Optimal. Leaf size=26 \[ \frac{1}{2} \log \left (x^2+2 x+5\right )-\frac{1}{2} \tan ^{-1}\left (\frac{x+1}{2}\right ) \]
[Out]
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Rubi [A] time = 0.0329877, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{1}{2} \log \left (x^2+2 x+5\right )-\frac{1}{2} \tan ^{-1}\left (\frac{x+1}{2}\right ) \]
Antiderivative was successfully verified.
[In] Int[x/(5 + 2*x + x^2),x]
[Out]
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Rubi in Sympy [A] time = 5.10193, size = 20, normalized size = 0.77 \[ \frac{\log{\left (x^{2} + 2 x + 5 \right )}}{2} - \frac{\operatorname{atan}{\left (\frac{x}{2} + \frac{1}{2} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(x**2+2*x+5),x)
[Out]
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Mathematica [A] time = 0.0064435, size = 26, normalized size = 1. \[ \frac{1}{2} \log \left (x^2+2 x+5\right )-\frac{1}{2} \tan ^{-1}\left (\frac{x+1}{2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x/(5 + 2*x + x^2),x]
[Out]
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Maple [A] time = 0.004, size = 21, normalized size = 0.8 \[ -{\frac{1}{2}\arctan \left ({\frac{1}{2}}+{\frac{x}{2}} \right ) }+{\frac{\ln \left ({x}^{2}+2\,x+5 \right ) }{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(x^2+2*x+5),x)
[Out]
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Maxima [A] time = 0.755293, size = 27, normalized size = 1.04 \[ -\frac{1}{2} \, \arctan \left (\frac{1}{2} \, x + \frac{1}{2}\right ) + \frac{1}{2} \, \log \left (x^{2} + 2 \, x + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x^2 + 2*x + 5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.204423, size = 27, normalized size = 1.04 \[ -\frac{1}{2} \, \arctan \left (\frac{1}{2} \, x + \frac{1}{2}\right ) + \frac{1}{2} \, \log \left (x^{2} + 2 \, x + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x^2 + 2*x + 5),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.189723, size = 20, normalized size = 0.77 \[ \frac{\log{\left (x^{2} + 2 x + 5 \right )}}{2} - \frac{\operatorname{atan}{\left (\frac{x}{2} + \frac{1}{2} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x**2+2*x+5),x)
[Out]
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GIAC/XCAS [A] time = 0.205571, size = 27, normalized size = 1.04 \[ -\frac{1}{2} \, \arctan \left (\frac{1}{2} \, x + \frac{1}{2}\right ) + \frac{1}{2} \,{\rm ln}\left (x^{2} + 2 \, x + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x^2 + 2*x + 5),x, algorithm="giac")
[Out]