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

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

[Out]

E^x^2/(2*(1 + x^2))

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Rubi [A]  time = 0.0823358, 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 \[ \frac{e^{x^2}}{2 \left (x^2+1\right )} \]

Antiderivative was successfully verified.

[In]  Int[(E^x^2*x^3)/(1 + x^2)^2,x]

[Out]

E^x^2/(2*(1 + x^2))

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Rubi in Sympy [A]  time = 9.8791, size = 10, normalized size = 0.62 \[ \frac{e^{x^{2}}}{2 \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(x**2)*x**3/(x**2+1)**2,x)

[Out]

exp(x**2)/(2*(x**2 + 1))

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Mathematica [A]  time = 0.0100631, 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))

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Maple [A]  time = 0.008, size = 14, normalized size = 0.9 \[{\frac{{{\rm e}^{{x}^{2}}}}{2\,{x}^{2}+2}} \]

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)

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Maxima [A]  time = 0.772056, size = 18, normalized size = 1.12 \[ \frac{e^{\left (x^{2}\right )}}{2 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*e^(x^2)/(x^2 + 1)^2,x, algorithm="maxima")

[Out]

1/2*e^(x^2)/(x^2 + 1)

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Fricas [A]  time = 0.216306, size = 18, normalized size = 1.12 \[ \frac{e^{\left (x^{2}\right )}}{2 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*e^(x^2)/(x^2 + 1)^2,x, algorithm="fricas")

[Out]

1/2*e^(x^2)/(x^2 + 1)

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Sympy [A]  time = 0.07536, size = 10, normalized size = 0.62 \[ \frac{e^{x^{2}}}{2 x^{2} + 2} \]

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/XCAS [A]  time = 0.227179, size = 18, normalized size = 1.12 \[ \frac{e^{\left (x^{2}\right )}}{2 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*e^(x^2)/(x^2 + 1)^2,x, algorithm="giac")

[Out]

1/2*e^(x^2)/(x^2 + 1)