3.97 \(\int \frac{\sqrt{1-x^2}}{\left (-1+x^2\right ) \sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=36 \[ -\frac{\sqrt{\frac{b x^2}{a}+1} F\left (\sin ^{-1}(x)|-\frac{b}{a}\right )}{\sqrt{a+b x^2}} \]

[Out]

-((Sqrt[1 + (b*x^2)/a]*EllipticF[ArcSin[x], -(b/a)])/Sqrt[a + b*x^2])

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Rubi [A]  time = 0.0888049, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{\sqrt{\frac{b x^2}{a}+1} F\left (\sin ^{-1}(x)|-\frac{b}{a}\right )}{\sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - x^2]/((-1 + x^2)*Sqrt[a + b*x^2]),x]

[Out]

-((Sqrt[1 + (b*x^2)/a]*EllipticF[ArcSin[x], -(b/a)])/Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 19.6536, size = 31, normalized size = 0.86 \[ - \frac{\sqrt{1 + \frac{b x^{2}}{a}} F\left (\operatorname{asin}{\left (x \right )}\middle | - \frac{b}{a}\right )}{\sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0699204, size = 37, normalized size = 1.03 \[ -\frac{\sqrt{\frac{a+b x^2}{a}} F\left (\sin ^{-1}(x)|-\frac{b}{a}\right )}{\sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - x^2]/((-1 + x^2)*Sqrt[a + b*x^2]),x]

[Out]

-((Sqrt[(a + b*x^2)/a]*EllipticF[ArcSin[x], -(b/a)])/Sqrt[a + b*x^2])

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Maple [A]  time = 0.042, size = 35, normalized size = 1. \[ -{1\sqrt{{\frac{b{x}^{2}+a}{a}}}{\it EllipticF} \left ( x,\sqrt{-{\frac{b}{a}}} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(b*x^2+a)^(1/2)*((b*x^2+a)/a)^(1/2)*EllipticF(x,(-b/a)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-x^{2} + 1}}{\sqrt{b x^{2} + a}{\left (x^{2} - 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + 1)/(sqrt(b*x^2 + a)*(x^2 - 1)),x, algorithm="maxima")

[Out]

integrate(sqrt(-x^2 + 1)/(sqrt(b*x^2 + a)*(x^2 - 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{-x^{2} + 1}}{\sqrt{b x^{2} + a}{\left (x^{2} - 1\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + 1)/(sqrt(b*x^2 + a)*(x^2 - 1)),x, algorithm="fricas")

[Out]

integral(sqrt(-x^2 + 1)/(sqrt(b*x^2 + a)*(x^2 - 1)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}{\sqrt{a + b x^{2}} \left (x - 1\right ) \left (x + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(-(x - 1)*(x + 1))/(sqrt(a + b*x**2)*(x - 1)*(x + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-x^{2} + 1}}{\sqrt{b x^{2} + a}{\left (x^{2} - 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + 1)/(sqrt(b*x^2 + a)*(x^2 - 1)),x, algorithm="giac")

[Out]

integrate(sqrt(-x^2 + 1)/(sqrt(b*x^2 + a)*(x^2 - 1)), x)