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

Optimal. Leaf size=19 \[ -\frac{2}{\sqrt{x+1}}-\frac{2}{\sqrt{x-1}} \]

[Out]

-2/Sqrt[-1 + x] - 2/Sqrt[1 + x]

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Rubi [A]  time = 0.434125, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033 \[ -\frac{2}{\sqrt{x+1}}-\frac{2}{\sqrt{x-1}} \]

Antiderivative was successfully verified.

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

[Out]

-2/Sqrt[-1 + x] - 2/Sqrt[1 + x]

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Rubi in Sympy [A]  time = 11.2729, size = 17, normalized size = 0.89 \[ - \frac{2}{\sqrt{x + 1}} - \frac{2}{\sqrt{x - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/sqrt(x + 1) - 2/sqrt(x - 1)

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Mathematica [A]  time = 0.0247948, size = 19, normalized size = 1. \[ -\frac{2}{\sqrt{x+1}}-\frac{2}{\sqrt{x-1}} \]

Antiderivative was successfully verified.

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

[Out]

-2/Sqrt[-1 + x] - 2/Sqrt[1 + x]

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Maple [A]  time = 0.004, size = 16, normalized size = 0.8 \[ -2\,{\frac{1}{\sqrt{-1+x}}}-2\,{\frac{1}{\sqrt{1+x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34085, size = 20, normalized size = 1.05 \[ -\frac{2}{\sqrt{x + 1}} - \frac{2}{\sqrt{x - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/sqrt(x + 1) - 2/sqrt(x - 1)

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Fricas [A]  time = 0.203075, size = 31, normalized size = 1.63 \[ -\frac{2 \,{\left (\sqrt{x + 1} + \sqrt{x - 1}\right )}}{\sqrt{x + 1} \sqrt{x - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 12.4909, size = 56, normalized size = 2.95 \[ - \frac{2 x \sqrt{x - 1}}{x^{2} - 1} - \frac{2 x \sqrt{x + 1}}{x^{2} - 1} - \frac{2 \sqrt{x - 1}}{x^{2} - 1} + \frac{2 \sqrt{x + 1}}{x^{2} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.213949, size = 20, normalized size = 1.05 \[ -\frac{2}{\sqrt{x + 1}} - \frac{2}{\sqrt{x - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/sqrt(x + 1) - 2/sqrt(x - 1)