3.588 \(\int \sqrt{\frac{-a+x}{a+x}} \, dx\)

Optimal. Leaf size=41 \[ \sqrt{-\frac{a-x}{a+x}} (a+x)-2 a \tanh ^{-1}\left (\sqrt{-\frac{a-x}{a+x}}\right ) \]

[Out]

Sqrt[-((a - x)/(a + x))]*(a + x) - 2*a*ArcTanh[Sqrt[-((a - x)/(a + x))]]

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Rubi [A]  time = 0.0478819, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \sqrt{-\frac{a-x}{a+x}} (a+x)-2 a \tanh ^{-1}\left (\sqrt{-\frac{a-x}{a+x}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[(-a + x)/(a + x)],x]

[Out]

Sqrt[-((a - x)/(a + x))]*(a + x) - 2*a*ArcTanh[Sqrt[-((a - x)/(a + x))]]

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Rubi in Sympy [A]  time = 2.17006, size = 36, normalized size = 0.88 \[ \frac{2 a \sqrt{\frac{- a + x}{a + x}}}{- \frac{- a + x}{a + x} + 1} - 2 a \operatorname{atanh}{\left (\sqrt{\frac{- a + x}{a + x}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((-a+x)/(a+x))**(1/2),x)

[Out]

2*a*sqrt((-a + x)/(a + x))/(-(-a + x)/(a + x) + 1) - 2*a*atanh(sqrt((-a + x)/(a
+ x)))

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Mathematica [A]  time = 0.0487526, size = 69, normalized size = 1.68 \[ \frac{\sqrt{\frac{x-a}{a+x}} \left (\sqrt{x-a} (a+x)-a \sqrt{a+x} \log \left (\sqrt{x-a} \sqrt{a+x}+x\right )\right )}{\sqrt{x-a}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[(-a + x)/(a + x)],x]

[Out]

(Sqrt[(-a + x)/(a + x)]*(Sqrt[-a + x]*(a + x) - a*Sqrt[a + x]*Log[x + Sqrt[-a +
x]*Sqrt[a + x]]))/Sqrt[-a + x]

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Maple [A]  time = 0.017, size = 60, normalized size = 1.5 \[ -{(a+x)\sqrt{{\frac{-a+x}{a+x}}} \left ( a\ln \left ( x+\sqrt{-{a}^{2}+{x}^{2}} \right ) -\sqrt{-{a}^{2}+{x}^{2}} \right ){\frac{1}{\sqrt{ \left ( a+x \right ) \left ( -a+x \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((-a+x)/(a+x))^(1/2),x)

[Out]

-((-a+x)/(a+x))^(1/2)*(a+x)*(a*ln(x+(-a^2+x^2)^(1/2))-(-a^2+x^2)^(1/2))/((a+x)*(
-a+x))^(1/2)

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Maxima [A]  time = 0.71399, size = 95, normalized size = 2.32 \[ a{\left (\frac{2 \, \sqrt{-\frac{a - x}{a + x}}}{\frac{a - x}{a + x} + 1} - \log \left (\sqrt{-\frac{a - x}{a + x}} + 1\right ) + \log \left (\sqrt{-\frac{a - x}{a + x}} - 1\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-(a - x)/(a + x)),x, algorithm="maxima")

[Out]

a*(2*sqrt(-(a - x)/(a + x))/((a - x)/(a + x) + 1) - log(sqrt(-(a - x)/(a + x)) +
1) + log(sqrt(-(a - x)/(a + x)) - 1))

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Fricas [A]  time = 0.275215, size = 78, normalized size = 1.9 \[ -a \log \left (\sqrt{-\frac{a - x}{a + x}} + 1\right ) + a \log \left (\sqrt{-\frac{a - x}{a + x}} - 1\right ) +{\left (a + x\right )} \sqrt{-\frac{a - x}{a + x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-(a - x)/(a + x)),x, algorithm="fricas")

[Out]

-a*log(sqrt(-(a - x)/(a + x)) + 1) + a*log(sqrt(-(a - x)/(a + x)) - 1) + (a + x)
*sqrt(-(a - x)/(a + x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{- a + x}{a + x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((-a+x)/(a+x))**(1/2),x)

[Out]

Integral(sqrt((-a + x)/(a + x)), x)

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GIAC/XCAS [A]  time = 0.270772, size = 54, normalized size = 1.32 \[ a{\rm ln}\left ({\left | -x + \sqrt{-a^{2} + x^{2}} \right |}\right ){\rm sign}\left (a + x\right ) + \sqrt{-a^{2} + x^{2}}{\rm sign}\left (a + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-(a - x)/(a + x)),x, algorithm="giac")

[Out]

a*ln(abs(-x + sqrt(-a^2 + x^2)))*sign(a + x) + sqrt(-a^2 + x^2)*sign(a + x)