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

Optimal. Leaf size=36 \[ \tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right )-\frac{\sqrt{x-1} \sqrt{x+1}}{x} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0438559, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right )-\frac{\sqrt{x-1} \sqrt{x+1}}{x} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.1202, size = 29, normalized size = 0.81 \[ \operatorname{atan}{\left (\sqrt{x - 1} \sqrt{x + 1} \right )} - \frac{\sqrt{x - 1} \sqrt{x + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0458833, size = 62, normalized size = 1.72 \[ -\frac{\sqrt{\frac{x-1}{x+1}} \left (\sqrt{x-1} (x+1)+x \sqrt{x+1} \tan ^{-1}\left (\frac{1}{\sqrt{x-1} \sqrt{x+1}}\right )\right )}{\sqrt{x-1} x} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.022, size = 43, normalized size = 1.2 \[{\frac{1}{x} \left ( -\arctan \left ({\frac{1}{\sqrt{{x}^{2}-1}}} \right ) x-\sqrt{{x}^{2}-1} \right ) \sqrt{-1+x}\sqrt{1+x}{\frac{1}{\sqrt{{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-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)

_______________________________________________________________________________________

Maxima [A]  time = 0.79807, size = 27, normalized size = 0.75 \[ -\frac{\sqrt{x^{2} - 1}}{x} - \arcsin \left (\frac{1}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(x^2 - 1)/x - arcsin(1/abs(x))

_______________________________________________________________________________________

Fricas [A]  time = 0.295791, size = 80, normalized size = 2.22 \[ \frac{2 \,{\left (\sqrt{x + 1} \sqrt{x - 1} x - x^{2}\right )} \arctan \left (\sqrt{x + 1} \sqrt{x - 1} - x\right ) + 1}{\sqrt{x + 1} \sqrt{x - 1} x - x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.271331, size = 57, normalized size = 1.58 \[ -\frac{8}{{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} + 4} - 2 \, \arctan \left (\frac{1}{2} \,{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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