3.8.30 \(\int \frac {e^{x^2} x^3}{(1+x^2)^2} \, dx\) [730]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 = {2327} \begin {gather*} \frac {e^{x^2}}{2 \left (x^2+1\right )} \end {gather*}

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 2327

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]]/Coefficient[z, x, Exponent[z, x]])*F^u*v^(n + 1), 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*}

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Mathematica [A]
time = 0.08, size = 16, normalized size = 1.00 \begin {gather*} \frac {e^{x^2}}{2 \left (1+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x^2*x^3)/(1 + x^2)^2,x]

[Out]

E^x^2/(2*(1 + x^2))

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Maple [A]
time = 0.07, size = 14, normalized size = 0.88

method	result	size
gosper	\(\frac {{\mathrm e}^{x^{2}}}{2 x^{2}+2}\)	\(14\)
derivativedivides	\(\frac {{\mathrm e}^{x^{2}}}{2 x^{2}+2}\)	\(14\)
default	\(\frac {{\mathrm e}^{x^{2}}}{2 x^{2}+2}\)	\(14\)
norman	\(\frac {{\mathrm e}^{x^{2}}}{2 x^{2}+2}\)	\(14\)
risch	\(\frac {{\mathrm e}^{x^{2}}}{2 x^{2}+2}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 13, normalized size = 0.81 \begin {gather*} \frac {e^{\left (x^{2}\right )}}{2 \, {\left (x^{2} + 1\right )}} \end {gather*}

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)

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Fricas [A]
time = 0.35, size = 13, normalized size = 0.81 \begin {gather*} \frac {e^{\left (x^{2}\right )}}{2 \, {\left (x^{2} + 1\right )}} \end {gather*}

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)

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Sympy [A]
time = 0.03, size = 10, normalized size = 0.62 \begin {gather*} \frac {e^{x^{2}}}{2 x^{2} + 2} \end {gather*}

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)

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Giac [A]
time = 6.02, size = 13, normalized size = 0.81 \begin {gather*} \frac {e^{\left (x^{2}\right )}}{2 \, {\left (x^{2} + 1\right )}} \end {gather*}

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)

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Mupad [B]
time = 3.48, size = 14, normalized size = 0.88 \begin {gather*} \frac {{\mathrm {e}}^{x^2}}{2\,\left (x^2+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*exp(x^2))/(x^2 + 1)^2,x)

[Out]

exp(x^2)/(2*(x^2 + 1))

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