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