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

Optimal. Leaf size=10 \[ \frac{1}{x^2+1}+\log (x) \]

[Out]

(1 + x^2)^(-1) + Log[x]

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Rubi [A]  time = 0.046195, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{1}{x^2+1}+\log (x) \]

Antiderivative was successfully verified.

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

[Out]

(1 + x^2)^(-1) + Log[x]

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Rubi in Sympy [A]  time = 3.27187, size = 12, normalized size = 1.2 \[ \frac{\log{\left (x^{2} \right )}}{2} + \frac{1}{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**2)/2 + 1/(x**2 + 1)

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Mathematica [A]  time = 0.00782358, size = 10, normalized size = 1. \[ \frac{1}{x^2+1}+\log (x) \]

Antiderivative was successfully verified.

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

[Out]

(1 + x^2)^(-1) + Log[x]

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Maple [A]  time = 0.009, size = 11, normalized size = 1.1 \[ \left ({x}^{2}+1 \right ) ^{-1}+\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4+1)/x/(x^2+1)^2,x)

[Out]

1/(x^2+1)+ln(x)

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Maxima [A]  time = 1.3425, size = 19, normalized size = 1.9 \[ \frac{1}{x^{2} + 1} + \frac{1}{2} \, \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(x^2 + 1) + 1/2*log(x^2)

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Fricas [A]  time = 0.195865, size = 24, normalized size = 2.4 \[ \frac{{\left (x^{2} + 1\right )} \log \left (x\right ) + 1}{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((x^2 + 1)*log(x) + 1)/(x^2 + 1)

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Sympy [A]  time = 0.102103, size = 8, normalized size = 0.8 \[ \log{\left (x \right )} + \frac{1}{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x) + 1/(x**2 + 1)

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GIAC/XCAS [A]  time = 0.222716, size = 19, normalized size = 1.9 \[ \frac{1}{x^{2} + 1} + \frac{1}{2} \,{\rm ln}\left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(x^2 + 1) + 1/2*ln(x^2)