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

Optimal. Leaf size=16 \[ \frac{e^{x^2}}{2 \left (x^2+1\right )} \]

[Out]

E^x^2/(2*(1 + x^2))

________________________________________________________________________________________

Rubi [A]  time = 0.0576866, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {2289} \[ \frac{e^{x^2}}{2 \left (x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Int[(E^x^2*x^3)/(1 + x^2)^2,x]

[Out]

E^x^2/(2*(1 + x^2))

Rule 2289

Int[(F_)^(u_)*(v_)^(n_.)*(w_), x_Symbol] :> With[{z = Log[F]*v*D[u, x] + (n + 1)*D[v, x]}, Simp[(Coefficient[w
, x, Exponent[w, x]]*F^u*v^(n + 1))/Coefficient[z, x, Exponent[z, x]], x] /; EqQ[Exponent[w, x], Exponent[z, x
]] && EqQ[w*Coefficient[z, x, Exponent[z, x]], z*Coefficient[w, x, Exponent[w, x]]]] /; FreeQ[{F, n}, x] && Po
lynomialQ[u, x] && PolynomialQ[v, x] && PolynomialQ[w, x]

Rubi steps

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

Mathematica [A]  time = 0.0345679, size = 16, normalized size = 1. \[ \frac{e^{x^2}}{2 \left (x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^x^2*x^3)/(1 + x^2)^2,x]

[Out]

E^x^2/(2*(1 + x^2))

________________________________________________________________________________________

Maple [A]  time = 0.02, size = 14, normalized size = 0.9 \begin{align*}{\frac{{{\rm e}^{{x}^{2}}}}{2\,{x}^{2}+2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.975302, size = 18, normalized size = 1.12 \begin{align*} \frac{e^{\left (x^{2}\right )}}{2 \,{\left (x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*e^(x^2)/(x^2 + 1)

________________________________________________________________________________________

Fricas [A]  time = 0.883296, size = 31, normalized size = 1.94 \begin{align*} \frac{e^{\left (x^{2}\right )}}{2 \,{\left (x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*e^(x^2)/(x^2 + 1)

________________________________________________________________________________________

Sympy [A]  time = 0.091789, size = 10, normalized size = 0.62 \begin{align*} \frac{e^{x^{2}}}{2 x^{2} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x**2)/(2*x**2 + 2)

________________________________________________________________________________________

Giac [A]  time = 1.3054, size = 18, normalized size = 1.12 \begin{align*} \frac{e^{\left (x^{2}\right )}}{2 \,{\left (x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*e^(x^2)/(x^2 + 1)