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

Optimal. Leaf size=13 \[ -\frac{1}{3 x^3}+\frac{1}{x}+\tan ^{-1}(x) \]

[Out]

-1/(3*x^3) + x^(-1) + ArcTan[x]

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Rubi [A]  time = 0.0172013, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{1}{3 x^3}+\frac{1}{x}+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(1 + x^2)),x]

[Out]

-1/(3*x^3) + x^(-1) + ArcTan[x]

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Rubi in Sympy [A]  time = 2.24829, size = 12, normalized size = 0.92 \[ \operatorname{atan}{\left (x \right )} + \frac{1}{x} - \frac{1}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(x) + 1/x - 1/(3*x**3)

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Mathematica [A]  time = 0.00598656, size = 13, normalized size = 1. \[ -\frac{1}{3 x^3}+\frac{1}{x}+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(1 + x^2)),x]

[Out]

-1/(3*x^3) + x^(-1) + ArcTan[x]

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Maple [A]  time = 0.007, size = 12, normalized size = 0.9 \[ -{\frac{1}{3\,{x}^{3}}}+{x}^{-1}+\arctan \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(x^2+1),x)

[Out]

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

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Maxima [A]  time = 1.484, size = 20, normalized size = 1.54 \[ \frac{3 \, x^{2} - 1}{3 \, x^{3}} + \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(3*x^2 - 1)/x^3 + arctan(x)

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Fricas [A]  time = 0.210309, size = 26, normalized size = 2. \[ \frac{3 \, x^{3} \arctan \left (x\right ) + 3 \, x^{2} - 1}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(3*x^3*arctan(x) + 3*x^2 - 1)/x^3

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Sympy [A]  time = 0.116823, size = 14, normalized size = 1.08 \[ \operatorname{atan}{\left (x \right )} + \frac{3 x^{2} - 1}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(x**2+1),x)

[Out]

atan(x) + (3*x**2 - 1)/(3*x**3)

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GIAC/XCAS [A]  time = 0.202067, size = 20, normalized size = 1.54 \[ \frac{3 \, x^{2} - 1}{3 \, x^{3}} + \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(3*x^2 - 1)/x^3 + arctan(x)