3.275 \(\int \frac{x^3}{1+2 x^4+x^8} \, dx\)

Optimal. Leaf size=11 \[ -\frac{1}{4 \left (x^4+1\right )} \]

[Out]

-1/(4*(1 + x^4))

_______________________________________________________________________________________

Rubi [A]  time = 0.00693275, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{1}{4 \left (x^4+1\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(1 + 2*x^4 + x^8),x]

[Out]

-1/(4*(1 + x^4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.85374, size = 8, normalized size = 0.73 \[ - \frac{1}{4 \left (x^{4} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(x**8+2*x**4+1),x)

[Out]

-1/(4*(x**4 + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.00306416, size = 11, normalized size = 1. \[ -\frac{1}{4 \left (x^4+1\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(1 + 2*x^4 + x^8),x]

[Out]

-1/(4*(1 + x^4))

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 10, normalized size = 0.9 \[ -{\frac{1}{4\,{x}^{4}+4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(x^8+2*x^4+1),x)

[Out]

-1/4/(x^4+1)

_______________________________________________________________________________________

Maxima [A]  time = 0.766338, size = 12, normalized size = 1.09 \[ -\frac{1}{4 \,{\left (x^{4} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(x^8 + 2*x^4 + 1),x, algorithm="maxima")

[Out]

-1/4/(x^4 + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.248724, size = 12, normalized size = 1.09 \[ -\frac{1}{4 \,{\left (x^{4} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(x^8 + 2*x^4 + 1),x, algorithm="fricas")

[Out]

-1/4/(x^4 + 1)

_______________________________________________________________________________________

Sympy [A]  time = 0.227761, size = 8, normalized size = 0.73 \[ - \frac{1}{4 x^{4} + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(x**8+2*x**4+1),x)

[Out]

-1/(4*x**4 + 4)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.261829, size = 12, normalized size = 1.09 \[ -\frac{1}{4 \,{\left (x^{4} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(x^8 + 2*x^4 + 1),x, algorithm="giac")

[Out]

-1/4/(x^4 + 1)