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

Optimal. Leaf size=41 \[ \frac{\sqrt{1-a^2 x^2}}{1-a x}-\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]

[Out]

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

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Rubi [A]  time = 0.131529, antiderivative size = 41, 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 \[ \frac{\sqrt{1-a^2 x^2}}{1-a x}-\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(1 - a*x)*Sqrt[1 - a^2*x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.0707, size = 29, normalized size = 0.71 \[ \frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}} - \operatorname{atanh}{\left (\sqrt{- a^{2} x^{2} + 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0620213, size = 45, normalized size = 1.1 \[ -\frac{\sqrt{1-a^2 x^2}}{a x-1}-\log \left (\sqrt{1-a^2 x^2}+1\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(1 - a*x)*Sqrt[1 - a^2*x^2]),x]

[Out]

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

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Maple [A]  time = 0.016, size = 58, normalized size = 1.4 \[ -{\frac{1}{a}\sqrt{- \left ( x-{a}^{-1} \right ) ^{2}{a}^{2}-2\, \left ( x-{a}^{-1} \right ) a} \left ( x-{a}^{-1} \right ) ^{-1}}-{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.89859, size = 78, normalized size = 1.9 \[ -a{\left (\frac{\log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right )}{a} + \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{2} x - a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.285353, size = 84, normalized size = 2.05 \[ \frac{2 \, a x +{\left (a x + \sqrt{-a^{2} x^{2} + 1} - 1\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right )}{a x + \sqrt{-a^{2} x^{2} + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.300939, size = 100, normalized size = 2.44 \[ -\frac{a{\rm ln}\left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{{\left | a \right |}} + \frac{2 \, a}{{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}{\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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