3.155 \(\int \frac{1}{x^2 (1-a x) \sqrt{1-a^2 x^2}} \, dx\)

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]

(-2*Sqrt[1 - a^2*x^2])/x + Sqrt[1 - a^2*x^2]/(x*(1 - a*x)) - a*ArcTanh[Sqrt[1 -
a^2*x^2]]

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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]

(-2*Sqrt[1 - a^2*x^2])/x + Sqrt[1 - a^2*x^2]/(x*(1 - a*x)) - a*ArcTanh[Sqrt[1 -
a^2*x^2]]

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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]

-a*atanh(sqrt(-a**2*x**2 + 1)) + (a*x + 1)/(x*sqrt(-a**2*x**2 + 1)) - 2*sqrt(-a*
*2*x**2 + 1)/x

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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]

-(Sqrt[1 - a^2*x^2]*(x^(-1) + a/(-1 + a*x))) + a*Log[x] - a*Log[1 + Sqrt[1 - a^2
*x^2]]

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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]

-1/(x-1/a)*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2)-(-a^2*x^2+1)^(1/2)/x-a*arctanh(1/(
-a^2*x^2+1)^(1/2))

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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]

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

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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]

-(a^3*x^3 - 4*a^2*x^2 - a*x - (a^3*x^3 + a^2*x^2 - 2*a*x - (a^2*x^2 - 2*a*x)*sqr
t(-a^2*x^2 + 1))*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (3*a^2*x^2 + a*x - 2)*sqrt(-a
^2*x^2 + 1) + 2)/(a^2*x^3 + a*x^2 - sqrt(-a^2*x^2 + 1)*(a*x^2 - 2*x) - 2*x)

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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]

-Integral(1/(a*x**3*sqrt(-a**2*x**2 + 1) - x**2*sqrt(-a**2*x**2 + 1)), x)

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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]

-a^2*ln(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) - 1/2*(
a^2 - 5*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/x)*a^2*x/((sqrt(-a^2*x^2 + 1)*abs(a) + a
)*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)*abs(a)) - 1/2*(sqrt(-a^2*x^2 + 1
)*abs(a) + a)/(x*abs(a))