Optimal. Leaf size=39 \[ \frac{\sqrt{\frac{d x^2}{c}+1} F\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{\sqrt{c+d x^2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.07319, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{\sqrt{\frac{d x^2}{c}+1} F\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{\sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[4 - x^2]*Sqrt[c + d*x^2]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 15.8944, size = 32, normalized size = 0.82 \[ \frac{\sqrt{1 + \frac{d x^{2}}{c}} F\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | - \frac{4 d}{c}\right )}{\sqrt{c + d x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-x**2+4)**(1/2)/(d*x**2+c)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0560501, size = 40, normalized size = 1.03 \[ \frac{\sqrt{\frac{c+d x^2}{c}} F\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{\sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[4 - x^2]*Sqrt[c + d*x^2]),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.032, size = 38, normalized size = 1. \[{1\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ({\frac{x}{2}},2\,\sqrt{-{\frac{d}{c}}} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-x^2+4)^(1/2)/(d*x^2+c)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{d x^{2} + c} \sqrt{-x^{2} + 4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(d*x^2 + c)*sqrt(-x^2 + 4)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{d x^{2} + c} \sqrt{-x^{2} + 4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(d*x^2 + c)*sqrt(-x^2 + 4)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- \left (x - 2\right ) \left (x + 2\right )} \sqrt{c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-x**2+4)**(1/2)/(d*x**2+c)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{d x^{2} + c} \sqrt{-x^{2} + 4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(d*x^2 + c)*sqrt(-x^2 + 4)),x, algorithm="giac")
[Out]