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

Optimal. Leaf size=24 \[ \frac{1}{2 \left (x^2+1\right )}-\frac{1}{2} \log \left (x^2+1\right )+\log (x) \]

[Out]

1/(2*(1 + x^2)) + Log[x] - Log[1 + x^2]/2

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Rubi [A]  time = 0.0255199, antiderivative size = 24, 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 \[ \frac{1}{2 \left (x^2+1\right )}-\frac{1}{2} \log \left (x^2+1\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(1 + x^2)^2),x]

[Out]

1/(2*(1 + x^2)) + Log[x] - Log[1 + x^2]/2

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Rubi in Sympy [A]  time = 1.95417, size = 22, normalized size = 0.92 \[ \frac{\log{\left (x^{2} \right )}}{2} - \frac{\log{\left (x^{2} + 1 \right )}}{2} + \frac{1}{2 \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**2)/2 - log(x**2 + 1)/2 + 1/(2*(x**2 + 1))

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Mathematica [A]  time = 0.0125833, size = 24, normalized size = 1. \[ \frac{1}{2 \left (x^2+1\right )}-\frac{1}{2} \log \left (x^2+1\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(1 + x^2)^2),x]

[Out]

1/(2*(1 + x^2)) + Log[x] - Log[1 + x^2]/2

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Maple [A]  time = 0.016, size = 21, normalized size = 0.9 \[{\frac{1}{2\,{x}^{2}+2}}+\ln \left ( x \right ) -{\frac{\ln \left ({x}^{2}+1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(x^2+1)^2,x)

[Out]

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

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Maxima [A]  time = 1.34761, size = 32, normalized size = 1.33 \[ \frac{1}{2 \,{\left (x^{2} + 1\right )}} - \frac{1}{2} \, \log \left (x^{2} + 1\right ) + \frac{1}{2} \, \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/(x^2 + 1) - 1/2*log(x^2 + 1) + 1/2*log(x^2)

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Fricas [A]  time = 0.22982, size = 43, normalized size = 1.79 \[ -\frac{{\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) - 2 \,{\left (x^{2} + 1\right )} \log \left (x\right ) - 1}{2 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*((x^2 + 1)*log(x^2 + 1) - 2*(x^2 + 1)*log(x) - 1)/(x^2 + 1)

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Sympy [A]  time = 0.110324, size = 19, normalized size = 0.79 \[ \log{\left (x \right )} - \frac{\log{\left (x^{2} + 1 \right )}}{2} + \frac{1}{2 x^{2} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(x**2+1)**2,x)

[Out]

log(x) - log(x**2 + 1)/2 + 1/(2*x**2 + 2)

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GIAC/XCAS [A]  time = 0.217808, size = 39, normalized size = 1.62 \[ \frac{x^{2} + 2}{2 \,{\left (x^{2} + 1\right )}} - \frac{1}{2} \,{\rm ln}\left (x^{2} + 1\right ) + \frac{1}{2} \,{\rm ln}\left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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