3.1371 \(\int \frac{1}{x^4 \left (1+x^6\right )} \, dx\)

Optimal. Leaf size=16 \[ -\frac{1}{3 x^3}-\frac{1}{3} \tan ^{-1}\left (x^3\right ) \]

[Out]

-1/(3*x^3) - ArcTan[x^3]/3

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Rubi [A]  time = 0.0213406, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{1}{3 x^3}-\frac{1}{3} \tan ^{-1}\left (x^3\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(1 + x^6)),x]

[Out]

-1/(3*x^3) - ArcTan[x^3]/3

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Rubi in Sympy [A]  time = 4.49828, size = 14, normalized size = 0.88 \[ - \frac{\operatorname{atan}{\left (x^{3} \right )}}{3} - \frac{1}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(x**6+1),x)

[Out]

-atan(x**3)/3 - 1/(3*x**3)

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Mathematica [A]  time = 0.00790838, size = 16, normalized size = 1. \[ \frac{1}{3} \tan ^{-1}\left (\frac{1}{x^3}\right )-\frac{1}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(1 + x^6)),x]

[Out]

-1/(3*x^3) + ArcTan[x^(-3)]/3

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Maple [B]  time = 0.02, size = 33, normalized size = 2.1 \[ -{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{3}}-{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{3}}-{\frac{1}{3\,{x}^{3}}}+{\frac{\arctan \left ( x \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(x^6+1),x)

[Out]

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

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Maxima [A]  time = 1.58269, size = 16, normalized size = 1. \[ -\frac{1}{3 \, x^{3}} - \frac{1}{3} \, \arctan \left (x^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 1)*x^4),x, algorithm="maxima")

[Out]

-1/3/x^3 - 1/3*arctan(x^3)

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Fricas [A]  time = 0.210705, size = 20, normalized size = 1.25 \[ -\frac{x^{3} \arctan \left (x^{3}\right ) + 1}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 1)*x^4),x, algorithm="fricas")

[Out]

-1/3*(x^3*arctan(x^3) + 1)/x^3

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Sympy [A]  time = 0.315279, size = 14, normalized size = 0.88 \[ - \frac{\operatorname{atan}{\left (x^{3} \right )}}{3} - \frac{1}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(x**6+1),x)

[Out]

-atan(x**3)/3 - 1/(3*x**3)

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GIAC/XCAS [A]  time = 0.222159, size = 16, normalized size = 1. \[ -\frac{1}{3 \, x^{3}} - \frac{1}{3} \, \arctan \left (x^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 1)*x^4),x, algorithm="giac")

[Out]

-1/3/x^3 - 1/3*arctan(x^3)