Optimal. Leaf size=23 \[ \frac{2}{x^2+1}-\frac{1}{4 \left (x^2+1\right )^2}+\tan ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.0407249, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{2}{x^2+1}-\frac{1}{4 \left (x^2+1\right )^2}+\tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[(1 - 3*x + 2*x^2 - 4*x^3 + x^4)/(1 + x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 15.5979, size = 22, normalized size = 0.96 \[ \frac{x^{2}}{4 \left (x^{2} + 1\right )^{2}} + \operatorname{atan}{\left (x \right )} + \frac{7}{4 \left (x^{2} + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**4-4*x**3+2*x**2-3*x+1)/(x**2+1)**3,x)
[Out]
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Mathematica [A] time = 0.0189779, size = 23, normalized size = 1. \[ \frac{2}{x^2+1}-\frac{1}{4 \left (x^2+1\right )^2}+\tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Integrate[(1 - 3*x + 2*x^2 - 4*x^3 + x^4)/(1 + x^2)^3,x]
[Out]
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Maple [A] time = 0.007, size = 19, normalized size = 0.8 \[{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( 2\,{x}^{2}+{\frac{7}{4}} \right ) }+\arctan \left ( x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^4-4*x^3+2*x^2-3*x+1)/(x^2+1)^3,x)
[Out]
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Maxima [A] time = 0.893775, size = 32, normalized size = 1.39 \[ \frac{8 \, x^{2} + 7}{4 \,{\left (x^{4} + 2 \, x^{2} + 1\right )}} + \arctan \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 - 4*x^3 + 2*x^2 - 3*x + 1)/(x^2 + 1)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.253124, size = 47, normalized size = 2.04 \[ \frac{8 \, x^{2} + 4 \,{\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (x\right ) + 7}{4 \,{\left (x^{4} + 2 \, x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 - 4*x^3 + 2*x^2 - 3*x + 1)/(x^2 + 1)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.32248, size = 20, normalized size = 0.87 \[ \frac{8 x^{2} + 7}{4 x^{4} + 8 x^{2} + 4} + \operatorname{atan}{\left (x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**4-4*x**3+2*x**2-3*x+1)/(x**2+1)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.261317, size = 26, normalized size = 1.13 \[ \frac{8 \, x^{2} + 7}{4 \,{\left (x^{2} + 1\right )}^{2}} + \arctan \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 - 4*x^3 + 2*x^2 - 3*x + 1)/(x^2 + 1)^3,x, algorithm="giac")
[Out]