Optimal. Leaf size=10 \[ \frac{E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c} \]
[Out]
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Rubi [A] time = 0.0342907, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 + c^2*x^2]/Sqrt[1 - c^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 9.07964, size = 8, normalized size = 0.8 \[ \frac{E\left (\operatorname{asin}{\left (c x \right )}\middle | -1\right )}{c} \]
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]
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Mathematica [A] time = 0.0366889, size = 10, normalized size = 1. \[ \frac{E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 + c^2*x^2]/Sqrt[1 - c^2*x^2],x]
[Out]
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Maple [C] time = 0.035, size = 15, normalized size = 1.5 \[{\frac{{\it EllipticE} \left ( x{\it csgn} \left ( c \right ) c,i \right ){\it csgn} \left ( c \right ) }{c}} \]
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]
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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]
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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]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c^{2} x^{2} + 1}}{\sqrt{- \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
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]
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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]