Optimal. Leaf size=64 \[ -\frac{2 \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x (1-a x)}-a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.199951, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{2 \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x (1-a x)}-a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(1 - a*x)*Sqrt[1 - a^2*x^2]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 15.8018, size = 48, normalized size = 0.75 \[ - a \operatorname{atanh}{\left (\sqrt{- a^{2} x^{2} + 1} \right )} + \frac{a x + 1}{x \sqrt{- a^{2} x^{2} + 1}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(-a*x+1)/(-a**2*x**2+1)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0770596, size = 54, normalized size = 0.84 \[ -\sqrt{1-a^2 x^2} \left (\frac{a}{a x-1}+\frac{1}{x}\right )-a \log \left (\sqrt{1-a^2 x^2}+1\right )+a \log (x) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(1 - a*x)*Sqrt[1 - a^2*x^2]),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.016, size = 73, normalized size = 1.1 \[ -{1\sqrt{- \left ( x-{a}^{-1} \right ) ^{2}{a}^{2}-2\, \left ( x-{a}^{-1} \right ) a} \left ( x-{a}^{-1} \right ) ^{-1}}-{\frac{1}{x}\sqrt{-{a}^{2}{x}^{2}+1}}-a{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(-a*x+1)/(-a^2*x^2+1)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{1}{\sqrt{-a^{2} x^{2} + 1}{\left (a x - 1\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(sqrt(-a^2*x^2 + 1)*(a*x - 1)*x^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.284937, size = 211, normalized size = 3.3 \[ -\frac{a^{3} x^{3} - 4 \, a^{2} x^{2} - a x -{\left (a^{3} x^{3} + a^{2} x^{2} - 2 \, a x -{\left (a^{2} x^{2} - 2 \, a x\right )} \sqrt{-a^{2} x^{2} + 1}\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) +{\left (3 \, a^{2} x^{2} + a x - 2\right )} \sqrt{-a^{2} x^{2} + 1} + 2}{a^{2} x^{3} + a x^{2} - \sqrt{-a^{2} x^{2} + 1}{\left (a x^{2} - 2 \, x\right )} - 2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(sqrt(-a^2*x^2 + 1)*(a*x - 1)*x^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{1}{a x^{3} \sqrt{- a^{2} x^{2} + 1} - x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(-a*x+1)/(-a**2*x**2+1)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.288667, size = 203, normalized size = 3.17 \[ -\frac{a^{2}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{{\left | a \right |}} - \frac{{\left (a^{2} - \frac{5 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{x}\right )} a^{2} x}{2 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{2 \, x{\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(sqrt(-a^2*x^2 + 1)*(a*x - 1)*x^2),x, algorithm="giac")
[Out]