3.718 \(\int \frac{e^{x^2} \left (1+4 x^4\right )}{x^2} \, dx\)

Optimal. Leaf size=19 \[ 2 e^{x^2} x-\frac{e^{x^2}}{x} \]

[Out]

-(E^x^2/x) + 2*E^x^2*x

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Rubi [A]  time = 0.153611, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ 2 e^{x^2} x-\frac{e^{x^2}}{x} \]

Antiderivative was successfully verified.

[In]  Int[(E^x^2*(1 + 4*x^4))/x^2,x]

[Out]

-(E^x^2/x) + 2*E^x^2*x

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (4 x^{4} + 1\right ) e^{x^{2}}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(x**2)*(4*x**4+1)/x**2,x)

[Out]

Integral((4*x**4 + 1)*exp(x**2)/x**2, x)

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Mathematica [A]  time = 0.00785526, size = 15, normalized size = 0.79 \[ e^{x^2} \left (2 x-\frac{1}{x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(E^x^2*(1 + 4*x^4))/x^2,x]

[Out]

E^x^2*(-x^(-1) + 2*x)

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Maple [A]  time = 0.007, size = 16, normalized size = 0.8 \[{\frac{{{\rm e}^{{x}^{2}}} \left ( 2\,{x}^{2}-1 \right ) }{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x^2)*(4*x^4+1)/x^2,x)

[Out]

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

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Maxima [A]  time = 0.839024, size = 49, normalized size = 2.58 \[ 2 \, x e^{\left (x^{2}\right )} + i \, \sqrt{\pi } \operatorname{erf}\left (i \, x\right ) - \frac{\sqrt{-x^{2}} \Gamma \left (-\frac{1}{2}, -x^{2}\right )}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^4 + 1)*e^(x^2)/x^2,x, algorithm="maxima")

[Out]

2*x*e^(x^2) + I*sqrt(pi)*erf(I*x) - 1/2*sqrt(-x^2)*gamma(-1/2, -x^2)/x

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Fricas [A]  time = 0.246013, size = 20, normalized size = 1.05 \[ \frac{{\left (2 \, x^{2} - 1\right )} e^{\left (x^{2}\right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^4 + 1)*e^(x^2)/x^2,x, algorithm="fricas")

[Out]

(2*x^2 - 1)*e^(x^2)/x

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Sympy [A]  time = 0.070949, size = 12, normalized size = 0.63 \[ \frac{\left (2 x^{2} - 1\right ) e^{x^{2}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(x**2)*(4*x**4+1)/x**2,x)

[Out]

(2*x**2 - 1)*exp(x**2)/x

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GIAC/XCAS [A]  time = 0.219867, size = 27, normalized size = 1.42 \[ \frac{2 \, x^{2} e^{\left (x^{2}\right )} - e^{\left (x^{2}\right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^4 + 1)*e^(x^2)/x^2,x, algorithm="giac")

[Out]

(2*x^2*e^(x^2) - e^(x^2))/x