Optimal. Leaf size=16 \[ -\log \left (\sqrt{4-x^2}+1\right ) \]
[Out]
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Rubi [A] time = 0.0811877, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\log \left (\sqrt{4-x^2}+1\right ) \]
Antiderivative was successfully verified.
[In] Int[x/(4 - x^2 + Sqrt[4 - x^2]),x]
[Out]
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Rubi in Sympy [A] time = 3.02601, size = 12, normalized size = 0.75 \[ - \log{\left (\sqrt{- x^{2} + 4} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(4-x**2+(-x**2+4)**(1/2)),x)
[Out]
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Mathematica [A] time = 0.0129228, size = 16, normalized size = 1. \[ -\log \left (\sqrt{4-x^2}+1\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x/(4 - x^2 + Sqrt[4 - x^2]),x]
[Out]
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Maple [B] time = 0.089, size = 266, normalized size = 16.6 \[ -{\frac{\ln \left ({x}^{2}-3 \right ) }{2}}+{\frac{1}{ \left ( 4+2\,\sqrt{3} \right ) \left ( -2+\sqrt{3} \right ) }\sqrt{- \left ( -2+x \right ) ^{2}-4\,x+8}}+{\frac{1}{ \left ( 4+2\,\sqrt{3} \right ) \left ( -2+\sqrt{3} \right ) }\sqrt{- \left ( 2+x \right ) ^{2}+4\,x+8}}+{\frac{1}{ \left ( 4+2\,\sqrt{3} \right ) \left ( -2+\sqrt{3} \right ) }{\it Artanh} \left ({\frac{2-2\,\sqrt{3} \left ( x-\sqrt{3} \right ) }{2}{\frac{1}{\sqrt{- \left ( x-\sqrt{3} \right ) ^{2}-2\,\sqrt{3} \left ( x-\sqrt{3} \right ) +1}}}} \right ) }-{\frac{1}{ \left ( 4+2\,\sqrt{3} \right ) \left ( -2+\sqrt{3} \right ) }\sqrt{- \left ( x-\sqrt{3} \right ) ^{2}-2\,\sqrt{3} \left ( x-\sqrt{3} \right ) +1}}+{\frac{1}{ \left ( 4+2\,\sqrt{3} \right ) \left ( -2+\sqrt{3} \right ) }{\it Artanh} \left ({\frac{2+2\,\sqrt{3} \left ( x+\sqrt{3} \right ) }{2}{\frac{1}{\sqrt{- \left ( x+\sqrt{3} \right ) ^{2}+2\,\sqrt{3} \left ( x+\sqrt{3} \right ) +1}}}} \right ) }-{\frac{1}{ \left ( 4+2\,\sqrt{3} \right ) \left ( -2+\sqrt{3} \right ) }\sqrt{- \left ( x+\sqrt{3} \right ) ^{2}+2\,\sqrt{3} \left ( x+\sqrt{3} \right ) +1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(4-x^2+(-x^2+4)^(1/2)),x)
[Out]
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Maxima [A] time = 1.34782, size = 19, normalized size = 1.19 \[ -\log \left (\sqrt{-x^{2} + 4} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x/(x^2 - sqrt(-x^2 + 4) - 4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.21199, size = 74, normalized size = 4.62 \[ -\frac{1}{2} \, \log \left (x^{2} - 3\right ) + \frac{1}{2} \, \log \left (-\frac{x^{2} + 3 \, \sqrt{-x^{2} + 4} - 6}{x^{2}}\right ) - \frac{1}{2} \, \log \left (-\frac{x^{2} + \sqrt{-x^{2} + 4} - 2}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x/(x^2 - sqrt(-x^2 + 4) - 4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.16125, size = 17, normalized size = 1.06 \[ - \begin{cases} \log{\left (\sqrt{- x^{2} + 4} + 1 \right )} & \text{for}\: x > -2 \wedge x < 2 \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(4-x**2+(-x**2+4)**(1/2)),x)
[Out]
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GIAC/XCAS [A] time = 0.209149, size = 19, normalized size = 1.19 \[ -{\rm ln}\left (\sqrt{-x^{2} + 4} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x/(x^2 - sqrt(-x^2 + 4) - 4),x, algorithm="giac")
[Out]