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

Optimal. Leaf size=16 \[ \tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right ) \]

[Out]

ArcTan[Sqrt[-1 + x]*Sqrt[1 + x]]

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Rubi [A]  time = 0.0238541, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[Sqrt[-1 + x]*Sqrt[1 + x]]

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Rubi in Sympy [A]  time = 1.77364, size = 14, normalized size = 0.88 \[ \operatorname{atan}{\left (\sqrt{x - 1} \sqrt{x + 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(sqrt(x - 1)*sqrt(x + 1))

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Mathematica [A]  time = 0.0244243, size = 18, normalized size = 1.12 \[ 2 \tan ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{x+1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

2*ArcTan[Sqrt[-1 + x]/Sqrt[1 + x]]

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Maple [B]  time = 0.018, size = 28, normalized size = 1.8 \[ -{1\sqrt{-1+x}\sqrt{1+x}\arctan \left ({\frac{1}{\sqrt{{x}^{2}-1}}} \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.50362, size = 9, normalized size = 0.56 \[ -\arcsin \left (\frac{1}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-arcsin(1/abs(x))

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Fricas [A]  time = 0.220935, size = 24, normalized size = 1.5 \[ 2 \, \arctan \left (\sqrt{x + 1} \sqrt{x - 1} - x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*arctan(sqrt(x + 1)*sqrt(x - 1) - x)

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Sympy [A]  time = 13.1189, size = 56, normalized size = 3.5 \[ - \frac{{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{1}{x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-meijerg(((3/4, 5/4, 1), (1, 1, 3/2)), ((1/2, 3/4, 1, 5/4, 3/2), (0,)), x**(-2))
/(4*pi**(3/2)) + I*meijerg(((0, 1/4, 1/2, 3/4, 1, 1), ()), ((1/4, 3/4), (0, 1/2,
1/2, 0)), exp_polar(2*I*pi)/x**2)/(4*pi**(3/2))

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GIAC/XCAS [A]  time = 0.221403, size = 27, normalized size = 1.69 \[ -2 \, \arctan \left (\frac{1}{2} \,{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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