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

Optimal. Leaf size=22 \[ x \left (-\sqrt{\frac{a}{x^2}}\right ) \tanh ^{-1}\left (\sqrt{x^2+1}\right ) \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0249561, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ x \left (-\sqrt{\frac{a}{x^2}}\right ) \tanh ^{-1}\left (\sqrt{x^2+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a/x^2]/Sqrt[1 + x^2],x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.99198, size = 20, normalized size = 0.91 \[ - x \sqrt{\frac{a}{x^{2}}} \operatorname{atanh}{\left (\sqrt{x^{2} + 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*sqrt(a/x**2)*atanh(sqrt(x**2 + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0122486, size = 28, normalized size = 1.27 \[ x \sqrt{\frac{a}{x^2}} \left (\log (x)-\log \left (\sqrt{x^2+1}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a/x^2]/Sqrt[1 + x^2],x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 19, normalized size = 0.9 \[ -\sqrt{{\frac{a}{{x}^{2}}}}x{\it Artanh} \left ({\frac{1}{\sqrt{{x}^{2}+1}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.783211, size = 14, normalized size = 0.64 \[ -\sqrt{a} \operatorname{arsinh}\left (\frac{1}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a)*arcsinh(1/abs(x))

_______________________________________________________________________________________

Fricas [A]  time = 0.280684, size = 1, normalized size = 0.05 \[ \left [x \sqrt{\frac{a}{x^{2}}} \log \left (\frac{x^{2} - \sqrt{x^{2} + 1}{\left (x + 1\right )} + x + 1}{x^{2} - \sqrt{x^{2} + 1} x}\right ), -2 \, \sqrt{-a} \arctan \left (-\frac{a x - \sqrt{x^{2} + 1} a}{\sqrt{-a} x \sqrt{\frac{a}{x^{2}}}}\right )\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.266763, size = 41, normalized size = 1.86 \[ -\frac{1}{2} \, \sqrt{a}{\left ({\rm ln}\left (\sqrt{x^{2} + 1} + 1\right ) -{\rm ln}\left (\sqrt{x^{2} + 1} - 1\right )\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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