Optimal. Leaf size=33 \[ -\frac{2 x+1}{2 \left (x^2+1\right )}-\frac{1}{2} \log \left (x^2+1\right )+\log (x)-2 \tan ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.0743529, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{2 x+1}{2 \left (x^2+1\right )}-\frac{1}{2} \log \left (x^2+1\right )+\log (x)-2 \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[(1 - 3*x + 2*x^2 - x^3)/(x*(1 + x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 7.3762, size = 29, normalized size = 0.88 \[ - \frac{x \left (2 + \frac{1}{x}\right )}{2 \left (x^{2} + 1\right )} + 2 \log{\left (x \right )} - \log{\left (x^{2} + 1 \right )} - 2 \operatorname{atan}{\left (x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x**3+2*x**2-3*x+1)/x/(x**2+1)**2,x)
[Out]
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Mathematica [A] time = 0.0352772, size = 33, normalized size = 1. \[ \frac{-2 x-1}{2 \left (x^2+1\right )}-\frac{1}{2} \log \left (x^2+1\right )+\log (x)-2 \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Integrate[(1 - 3*x + 2*x^2 - x^3)/(x*(1 + x^2)^2),x]
[Out]
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Maple [A] time = 0.011, size = 28, normalized size = 0.9 \[ -{\frac{1}{{x}^{2}+1} \left ( x+{\frac{1}{2}} \right ) }-{\frac{\ln \left ({x}^{2}+1 \right ) }{2}}-2\,\arctan \left ( x \right ) +\ln \left ( x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x^3+2*x^2-3*x+1)/x/(x^2+1)^2,x)
[Out]
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Maxima [A] time = 1.54702, size = 39, normalized size = 1.18 \[ -\frac{2 \, x + 1}{2 \,{\left (x^{2} + 1\right )}} - 2 \, \arctan \left (x\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) + \log \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^3 - 2*x^2 + 3*x - 1)/((x^2 + 1)^2*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.209482, size = 59, normalized size = 1.79 \[ -\frac{4 \,{\left (x^{2} + 1\right )} \arctan \left (x\right ) +{\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) - 2 \,{\left (x^{2} + 1\right )} \log \left (x\right ) + 2 \, x + 1}{2 \,{\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^3 - 2*x^2 + 3*x - 1)/((x^2 + 1)^2*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.180143, size = 27, normalized size = 0.82 \[ - \frac{2 x + 1}{2 x^{2} + 2} + \log{\left (x \right )} - \frac{\log{\left (x^{2} + 1 \right )}}{2} - 2 \operatorname{atan}{\left (x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x**3+2*x**2-3*x+1)/x/(x**2+1)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.205696, size = 41, normalized size = 1.24 \[ -\frac{2 \, x + 1}{2 \,{\left (x^{2} + 1\right )}} - 2 \, \arctan \left (x\right ) - \frac{1}{2} \,{\rm ln}\left (x^{2} + 1\right ) +{\rm ln}\left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^3 - 2*x^2 + 3*x - 1)/((x^2 + 1)^2*x),x, algorithm="giac")
[Out]