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

Optimal. Leaf size=27 \[ -\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )}{a} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0634955, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.90686, size = 24, normalized size = 0.89 \[ - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{- a x + 1}}{2} \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0398769, size = 53, normalized size = 1.96 \[ \frac{\sqrt{a x+1} \sqrt{2 a x-2} \tan ^{-1}\left (\frac{\sqrt{a x-1}}{\sqrt{2}}\right )}{a \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.018, size = 50, normalized size = 1.9 \[ -{\frac{\sqrt{2}}{a}\sqrt{-{a}^{2}{x}^{2}+1}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{-ax+1}} \right ){\frac{1}{\sqrt{ax+1}}}{\frac{1}{\sqrt{-ax+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.801465, size = 55, normalized size = 2.04 \[ \frac{\sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - \sqrt{-a x + 1}\right )}}{2 \, \sqrt{2} + 2 \, \sqrt{-a x + 1}}\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.229958, size = 85, normalized size = 3.15 \[ \frac{\sqrt{2} \log \left (-\frac{a^{2} x^{2} - 2 \, a x + 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{a x + 1} - 3}{a^{2} x^{2} + 2 \, a x + 1}\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.21382, size = 58, normalized size = 2.15 \[ -\frac{\sqrt{2}{\rm ln}\left (\sqrt{2} + \sqrt{-a x + 1}\right ) - \sqrt{2}{\rm ln}\left (\sqrt{2} - \sqrt{-a x + 1}\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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