x^2} (-12+2 x)+e^{x^2} \left (36-12 x+x^2\right )} \, dx=\int \frac {-3+36 x-12 e^{x^2} x-6 x^2}{e^{3 x^2}+e^{2 x^2} (-12+2 x)+e^{x^2} \left (36-12 x+x^2\right )} \, dx \]

[In]

Int[(-3 + 36*x - 12*E^x^2*x - 6*x^2)/(E^(3*x^2) + E^(2*x^2)*(-12 + 2*x) + E^x^2*(36 - 12*x + x^2)),x]

[Out]

-3*Defer[Int][1/(E^x^2*(-6 + E^x^2 + x)^2), x] - 36*Defer[Int][x/(E^x^2*(-6 + E^x^2 + x)^2), x] + 6*Defer[Int]
[x^2/(E^x^2*(-6 + E^x^2 + x)^2), x] - 12*Defer[Int][x/(E^x^2*(-6 + E^x^2 + x)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {3 e^{-x^2} \left (-1-4 \left (-3+e^{x^2}\right ) x-2 x^2\right )}{\left (6-e^{x^2}-x\right )^2} \, dx \\ & = 3 \int \frac {e^{-x^2} \left (-1-4 \left (-3+e^{x^2}\right ) x-2 x^2\right )}{\left (6-e^{x^2}-x\right )^2} \, dx \\ & = 3 \int \left (-\frac {4 e^{-x^2} x}{-6+e^{x^2}+x}+\frac {e^{-x^2} \left (-1-12 x+2 x^2\right )}{\left (-6+e^{x^2}+x\right )^2}\right ) \, dx \\ & = 3 \int \frac {e^{-x^2} \left (-1-12 x+2 x^2\right )}{\left (-6+e^{x^2}+x\right )^2} \, dx-12 \int \frac {e^{-x^2} x}{-6+e^{x^2}+x} \, dx \\ & = 3 \int \left (-\frac {e^{-x^2}}{\left (-6+e^{x^2}+x\right )^2}-\frac {12 e^{-x^2} x}{\left (-6+e^{x^2}+x\right )^2}+\frac {2 e^{-x^2} x^2}{\left (-6+e^{x^2}+x\right )^2}\right ) \, dx-12 \int \frac {e^{-x^2} x}{-6+e^{x^2}+x} \, dx \\ & = -\left (3 \int \frac {e^{-x^2}}{\left (-6+e^{x^2}+x\right )^2} \, dx\right )+6 \int \frac {e^{-x^2} x^2}{\left (-6+e^{x^2}+x\right )^2} \, dx-12 \int \frac {e^{-x^2} x}{-6+e^{x^2}+x} \, dx-36 \int \frac {e^{-x^2} x}{\left (-6+e^{x^2}+x\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-3+36 x-12 e^{x^2} x-6 x^2}{e^{3 x^2}+e^{2 x^2} (-12+2 x)+e^{x^2} \left (36-12 x+x^2\right )} \, dx=\frac {3 e^{-x^2}}{-6+e^{x^2}+x} \]

[In]

Integrate[(-3 + 36*x - 12*E^x^2*x - 6*x^2)/(E^(3*x^2) + E^(2*x^2)*(-12 + 2*x) + E^x^2*(36 - 12*x + x^2)),x]

[Out]

3/(E^x^2*(-6 + E^x^2 + x))

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

method	result	size
norman	\(\frac {3 \,{\mathrm e}^{-x^{2}}}{{\mathrm e}^{x^{2}}+x -6}\)	\(18\)
parallelrisch	\(\frac {3 \,{\mathrm e}^{-x^{2}}}{{\mathrm e}^{x^{2}}+x -6}\)	\(18\)
risch	\(\frac {3 \,{\mathrm e}^{-x^{2}}}{-6+x}-\frac {3}{\left (-6+x \right ) \left ({\mathrm e}^{x^{2}}+x -6\right )}\)	\(31\)

[In]

int((-12*exp(x^2)*x-6*x^2+36*x-3)/(exp(x^2)^3+(2*x-12)*exp(x^2)^2+(x^2-12*x+36)*exp(x^2)),x,method=_RETURNVERB
OSE)

