3.156 \(\int \frac{1}{x^3 (1-a x) \sqrt{1-a^2 x^2}} \, dx\)

Optimal. Leaf size=90 \[ -\frac{2 a \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x^2 (1-a x)}-\frac{3 \sqrt{1-a^2 x^2}}{2 x^2}-\frac{3}{2} a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]

[Out]

(-3*Sqrt[1 - a^2*x^2])/(2*x^2) - (2*a*Sqrt[1 - a^2*x^2])/x + Sqrt[1 - a^2*x^2]/(
x^2*(1 - a*x)) - (3*a^2*ArcTanh[Sqrt[1 - a^2*x^2]])/2

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Rubi [A]  time = 0.274768, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{2 a \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x^2 (1-a x)}-\frac{3 \sqrt{1-a^2 x^2}}{2 x^2}-\frac{3}{2} a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(1 - a*x)*Sqrt[1 - a^2*x^2]),x]

[Out]

(-3*Sqrt[1 - a^2*x^2])/(2*x^2) - (2*a*Sqrt[1 - a^2*x^2])/x + Sqrt[1 - a^2*x^2]/(
x^2*(1 - a*x)) - (3*a^2*ArcTanh[Sqrt[1 - a^2*x^2]])/2

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Rubi in Sympy [A]  time = 22.9582, size = 75, normalized size = 0.83 \[ - \frac{3 a^{2} \operatorname{atanh}{\left (\sqrt{- a^{2} x^{2} + 1} \right )}}{2} - \frac{2 a \sqrt{- a^{2} x^{2} + 1}}{x} + \frac{a x + 1}{x^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{3 \sqrt{- a^{2} x^{2} + 1}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(-a*x+1)/(-a**2*x**2+1)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.142581, size = 74, normalized size = 0.82 \[ -\sqrt{1-a^2 x^2} \left (\frac{a^2}{a x-1}+\frac{a}{x}+\frac{1}{2 x^2}\right )-\frac{3}{2} a^2 \log \left (\sqrt{1-a^2 x^2}+1\right )+\frac{3}{2} a^2 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(1 - a*x)*Sqrt[1 - a^2*x^2]),x]

[Out]

-(Sqrt[1 - a^2*x^2]*(1/(2*x^2) + a/x + a^2/(-1 + a*x))) + (3*a^2*Log[x])/2 - (3*
a^2*Log[1 + Sqrt[1 - a^2*x^2]])/2

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Maple [A]  time = 0.017, size = 94, normalized size = 1. \[ -{a\sqrt{- \left ( x-{a}^{-1} \right ) ^{2}{a}^{2}-2\, \left ( x-{a}^{-1} \right ) a} \left ( x-{a}^{-1} \right ) ^{-1}}-{\frac{1}{2\,{x}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{3\,{a}^{2}}{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }-{\frac{a}{x}\sqrt{-{a}^{2}{x}^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(-a*x+1)/(-a^2*x^2+1)^(1/2),x)

[Out]

-a/(x-1/a)*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2)-1/2*(-a^2*x^2+1)^(1/2)/x^2-3/2*a^2
*arctanh(1/(-a^2*x^2+1)^(1/2))-a*(-a^2*x^2+1)^(1/2)/x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{1}{\sqrt{-a^{2} x^{2} + 1}{\left (a x - 1\right )} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(sqrt(-a^2*x^2 + 1)*(a*x - 1)*x^3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.287877, size = 317, normalized size = 3.52 \[ \frac{6 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 16 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 6 \, a x + 3 \,{\left (a^{5} x^{5} - 3 \, a^{4} x^{4} - 2 \, a^{3} x^{3} + 4 \, a^{2} x^{2} +{\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 4 \, a^{2} x^{2}\right )} \sqrt{-a^{2} x^{2} + 1}\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) -{\left (2 \, a^{4} x^{4} - 13 \, a^{3} x^{3} - 7 \, a^{2} x^{2} + 6 \, a x + 4\right )} \sqrt{-a^{2} x^{2} + 1} + 4}{2 \,{\left (a^{3} x^{5} - 3 \, a^{2} x^{4} - 2 \, a x^{3} + 4 \, x^{2} +{\left (a^{2} x^{4} + 2 \, a x^{3} - 4 \, x^{2}\right )} \sqrt{-a^{2} x^{2} + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(sqrt(-a^2*x^2 + 1)*(a*x - 1)*x^3),x, algorithm="fricas")

[Out]

1/2*(6*a^5*x^5 + 5*a^4*x^4 - 16*a^3*x^3 - 9*a^2*x^2 + 6*a*x + 3*(a^5*x^5 - 3*a^4
*x^4 - 2*a^3*x^3 + 4*a^2*x^2 + (a^4*x^4 + 2*a^3*x^3 - 4*a^2*x^2)*sqrt(-a^2*x^2 +
1))*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (2*a^4*x^4 - 13*a^3*x^3 - 7*a^2*x^2 + 6*a
*x + 4)*sqrt(-a^2*x^2 + 1) + 4)/(a^3*x^5 - 3*a^2*x^4 - 2*a*x^3 + 4*x^2 + (a^2*x^
4 + 2*a*x^3 - 4*x^2)*sqrt(-a^2*x^2 + 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{1}{a x^{4} \sqrt{- a^{2} x^{2} + 1} - x^{3} \sqrt{- a^{2} x^{2} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(-a*x+1)/(-a**2*x**2+1)**(1/2),x)

[Out]

-Integral(1/(a*x**4*sqrt(-a**2*x**2 + 1) - x**3*sqrt(-a**2*x**2 + 1)), x)

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GIAC/XCAS [A]  time = 0.287118, size = 288, normalized size = 3.2 \[ -\frac{{\left (a^{3} + \frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a}{x} - \frac{20 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a x^{2}}\right )} a^{4} x^{2}}{8 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}{\left | a \right |}} - \frac{3 \, a^{3}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{2 \,{\left | a \right |}} - \frac{\frac{4 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a{\left | a \right |}}{x} + \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}{\left | a \right |}}{a x^{2}}}{8 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(sqrt(-a^2*x^2 + 1)*(a*x - 1)*x^3),x, algorithm="giac")

[Out]

-1/8*(a^3 + 3*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a/x - 20*(sqrt(-a^2*x^2 + 1)*abs(a
) + a)^2/(a*x^2))*a^4*x^2/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*((sqrt(-a^2*x^2 + 1
)*abs(a) + a)/(a^2*x) - 1)*abs(a)) - 3/2*a^3*ln(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*ab
s(a) - 2*a)/(a^2*abs(x)))/abs(a) - 1/8*(4*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a*abs(
a)/x + (sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*abs(a)/(a*x^2))/a^2