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

Optimal. Leaf size=39 \[ 2 a \tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right )-\frac{\sqrt{x-1} \sqrt{x+1}}{x} \]

[Out]

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

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Rubi [A]  time = 0.115625, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.139 \[ 2 a \tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right )-\frac{\sqrt{x-1} \sqrt{x+1}}{x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.2261, size = 32, normalized size = 0.82 \[ 2 a \operatorname{atan}{\left (\sqrt{x - 1} \sqrt{x + 1} \right )} - \frac{\sqrt{x - 1} \sqrt{x + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0268408, size = 39, normalized size = 1. \[ -2 a \tan ^{-1}\left (\frac{1}{\sqrt{x-1} \sqrt{x+1}}\right )-\frac{\sqrt{x-1} \sqrt{x+1}}{x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 44, normalized size = 1.1 \[{\frac{1}{x} \left ( -2\,a\arctan \left ({\frac{1}{\sqrt{{x}^{2}-1}}} \right ) x-\sqrt{{x}^{2}-1} \right ) \sqrt{-1+x}\sqrt{1+x}{\frac{1}{\sqrt{{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.48854, size = 28, normalized size = 0.72 \[ -2 \, a \arcsin \left (\frac{1}{{\left | x \right |}}\right ) - \frac{\sqrt{x^{2} - 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.231873, size = 82, normalized size = 2.1 \[ \frac{4 \,{\left (a \sqrt{x + 1} \sqrt{x - 1} x - a x^{2}\right )} \arctan \left (\sqrt{x + 1} \sqrt{x - 1} - x\right ) + 1}{\sqrt{x + 1} \sqrt{x - 1} x - x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.216578, size = 58, normalized size = 1.49 \[ -4 \, a \arctan \left (\frac{1}{2} \,{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2}\right ) - \frac{8}{{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*a*arctan(1/2*(sqrt(x + 1) - sqrt(x - 1))^2) - 8/((sqrt(x + 1) - sqrt(x - 1))^
4 + 4)