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

Optimal. Leaf size=102 \[ -\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{x (c-a c x)^{3/2}}-\frac{a c \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}+a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.374969, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ -\frac{c^2 \left (1-a^2 x^2\right )^{3/2}}{x (c-a c x)^{3/2}}-\frac{a c \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}+a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.9082, size = 87, normalized size = 0.85 \[ a \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{\sqrt{- a c x + c}} \right )} - \frac{a c \sqrt{- a^{2} x^{2} + 1}}{\sqrt{- a c x + c}} - \frac{c^{2} \left (- a^{2} x^{2} + 1\right )^{\frac{3}{2}}}{x \left (- a c x + c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.173624, size = 92, normalized size = 0.9 \[ \sqrt{1-a^2 x^2} \left (\frac{3 a}{a x-1}-\frac{1}{x}\right ) \sqrt{-c (a x-1)}-a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{1-a^2 x^2} \sqrt{-c (a x-1)}}{\sqrt{c} (a x-1)}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.04, size = 95, normalized size = 0.9 \[{\frac{1}{ \left ( ax-1 \right ) x} \left ( -{\it Artanh} \left ({1\sqrt{c \left ( ax+1 \right ) }{\frac{1}{\sqrt{c}}}} \right ) xac+2\,xa\sqrt{c \left ( ax+1 \right ) }\sqrt{c}+\sqrt{c \left ( ax+1 \right ) }\sqrt{c} \right ) \sqrt{-c \left ( ax-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}{\frac{1}{\sqrt{c \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.796273, size = 72, normalized size = 0.71 \[ -\frac{1}{2} \, a \sqrt{c}{\left (4 \, \sqrt{a x + 1} + \frac{2 \, \sqrt{a x + 1}}{a x} - \log \left (\sqrt{a x + 1} + 1\right ) + \log \left (\sqrt{a x + 1} - 1\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*a*sqrt(c)*(4*sqrt(a*x + 1) + 2*sqrt(a*x + 1)/(a*x) - log(sqrt(a*x + 1) + 1)
+ log(sqrt(a*x + 1) - 1))

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Fricas [A]  time = 0.294326, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, a^{3} c x^{3} + 2 \, a^{2} c x^{2} + \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} a \sqrt{c} x \log \left (-\frac{a^{2} c x^{2} + a c x - 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right ) - 4 \, a c x - 2 \, c}{2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} x}, \frac{2 \, a^{3} c x^{3} + a^{2} c x^{2} - \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} a \sqrt{-c} x \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{{\left (a^{2} x^{2} - 1\right )} \sqrt{-c}}\right ) - 2 \, a c x - c}{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(4*a^3*c*x^3 + 2*a^2*c*x^2 + sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*a*sqrt(c)*
x*log(-(a^2*c*x^2 + a*c*x - 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)
/(a*x^2 - x)) - 4*a*c*x - 2*c)/(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*x), (2*a^3*c
*x^3 + a^2*c*x^2 - sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*a*sqrt(-c)*x*arctan(sqrt(
-a^2*x^2 + 1)*sqrt(-a*c*x + c)/((a^2*x^2 - 1)*sqrt(-c))) - 2*a*c*x - c)/(sqrt(-a
^2*x^2 + 1)*sqrt(-a*c*x + c)*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.303064, size = 154, normalized size = 1.51 \[ -\frac{{\left (2 \,{\left (\frac{c \arctan \left (\frac{\sqrt{a c x + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} + 2 \, \sqrt{a c x + c} + \frac{\sqrt{a c x + c}}{a x}\right )} a^{2} c^{2} - \frac{\sqrt{2}{\left (\sqrt{2} a^{2} c^{3} \arctan \left (\frac{\sqrt{2} \sqrt{c}}{\sqrt{-c}}\right ) + 6 \, a^{2} \sqrt{-c} c^{\frac{5}{2}}\right )}}{\sqrt{-c}}\right )}{\left | c \right |}}{2 \, a c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*(c*arctan(sqrt(a*c*x + c)/sqrt(-c))/sqrt(-c) + 2*sqrt(a*c*x + c) + sqrt(
a*c*x + c)/(a*x))*a^2*c^2 - sqrt(2)*(sqrt(2)*a^2*c^3*arctan(sqrt(2)*sqrt(c)/sqrt
(-c)) + 6*a^2*sqrt(-c)*c^(5/2))/sqrt(-c))*abs(c)/(a*c^3)