3.114 \(\int \frac{x}{4-x^2+\sqrt{4-x^2}} \, dx\)

Optimal. Leaf size=16 \[ -\log \left (\sqrt{4-x^2}+1\right ) \]

[Out]

-Log[1 + Sqrt[4 - x^2]]

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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]

-Log[1 + Sqrt[4 - x^2]]

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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]

-log(sqrt(-x**2 + 4) + 1)

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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]

-Log[1 + Sqrt[4 - x^2]]

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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]

-1/2*ln(x^2-3)+1/2/(2+3^(1/2))/(-2+3^(1/2))*(-(-2+x)^2-4*x+8)^(1/2)+1/2/(2+3^(1/
2))/(-2+3^(1/2))*(-(2+x)^2+4*x+8)^(1/2)+1/2/(2+3^(1/2))/(-2+3^(1/2))*arctanh(1/2
*(2-2*3^(1/2)*(x-3^(1/2)))/(-(x-3^(1/2))^2-2*3^(1/2)*(x-3^(1/2))+1)^(1/2))-1/2/(
2+3^(1/2))/(-2+3^(1/2))*(-(x-3^(1/2))^2-2*3^(1/2)*(x-3^(1/2))+1)^(1/2)+1/2/(2+3^
(1/2))/(-2+3^(1/2))*arctanh(1/2*(2+2*3^(1/2)*(x+3^(1/2)))/(-(x+3^(1/2))^2+2*3^(1
/2)*(x+3^(1/2))+1)^(1/2))-1/2/(2+3^(1/2))/(-2+3^(1/2))*(-(x+3^(1/2))^2+2*3^(1/2)
*(x+3^(1/2))+1)^(1/2)

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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]

-log(sqrt(-x^2 + 4) + 1)

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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]

-1/2*log(x^2 - 3) + 1/2*log(-(x^2 + 3*sqrt(-x^2 + 4) - 6)/x^2) - 1/2*log(-(x^2 +
sqrt(-x^2 + 4) - 2)/x^2)

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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]

-Piecewise((log(sqrt(-x**2 + 4) + 1), (x > -2) & (x < 2)))

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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]

-ln(sqrt(-x^2 + 4) + 1)