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

Optimal. Leaf size=42 \[ 2 a \tan ^{-1}\left (\sqrt{\frac{a+x}{a-x}}\right )-(a-x) \sqrt{\frac{a+x}{a-x}} \]

[Out]

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

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Rubi [A]  time = 0.033987, antiderivative size = 42, 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 \[ 2 a \tan ^{-1}\left (\sqrt{\frac{a+x}{a-x}}\right )-(a-x) \sqrt{\frac{a+x}{a-x}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[(a + x)/(a - x)],x]

[Out]

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

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Rubi in Sympy [A]  time = 1.85796, size = 36, normalized size = 0.86 \[ - \frac{2 a \sqrt{\frac{a + x}{a - x}}}{1 + \frac{a + x}{a - x}} + 2 a \operatorname{atan}{\left (\sqrt{\frac{a + x}{a - x}} \right )} \]

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

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Mathematica [A]  time = 0.0823051, size = 67, normalized size = 1.6 \[ \frac{\sqrt{\frac{a+x}{a-x}} \left (\sqrt{a+x} (x-a)+a \sqrt{a-x} \tan ^{-1}\left (\frac{x}{\sqrt{a-x} \sqrt{a+x}}\right )\right )}{\sqrt{a+x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.027, size = 62, normalized size = 1.5 \[{(-a+x)\sqrt{-{\frac{a+x}{-a+x}}} \left ( \sqrt{{a}^{2}-{x}^{2}}-a\arctan \left ({x{\frac{1}{\sqrt{{a}^{2}-{x}^{2}}}}} \right ) \right ){\frac{1}{\sqrt{- \left ( a+x \right ) \left ( -a+x \right ) }}}} \]

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

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Maxima [A]  time = 1.50089, size = 66, normalized size = 1.57 \[ -2 \, a{\left (\frac{\sqrt{\frac{a + x}{a - x}}}{\frac{a + x}{a - x} + 1} - \arctan \left (\sqrt{\frac{a + x}{a - x}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.209414, size = 51, normalized size = 1.21 \[ 2 \, a \arctan \left (\sqrt{\frac{a + x}{a - x}}\right ) -{\left (a - x\right )} \sqrt{\frac{a + x}{a - x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*arctan(sqrt((a + x)/(a - x))) - (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 = 0.217732, size = 49, normalized size = 1.17 \[ a \arcsin \left (\frac{x}{a}\right ){\rm sign}\left (a - x\right ){\rm sign}\left (a\right ) - \sqrt{a^{2} - x^{2}}{\rm sign}\left (a - x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*arcsin(x/a)*sign(a - x)*sign(a) - sqrt(a^2 - x^2)*sign(a - x)