3.836 \(\int \frac{1}{(-1+x)^{3/2} x (1+x)^{3/2}} \, dx\)

Optimal. Leaf size=35 \[ -\frac{1}{\sqrt{x-1} \sqrt{x+1}}-\tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0481706, antiderivative size = 35, 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 \[ -\frac{1}{\sqrt{x-1} \sqrt{x+1}}-\tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/((-1 + x)^(3/2)*x*(1 + x)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.43918, size = 31, normalized size = 0.89 \[ - \operatorname{atan}{\left (\sqrt{x - 1} \sqrt{x + 1} \right )} - \frac{1}{\sqrt{x - 1} \sqrt{x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-1+x)**(3/2)/x/(1+x)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0587028, size = 33, normalized size = 0.94 \[ -\frac{1}{\sqrt{x-1} \sqrt{x+1}}-2 \tan ^{-1}\left (\sqrt{\frac{x-1}{x+1}}\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((-1 + x)^(3/2)*x*(1 + x)^(3/2)),x]

[Out]

-(1/(Sqrt[-1 + x]*Sqrt[1 + x])) - 2*ArcTan[Sqrt[(-1 + x)/(1 + x)]]

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Maple [A]  time = 0.024, size = 51, normalized size = 1.5 \[{1 \left ( \arctan \left ({\frac{1}{\sqrt{{x}^{2}-1}}} \right ){x}^{2}-\arctan \left ({\frac{1}{\sqrt{{x}^{2}-1}}} \right ) -\sqrt{{x}^{2}-1} \right ){\frac{1}{\sqrt{{x}^{2}-1}}}{\frac{1}{\sqrt{1+x}}}{\frac{1}{\sqrt{-1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-1+x)^(3/2)/x/(1+x)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.52762, size = 20, normalized size = 0.57 \[ -\frac{1}{\sqrt{x^{2} - 1}} + \arcsin \left (\frac{1}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 1)^(3/2)*(x - 1)^(3/2)*x),x, algorithm="maxima")

[Out]

-1/sqrt(x^2 - 1) + arcsin(1/abs(x))

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Fricas [A]  time = 0.237944, size = 100, normalized size = 2.86 \[ -\frac{2 \,{\left (\sqrt{x + 1} \sqrt{x - 1} x - x^{2} + 1\right )} \arctan \left (\sqrt{x + 1} \sqrt{x - 1} - x\right ) - \sqrt{x + 1} \sqrt{x - 1} + x}{\sqrt{x + 1} \sqrt{x - 1} x - x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 1)^(3/2)*(x - 1)^(3/2)*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 31.7236, size = 58, normalized size = 1.66 \[ - \frac{{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{5}{4}, \frac{7}{4}, 1 & 1, 2, \frac{5}{2} \\\frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2, \frac{5}{2} & 0 \end{matrix} \middle |{\frac{1}{x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}}} - \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, 1 & \\\frac{3}{4}, \frac{5}{4} & 0, \frac{1}{2}, \frac{3}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-1+x)**(3/2)/x/(1+x)**(3/2),x)

[Out]

-meijerg(((5/4, 7/4, 1), (1, 2, 5/2)), ((5/4, 3/2, 7/4, 2, 5/2), (0,)), x**(-2))
/(2*pi**(3/2)) - I*meijerg(((0, 1/2, 3/4, 1, 5/4, 1), ()), ((3/4, 5/4), (0, 1/2,
3/2, 0)), exp_polar(2*I*pi)/x**2)/(2*pi**(3/2))

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GIAC/XCAS [A]  time = 0.219911, size = 73, normalized size = 2.09 \[ -\frac{\sqrt{x + 1}}{2 \, \sqrt{x - 1}} + \frac{2}{{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2} + 2} + 2 \, \arctan \left (\frac{1}{2} \,{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 1)^(3/2)*(x - 1)^(3/2)*x),x, algorithm="giac")

[Out]

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