3.1495 \(\int \frac{1}{x^5 \left (1+x^8\right )} \, dx\)

Optimal. Leaf size=16 \[ -\frac{1}{4 x^4}-\frac{1}{4} \tan ^{-1}\left (x^4\right ) \]

[Out]

-1/(4*x^4) - ArcTan[x^4]/4

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Rubi [A]  time = 0.0243344, 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 \[ -\frac{1}{4 x^4}-\frac{1}{4} \tan ^{-1}\left (x^4\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(1 + x^8)),x]

[Out]

-1/(4*x^4) - ArcTan[x^4]/4

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Rubi in Sympy [A]  time = 4.72229, size = 14, normalized size = 0.88 \[ - \frac{\operatorname{atan}{\left (x^{4} \right )}}{4} - \frac{1}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(x**8+1),x)

[Out]

-atan(x**4)/4 - 1/(4*x**4)

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Mathematica [A]  time = 0.00776439, size = 16, normalized size = 1. \[ \frac{1}{4} \tan ^{-1}\left (\frac{1}{x^4}\right )-\frac{1}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(1 + x^8)),x]

[Out]

-1/(4*x^4) + ArcTan[x^(-4)]/4

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Maple [A]  time = 0.008, size = 13, normalized size = 0.8 \[ -{\frac{1}{4\,{x}^{4}}}-{\frac{\arctan \left ({x}^{4} \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(x^8+1),x)

[Out]

-1/4/x^4-1/4*arctan(x^4)

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Maxima [A]  time = 1.56746, size = 16, normalized size = 1. \[ -\frac{1}{4 \, x^{4}} - \frac{1}{4} \, \arctan \left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 1)*x^5),x, algorithm="maxima")

[Out]

-1/4/x^4 - 1/4*arctan(x^4)

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Fricas [A]  time = 0.215091, size = 20, normalized size = 1.25 \[ -\frac{x^{4} \arctan \left (x^{4}\right ) + 1}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 1)*x^5),x, algorithm="fricas")

[Out]

-1/4*(x^4*arctan(x^4) + 1)/x^4

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Sympy [A]  time = 0.370067, size = 14, normalized size = 0.88 \[ - \frac{\operatorname{atan}{\left (x^{4} \right )}}{4} - \frac{1}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(x**8+1),x)

[Out]

-atan(x**4)/4 - 1/(4*x**4)

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GIAC/XCAS [A]  time = 0.233213, size = 16, normalized size = 1. \[ -\frac{1}{4 \, x^{4}} - \frac{1}{4} \, \arctan \left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 1)*x^5),x, algorithm="giac")

[Out]

-1/4/x^4 - 1/4*arctan(x^4)