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

Optimal. Leaf size=23 \[ \frac{1}{3} (x+1)^{3/2}-\frac{1}{3} (x-1)^{3/2} \]

[Out]

-(-1 + x)^(3/2)/3 + (1 + x)^(3/2)/3

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Rubi [A]  time = 0.0397659, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{1}{3} (x+1)^{3/2}-\frac{1}{3} (x-1)^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

-(-1 + x)^(3/2)/3 + (1 + x)^(3/2)/3

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Rubi in Sympy [A]  time = 2.57501, size = 15, normalized size = 0.65 \[ - \frac{\left (x - 1\right )^{\frac{3}{2}}}{3} + \frac{\left (x + 1\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(x - 1)**(3/2)/3 + (x + 1)**(3/2)/3

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Mathematica [A]  time = 0.0218059, size = 31, normalized size = 1.35 \[ \left (\frac{x-1}{3}+\frac{2}{3}\right ) \sqrt{x+1}-\frac{1}{3} (x-1)^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

-(-1 + x)^(3/2)/3 + (2/3 + (-1 + x)/3)*Sqrt[1 + x]

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Maple [A]  time = 0.002, size = 16, normalized size = 0.7 \[ -{\frac{1}{3} \left ( -1+x \right ) ^{{\frac{3}{2}}}}+{\frac{1}{3} \left ( 1+x \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(-1+x)^(3/2)+1/3*(1+x)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x + 1} + \sqrt{x - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x + 1) + sqrt(x - 1)), x)

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Fricas [A]  time = 0.264395, size = 20, normalized size = 0.87 \[ \frac{1}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} - \frac{1}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(x + 1)^(3/2) - 1/3*(x - 1)^(3/2)

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Sympy [A]  time = 1.48457, size = 51, normalized size = 2.22 \[ \frac{4 x}{3 \sqrt{x - 1} + 3 \sqrt{x + 1}} + \frac{2 \sqrt{x - 1} \sqrt{x + 1}}{3 \sqrt{x - 1} + 3 \sqrt{x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*x/(3*sqrt(x - 1) + 3*sqrt(x + 1)) + 2*sqrt(x - 1)*sqrt(x + 1)/(3*sqrt(x - 1) +
3*sqrt(x + 1))

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GIAC/XCAS [A]  time = 0.279648, size = 20, normalized size = 0.87 \[ \frac{1}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} - \frac{1}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(x + 1)^(3/2) - 1/3*(x - 1)^(3/2)