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

Optimal. Leaf size=18 \[ \frac{\sqrt{x-1} \sqrt{x+1}}{x} \]

[Out]

(Sqrt[-1 + x]*Sqrt[1 + x])/x

_______________________________________________________________________________________

Rubi [A]  time = 0.0247513, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{\sqrt{x-1} \sqrt{x+1}}{x} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[-1 + x]*Sqrt[1 + x])/x

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.60379, size = 14, normalized size = 0.78 \[ \frac{\sqrt{x - 1} \sqrt{x + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x - 1)*sqrt(x + 1)/x

_______________________________________________________________________________________

Mathematica [A]  time = 0.0137036, size = 18, normalized size = 1. \[ \frac{\sqrt{x-1} \sqrt{x+1}}{x} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[-1 + x]*Sqrt[1 + x])/x

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 15, normalized size = 0.8 \[{\frac{1}{x}\sqrt{-1+x}\sqrt{1+x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.48839, size = 15, normalized size = 0.83 \[ \frac{\sqrt{x^{2} - 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x^2 - 1)/x

_______________________________________________________________________________________

Fricas [A]  time = 0.226151, size = 30, normalized size = 1.67 \[ -\frac{1}{\sqrt{x + 1} \sqrt{x - 1} x - x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(sqrt(x + 1)*sqrt(x - 1)*x - x^2)

_______________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.21815, size = 28, normalized size = 1.56 \[ \frac{8}{{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8/((sqrt(x + 1) - sqrt(x - 1))^4 + 4)