3.825 \(\int \frac{x}{\sqrt{-1+x} \sqrt{1+x}} \, dx\)

Optimal. Leaf size=15 \[ \sqrt{x-1} \sqrt{x+1} \]

[Out]

Sqrt[-1 + x]*Sqrt[1 + x]

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Rubi [A]  time = 0.0107018, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \sqrt{x-1} \sqrt{x+1} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[-1 + x]*Sqrt[1 + x]

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Rubi in Sympy [A]  time = 1.90879, size = 12, normalized size = 0.8 \[ \sqrt{x - 1} \sqrt{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x - 1)*sqrt(x + 1)

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Mathematica [A]  time = 0.0107335, size = 15, normalized size = 1. \[ \sqrt{x-1} \sqrt{x+1} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[-1 + x]*Sqrt[1 + x]

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Maple [A]  time = 0.004, size = 12, normalized size = 0.8 \[ \sqrt{-1+x}\sqrt{1+x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.33749, size = 9, normalized size = 0.6 \[ \sqrt{x^{2} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x^2 - 1)

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Fricas [A]  time = 0.237103, size = 51, normalized size = 3.4 \[ -\frac{\sqrt{x + 1} \sqrt{x - 1} x - x^{2} + 1}{\sqrt{x + 1} \sqrt{x - 1} - x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 7.50807, size = 76, normalized size = 5.07 \[ \frac{{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

meijerg(((-1/4, 1/4), (0, 0, 1/2, 1)), ((-1/2, -1/4, 0, 1/4, 1/2, 0), ()), x**(-
2))/(4*pi**(3/2)) + I*meijerg(((-1, -3/4, -1/2, -1/4, 0, 1), ()), ((-3/4, -1/4),
(-1, -1/2, -1/2, 0)), exp_polar(2*I*pi)/x**2)/(4*pi**(3/2))

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GIAC/XCAS [A]  time = 0.238545, size = 15, normalized size = 1. \[ \sqrt{x + 1} \sqrt{x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x + 1)*sqrt(x - 1)