3.118 \(\int \frac{\left (1-a^2 x^2\right )^{3/2}}{x^2 (1-a x)} \, dx\)

Optimal. Leaf size=51 \[ -\frac{\sqrt{1-a^2 x^2} (1-a x)}{x}-a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-a \sin ^{-1}(a x) \]

[Out]

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

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Rubi [A]  time = 0.241016, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{\sqrt{1-a^2 x^2} (1-a x)}{x}-a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-a \sin ^{-1}(a x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.4625, size = 41, normalized size = 0.8 \[ - a \operatorname{asin}{\left (a x \right )} - a \operatorname{atanh}{\left (\sqrt{- a^{2} x^{2} + 1} \right )} - \frac{\left (- a x + 1\right ) \sqrt{- a^{2} x^{2} + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0640654, size = 54, normalized size = 1.06 \[ \sqrt{1-a^2 x^2} \left (a-\frac{1}{x}\right )-a \log \left (\sqrt{1-a^2 x^2}+1\right )+a \log (x)-a \sin ^{-1}(a x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.03, size = 238, normalized size = 4.7 \[ -{\frac{a}{3} \left ( - \left ( x-{a}^{-1} \right ) ^{2}{a}^{2}-2\, \left ( x-{a}^{-1} \right ) a \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}x}{2}\sqrt{- \left ( x-{a}^{-1} \right ) ^{2}{a}^{2}-2\, \left ( x-{a}^{-1} \right ) a}}+{\frac{{a}^{2}}{2}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{- \left ( x-{a}^{-1} \right ) ^{2}{a}^{2}-2\, \left ( x-{a}^{-1} \right ) a}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{1}{x} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}-{a}^{2}x \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}-{\frac{3\,{a}^{2}x}{2}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{3\,{a}^{2}}{2}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{a}{3} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+a\sqrt{-{a}^{2}{x}^{2}+1}-a{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*a*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(3/2)+1/2*a^2*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(
1/2)*x+1/2*a^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/
2))-1/x*(-a^2*x^2+1)^(5/2)-a^2*x*(-a^2*x^2+1)^(3/2)-3/2*a^2*x*(-a^2*x^2+1)^(1/2)
-3/2*a^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+1/3*a*(-a^2*x^2+1)
^(3/2)+a*(-a^2*x^2+1)^(1/2)-a*arctanh(1/(-a^2*x^2+1)^(1/2))

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Maxima [A]  time = 0.792898, size = 111, normalized size = 2.18 \[ -\frac{a^{2} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}} - a \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \sqrt{-a^{2} x^{2} + 1} a - \frac{\sqrt{-a^{2} x^{2} + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.297304, size = 242, normalized size = 4.75 \[ -\frac{a^{3} x^{3} - 2 \, a^{2} x^{2} - 2 \,{\left (a^{3} x^{3} + 2 \, \sqrt{-a^{2} x^{2} + 1} a x - 2 \, a x\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (a^{3} x^{3} + 2 \, \sqrt{-a^{2} x^{2} + 1} a x - 2 \, a x\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) -{\left (a^{3} x^{3} - a^{2} x^{2} + 2\right )} \sqrt{-a^{2} x^{2} + 1} + 2}{a^{2} x^{3} + 2 \, \sqrt{-a^{2} x^{2} + 1} x - 2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a^3*x^3 - 2*a^2*x^2 - 2*(a^3*x^3 + 2*sqrt(-a^2*x^2 + 1)*a*x - 2*a*x)*arctan((s
qrt(-a^2*x^2 + 1) - 1)/(a*x)) - (a^3*x^3 + 2*sqrt(-a^2*x^2 + 1)*a*x - 2*a*x)*log
((sqrt(-a^2*x^2 + 1) - 1)/x) - (a^3*x^3 - a^2*x^2 + 2)*sqrt(-a^2*x^2 + 1) + 2)/(
a^2*x^3 + 2*sqrt(-a^2*x^2 + 1)*x - 2*x)

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Sympy [A]  time = 12.0071, size = 170, normalized size = 3.33 \[ a \left (\begin{cases} i \sqrt{a^{2} x^{2} - 1} - \log{\left (a x \right )} + \frac{\log{\left (a^{2} x^{2} \right )}}{2} + i \operatorname{asin}{\left (\frac{1}{a x} \right )} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\sqrt{- a^{2} x^{2} + 1} + \frac{\log{\left (a^{2} x^{2} \right )}}{2} - \log{\left (\sqrt{- a^{2} x^{2} + 1} + 1 \right )} & \text{otherwise} \end{cases}\right ) + \begin{cases} - \frac{i a^{2} x}{\sqrt{a^{2} x^{2} - 1}} + i a \operatorname{acosh}{\left (a x \right )} + \frac{i}{x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{a^{2} x}{\sqrt{- a^{2} x^{2} + 1}} - a \operatorname{asin}{\left (a x \right )} - \frac{1}{x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*Piecewise((I*sqrt(a**2*x**2 - 1) - log(a*x) + log(a**2*x**2)/2 + I*asin(1/(a*x
)), Abs(a**2*x**2) > 1), (sqrt(-a**2*x**2 + 1) + log(a**2*x**2)/2 - log(sqrt(-a*
*2*x**2 + 1) + 1), True)) + Piecewise((-I*a**2*x/sqrt(a**2*x**2 - 1) + I*a*acosh
(a*x) + I/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (a**2*x/sqrt(-a**2*x**2
+ 1) - a*asin(a*x) - 1/(x*sqrt(-a**2*x**2 + 1)), True))

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GIAC/XCAS [A]  time = 0.289853, size = 169, normalized size = 3.31 \[ \frac{a^{4} x}{2 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}{\left | a \right |}} - \frac{a^{2} \arcsin \left (a x\right ){\rm sign}\left (a\right )}{{\left | a \right |}} - \frac{a^{2}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{{\left | a \right |}} + \sqrt{-a^{2} x^{2} + 1} a - \frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{2 \, x{\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a^4*x/((sqrt(-a^2*x^2 + 1)*abs(a) + a)*abs(a)) - a^2*arcsin(a*x)*sign(a)/abs
(a) - a^2*ln(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) +
sqrt(-a^2*x^2 + 1)*a - 1/2*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(x*abs(a))