Optimal. Leaf size=19 \[ \frac{x}{2 \left (x^2+1\right )}+\frac{5}{2} \tan ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.0167511, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x}{2 \left (x^2+1\right )}+\frac{5}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[(3 + 2*x^2)/(1 + x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 4.10227, size = 14, normalized size = 0.74 \[ \frac{x}{2 \left (x^{2} + 1\right )} + \frac{5 \operatorname{atan}{\left (x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*x**2+3)/(x**2+1)**2,x)
[Out]
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Mathematica [A] time = 0.0116605, size = 19, normalized size = 1. \[ \frac{x}{2 \left (x^2+1\right )}+\frac{5}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Integrate[(3 + 2*x^2)/(1 + x^2)^2,x]
[Out]
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Maple [A] time = 0.008, size = 16, normalized size = 0.8 \[{\frac{x}{2\,{x}^{2}+2}}+{\frac{5\,\arctan \left ( x \right ) }{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*x^2+3)/(x^2+1)^2,x)
[Out]
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Maxima [A] time = 1.48253, size = 20, normalized size = 1.05 \[ \frac{x}{2 \,{\left (x^{2} + 1\right )}} + \frac{5}{2} \, \arctan \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^2 + 3)/(x^2 + 1)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.236045, size = 27, normalized size = 1.42 \[ \frac{5 \,{\left (x^{2} + 1\right )} \arctan \left (x\right ) + x}{2 \,{\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^2 + 3)/(x^2 + 1)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.221471, size = 14, normalized size = 0.74 \[ \frac{x}{2 x^{2} + 2} + \frac{5 \operatorname{atan}{\left (x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x**2+3)/(x**2+1)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.221626, size = 20, normalized size = 1.05 \[ \frac{x}{2 \,{\left (x^{2} + 1\right )}} + \frac{5}{2} \, \arctan \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^2 + 3)/(x^2 + 1)^2,x, algorithm="giac")
[Out]