[Out]

3/(exp(x^2)+x-6)/exp(x^2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-3+36 x-12 e^{x^2} x-6 x^2}{e^{3 x^2}+e^{2 x^2} (-12+2 x)+e^{x^2} \left (36-12 x+x^2\right )} \, dx=\frac {3}{{\left (x - 6\right )} e^{\left (x^{2}\right )} + e^{\left (2 \, x^{2}\right )}} \]

[In]

integrate((-12*exp(x^2)*x-6*x^2+36*x-3)/(exp(x^2)^3+(2*x-12)*exp(x^2)^2+(x^2-12*x+36)*exp(x^2)),x, algorithm="
fricas")

[Out]

3/((x - 6)*e^(x^2) + e^(2*x^2))

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {-3+36 x-12 e^{x^2} x-6 x^2}{e^{3 x^2}+e^{2 x^2} (-12+2 x)+e^{x^2} \left (36-12 x+x^2\right )} \, dx=- \frac {3}{x^{2} - 12 x + \left (x - 6\right ) e^{x^{2}} + 36} + \frac {3 e^{- x^{2}}}{x - 6} \]

[In]

integrate((-12*exp(x**2)*x-6*x**2+36*x-3)/(exp(x**2)**3+(2*x-12)*exp(x**2)**2+(x**2-12*x+36)*exp(x**2)),x)

[Out]

-3/(x**2 - 12*x + (x - 6)*exp(x**2) + 36) + 3*exp(-x**2)/(x - 6)

Maxima [F]

\[ \int \frac {-3+36 x-12 e^{x^2} x-6 x^2}{e^{3 x^2}+e^{2 x^2} (-12+2 x)+e^{x^2} \left (36-12 x+x^2\right )} \, dx=\int { -\frac {3 \, {\left (2 \, x^{2} + 4 \, x e^{\left (x^{2}\right )} - 12 \, x + 1\right )}}{2 \, {\left (x - 6\right )} e^{\left (2 \, x^{2}\right )} + {\left (x^{2} - 12 \, x + 36\right )} e^{\left (x^{2}\right )} + e^{\left (3 \, x^{2}\right )}} \,d x } \]

[In]

integrate((-12*exp(x^2)*x-6*x^2+36*x-3)/(exp(x^2)^3+(2*x-12)*exp(x^2)^2+(x^2-12*x+36)*exp(x^2)),x, algorithm="
maxima")

[Out]

-3*integrate((2*x^2 + 4*x*e^(x^2) - 12*x + 1)/(2*(x - 6)*e^(2*x^2) + (x^2 - 12*x + 36)*e^(x^2) + e^(3*x^2)), x
)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {-3+36 x-12 e^{x^2} x-6 x^2}{e^{3 x^2}+e^{2 x^2} (-12+2 x)+e^{x^2} \left (36-12 x+x^2\right )} \, dx=\frac {3}{x e^{\left (x^{2}\right )} + e^{\left (2 \, x^{2}\right )} - 6 \, e^{\left (x^{2}\right )}} \]

[In]

integrate((-12*exp(x^2)*x-6*x^2+36*x-3)/(exp(x^2)^3+(2*x-12)*exp(x^2)^2+(x^2-12*x+36)*exp(x^2)),x, algorithm="
giac")

[Out]

3/(x*e^(x^2) + e^(2*x^2) - 6*e^(x^2))

Mupad [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-3+36 x-12 e^{x^2} x-6 x^2}{e^{3 x^2}+e^{2 x^2} (-12+2 x)+e^{x^2} \left (36-12 x+x^2\right )} \, dx=\frac {3}{{\mathrm {e}}^{2\,x^2}+{\mathrm {e}}^{x^2}\,\left (x-6\right )} \]

[In]

int(-(12*x*exp(x^2) - 36*x + 6*x^2 + 3)/(exp(3*x^2) + exp(2*x^2)*(2*x - 12) + exp(x^2)*(x^2 - 12*x + 36)),x)

[Out]

3/(exp(2*x^2) + exp(x^2)*(x - 6))