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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]