Optimal. Leaf size=24 \[ \frac{2 x+3}{2 \left (x^2+4 x+5\right )}+\tan ^{-1}(x+2) \]
[Out]
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Rubi [A] time = 0.0221041, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{2 x+3}{2 \left (x^2+4 x+5\right )}+\tan ^{-1}(x+2) \]
Antiderivative was successfully verified.
[In] Int[(4 + x)/(5 + 4*x + x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 4.39601, size = 19, normalized size = 0.79 \[ \frac{4 x + 6}{4 \left (x^{2} + 4 x + 5\right )} + \operatorname{atan}{\left (x + 2 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((4+x)/(x**2+4*x+5)**2,x)
[Out]
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Mathematica [A] time = 0.0200706, size = 24, normalized size = 1. \[ \frac{2 x+3}{2 \left (x^2+4 x+5\right )}+\tan ^{-1}(x+2) \]
Antiderivative was successfully verified.
[In] Integrate[(4 + x)/(5 + 4*x + x^2)^2,x]
[Out]
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Maple [A] time = 0.004, size = 23, normalized size = 1. \[{\frac{4\,x+6}{4\,{x}^{2}+16\,x+20}}+\arctan \left ( 2+x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((4+x)/(x^2+4*x+5)^2,x)
[Out]
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Maxima [A] time = 0.772106, size = 30, normalized size = 1.25 \[ \frac{2 \, x + 3}{2 \,{\left (x^{2} + 4 \, x + 5\right )}} + \arctan \left (x + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 4)/(x^2 + 4*x + 5)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.292004, size = 42, normalized size = 1.75 \[ \frac{2 \,{\left (x^{2} + 4 \, x + 5\right )} \arctan \left (x + 2\right ) + 2 \, x + 3}{2 \,{\left (x^{2} + 4 \, x + 5\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 4)/(x^2 + 4*x + 5)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.260082, size = 19, normalized size = 0.79 \[ \frac{2 x + 3}{2 x^{2} + 8 x + 10} + \operatorname{atan}{\left (x + 2 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4+x)/(x**2+4*x+5)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.268528, size = 30, normalized size = 1.25 \[ \frac{2 \, x + 3}{2 \,{\left (x^{2} + 4 \, x + 5\right )}} + \arctan \left (x + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 4)/(x^2 + 4*x + 5)^2,x, algorithm="giac")
[Out]