\(\int \frac {e^{1-e^{x^2} x+2 x^2} (x+2 x^3)}{(1-e^{x^2} x)^2} \, dx\) [169]

Optimal result

Integrand size = 37, antiderivative size = 25 \[ \int \frac {e^{1-e^{x^2} x+2 x^2} \left (x+2 x^3\right )}{\left (1-e^{x^2} x\right )^2} \, dx=-\frac {e^{1-e^{x^2} x}}{-1+e^{x^2} x} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^{1-e^{x^2} x+2 x^2} \left (x+2 x^3\right )}{\left (1-e^{x^2} x\right )^2} \, dx=-\frac {e^{1-e^{x^2} x}}{-1+e^{x^2} x} \]

[In]

Integrate[(E^(1 - E^x^2*x + 2*x^2)*(x + 2*x^3))/(1 - E^x^2*x)^2,x]

[Out]

-(E^(1 - E^x^2*x)/(-1 + E^x^2*x))

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

method	result	size
risch	\(-\frac {{\mathrm e}^{1-{\mathrm e}^{x^{2}} x}}{-1+{\mathrm e}^{x^{2}} x}\)	\(23\)

[In]

int(exp(1-exp(x^2)*x+2*x^2)*(2*x^3+x)/(1-exp(x^2)*x)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {e^{1-e^{x^2} x+2 x^2} \left (x+2 x^3\right )}{\left (1-e^{x^2} x\right )^2} \, dx=-\frac {e^{\left (2 \, x^{2} - x e^{\left (x^{2}\right )} + 1\right )}}{x e^{\left (3 \, x^{2}\right )} - e^{\left (2 \, x^{2}\right )}} \]

[In]

integrate(exp(1-exp(x^2)*x+2*x^2)*(2*x^3+x)/(1-exp(x^2)*x)^2,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {e^{1-e^{x^2} x+2 x^2} \left (x+2 x^3\right )}{\left (1-e^{x^2} x\right )^2} \, dx=- \frac {e^{2 x^{2} - x e^{x^{2}} + 1}}{x e^{3 x^{2}} - e^{2 x^{2}}} \]

[In]

integrate(exp(1-exp(x**2)*x+2*x**2)*(2*x**3+x)/(1-exp(x**2)*x)**2,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {e^{1-e^{x^2} x+2 x^2} \left (x+2 x^3\right )}{\left (1-e^{x^2} x\right )^2} \, dx=-\frac {e^{\left (-x e^{\left (x^{2}\right )} + 1\right )}}{x e^{\left (x^{2}\right )} - 1} \]

[In]

integrate(exp(1-exp(x^2)*x+2*x^2)*(2*x^3+x)/(1-exp(x^2)*x)^2,x, algorithm="maxima")

[Out]

-e^(-x*e^(x^2) + 1)/(x*e^(x^2) - 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {e^{1-e^{x^2} x+2 x^2} \left (x+2 x^3\right )}{\left (1-e^{x^2} x\right )^2} \, dx=-\frac {e^{\left (2 \, x^{2} - x e^{\left (x^{2}\right )} + 1\right )}}{x e^{\left (3 \, x^{2}\right )} - e^{\left (2 \, x^{2}\right )}} \]

[In]

integrate(exp(1-exp(x^2)*x+2*x^2)*(2*x^3+x)/(1-exp(x^2)*x)^2,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{1-e^{x^2} x+2 x^2} \left (x+2 x^3\right )}{\left (1-e^{x^2} x\right )^2} \, dx=\int \frac {{\mathrm {e}}^{2\,x^2-x\,{\mathrm {e}}^{x^2}+1}\,\left (2\,x^3+x\right )}{{\left (x\,{\mathrm {e}}^{x^2}-1\right )}^2} \,d x \]

[In]

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

[Out]

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