3.32 \(\int \frac{\sqrt{1-c^2 x^2}}{\sqrt{1+c^2 x^2}} \, dx\)

Optimal. Leaf size=23 \[ \frac{2 F\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c}-\frac{E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c} \]

[Out]

-(EllipticE[ArcSin[c*x], -1]/c) + (2*EllipticF[ArcSin[c*x], -1])/c

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Rubi [A]  time = 0.0944334, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{2 F\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c}-\frac{E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - c^2*x^2]/Sqrt[1 + c^2*x^2],x]

[Out]

-(EllipticE[ArcSin[c*x], -1]/c) + (2*EllipticF[ArcSin[c*x], -1])/c

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Rubi in Sympy [A]  time = 21.1797, size = 20, normalized size = 0.87 \[ - \frac{E\left (\operatorname{asin}{\left (c x \right )}\middle | -1\right )}{c} + \frac{2 F\left (\operatorname{asin}{\left (c x \right )}\middle | -1\right )}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-elliptic_e(asin(c*x), -1)/c + 2*elliptic_f(asin(c*x), -1)/c

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Mathematica [A]  time = 0.0348893, size = 24, normalized size = 1.04 \[ \frac{E\left (\left .\sin ^{-1}\left (\sqrt{-c^2} x\right )\right |-1\right )}{\sqrt{-c^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - c^2*x^2]/Sqrt[1 + c^2*x^2],x]

[Out]

EllipticE[ArcSin[Sqrt[-c^2]*x], -1]/Sqrt[-c^2]

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Maple [C]  time = 0.016, size = 28, normalized size = 1.2 \[{\frac{ \left ( 2\,{\it EllipticF} \left ( x{\it csgn} \left ( c \right ) c,i \right ) -{\it EllipticE} \left ( x{\it csgn} \left ( c \right ) c,i \right ) \right ){\it csgn} \left ( c \right ) }{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*EllipticF(x*csgn(c)*c,I)-EllipticE(x*csgn(c)*c,I))*csgn(c)/c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-c^{2} x^{2} + 1}}{\sqrt{c^{2} x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-c^{2} x^{2} + 1}}{\sqrt{c^{2} x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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