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

Optimal. Leaf size=25 \[ -\frac{e^{1-e^{x^2} x}}{e^{x^2} x-1} \]

[Out]

-(E^(1 - E^x^2*x)/(-1 + E^x^2*x))

________________________________________________________________________________________

Rubi [F]  time = 0.981599, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{e^{1-e^{x^2} x+2 x^2} \left (x+2 x^3\right )}{\left (1-e^{x^2} x\right )^2} \, dx \]

Verification is Not applicable to the result.

[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*} \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} 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]  time = 0.164539, size = 25, normalized size = 1. \[ -\frac{e^{1-e^{x^2} x}}{e^{x^2} x-1} \]

Antiderivative was successfully verified.

[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]  time = 0.06, size = 23, normalized size = 0.9 \begin{align*} -{\frac{{{\rm e}^{1-{{\rm e}^{{x}^{2}}}x}}}{-1+{{\rm e}^{{x}^{2}}}x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.27126, size = 30, normalized size = 1.2 \begin{align*} -\frac{e^{\left (-x e^{\left (x^{2}\right )} + 1\right )}}{x e^{\left (x^{2}\right )} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [A]  time = 1.79165, size = 73, normalized size = 2.92 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 0.406817, size = 31, normalized size = 1.24 \begin{align*} - \frac{e^{2 x^{2} - x e^{x^{2}} + 1}}{x e^{3 x^{2}} - e^{2 x^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (2 \, x^{3} + x\right )} e^{\left (2 \, x^{2} - x e^{\left (x^{2}\right )} + 1\right )}}{{\left (x e^{\left (x^{2}\right )} - 1\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

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