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

Optimal. Leaf size=30 \[ -\frac{3}{4 x^2}-\frac{3}{4} \tan ^{-1}\left (x^2\right )+\frac{1}{4 x^2 \left (x^4+1\right )} \]

[Out]

-3/(4*x^2) + 1/(4*x^2*(1 + x^4)) - (3*ArcTan[x^2])/4

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Rubi [A]  time = 0.0331186, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ -\frac{3}{4 x^2}-\frac{3}{4} \tan ^{-1}\left (x^2\right )+\frac{1}{4 x^2 \left (x^4+1\right )} \]

Antiderivative was successfully verified.

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

[Out]

-3/(4*x^2) + 1/(4*x^2*(1 + x^4)) - (3*ArcTan[x^2])/4

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Rubi in Sympy [A]  time = 7.58688, size = 26, normalized size = 0.87 \[ - \frac{3 \operatorname{atan}{\left (x^{2} \right )}}{4} - \frac{3}{4 x^{2}} + \frac{1}{4 x^{2} \left (x^{4} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*atan(x**2)/4 - 3/(4*x**2) + 1/(4*x**2*(x**4 + 1))

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Mathematica [A]  time = 0.0181117, size = 30, normalized size = 1. \[ -\frac{1}{2 x^2}+\frac{3}{4} \tan ^{-1}\left (\frac{1}{x^2}\right )-\frac{x^2}{4 \left (x^4+1\right )} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*x^2) - x^2/(4*(1 + x^4)) + (3*ArcTan[x^(-2)])/4

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Maple [A]  time = 0.015, size = 25, normalized size = 0.8 \[ -{\frac{{x}^{2}}{4\,{x}^{4}+4}}-{\frac{3\,\arctan \left ({x}^{2} \right ) }{4}}-{\frac{1}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(x^8+2*x^4+1),x)

[Out]

-1/4*x^2/(x^4+1)-3/4*arctan(x^2)-1/2/x^2

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Maxima [A]  time = 0.851052, size = 34, normalized size = 1.13 \[ -\frac{3 \, x^{4} + 2}{4 \,{\left (x^{6} + x^{2}\right )}} - \frac{3}{4} \, \arctan \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(3*x^4 + 2)/(x^6 + x^2) - 3/4*arctan(x^2)

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Fricas [A]  time = 0.251319, size = 42, normalized size = 1.4 \[ -\frac{3 \, x^{4} + 3 \,{\left (x^{6} + x^{2}\right )} \arctan \left (x^{2}\right ) + 2}{4 \,{\left (x^{6} + x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(3*x^4 + 3*(x^6 + x^2)*arctan(x^2) + 2)/(x^6 + x^2)

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Sympy [A]  time = 0.394401, size = 26, normalized size = 0.87 \[ - \frac{3 x^{4} + 2}{4 x^{6} + 4 x^{2}} - \frac{3 \operatorname{atan}{\left (x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(3*x**4 + 2)/(4*x**6 + 4*x**2) - 3*atan(x**2)/4

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GIAC/XCAS [A]  time = 0.299475, size = 34, normalized size = 1.13 \[ -\frac{3 \, x^{4} + 2}{4 \,{\left (x^{6} + x^{2}\right )}} - \frac{3}{4} \, \arctan \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(3*x^4 + 2)/(x^6 + x^2) - 3/4*arctan(x^2)