3.695 \(\int \sin ^{-1}\left (\sqrt{\frac{-a+x}{a+x}}\right ) \, dx\)

Optimal. Leaf size=55 \[ (a+x) \sin ^{-1}\left (\sqrt{\frac{x-a}{a+x}}\right )-\frac{\sqrt{2} a \sqrt{\frac{x-a}{a+x}}}{\sqrt{\frac{a}{a+x}}} \]

[Out]

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

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Rubi [B]  time = 1.34713, antiderivative size = 118, normalized size of antiderivative = 2.15, number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ -\sqrt{2} \sqrt{\frac{a}{a+x}} \sqrt{-\frac{a-x}{a+x}} (a+x)+x \sin ^{-1}\left (\sqrt{-\frac{a-x}{a+x}}\right )-\frac{a \sqrt{\frac{a}{a+x}} \tanh ^{-1}\left (\frac{\sqrt{-\frac{a-x}{a+x}}}{\sqrt{2} \sqrt{-\frac{a}{a+x}}}\right )}{\sqrt{-\frac{a}{a+x}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 78.8106, size = 80, normalized size = 1.45 \[ - \sqrt{a} \sqrt{\frac{a}{a + x}} \sqrt{a + x} \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{a}}{\sqrt{a + x}} \right )} + x \operatorname{asin}{\left (\sqrt{\frac{- a + x}{a + x}} \right )} - \sqrt{2} \sqrt{\frac{a}{a + x}} \left (a + x\right ) \sqrt{- \frac{2 a}{a + x} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.222995, size = 99, normalized size = 1.8 \[ x \sin ^{-1}\left (\sqrt{\frac{x-a}{a+x}}\right )+\frac{\sqrt{\frac{a}{a+x}} \left (\sqrt{2} \sqrt{a} \sqrt{x-a} \tan ^{-1}\left (\frac{\sqrt{x-a}}{\sqrt{2} \sqrt{a}}\right )+2 a-2 x\right )}{\sqrt{2} \sqrt{\frac{x-a}{a+x}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.045, size = 85, normalized size = 1.6 \[ x\arcsin \left ( \sqrt{{\frac{-a+x}{a+x}}} \right ) +{\frac{\sqrt{2}}{2}\sqrt{-a+x}\sqrt{{\frac{a}{a+x}}} \left ( \sqrt{a}\sqrt{2}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-a+x}{\frac{1}{\sqrt{a}}}} \right ) -2\,\sqrt{-a+x} \right ){\frac{1}{\sqrt{{\frac{-a+x}{a+x}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arcsin(((-a+x)/(a+x))^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.52682, size = 139, normalized size = 2.53 \[ a{\left (\frac{2 \, \arcsin \left (\sqrt{-\frac{a - x}{a + x}}\right )}{\frac{a - x}{a + x} + 1} + \frac{\sqrt{\frac{a - x}{a + x} + 1}}{\sqrt{-\frac{a - x}{a + x}} + 1} + \frac{\sqrt{\frac{a - x}{a + x} + 1}}{\sqrt{-\frac{a - x}{a + x}} - 1}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.230716, size = 69, normalized size = 1.25 \[ -\sqrt{2}{\left (a + x\right )} \sqrt{-\frac{a - x}{a + x}} \sqrt{\frac{a}{a + x}} +{\left (a + x\right )} \arcsin \left (\sqrt{-\frac{a - x}{a + x}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*(a + x)*sqrt(-(a - x)/(a + x))*sqrt(a/(a + x)) + (a + x)*arcsin(sqrt(-(
a - x)/(a + x)))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \arcsin \left (\sqrt{-\frac{a - x}{a + x}}\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(arcsin(sqrt(-(a - x)/(a + x))), x)