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

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Rubi [A]  time = 0.27, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {6688} \[ -\frac {2}{\sqrt {x+1}}-\frac {2}{\sqrt {x-1}} \]

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]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {align*} \int \frac {(-1+x)^{3/2}+(1+x)^{3/2}}{(-1+x)^{3/2} (1+x)^{3/2}} \, dx &=\int \left (\frac {1}{(-1+x)^{3/2}}+\frac {1}{(1+x)^{3/2}}\right ) \, dx\\ &=-\frac {2}{\sqrt {-1+x}}-\frac {2}{\sqrt {1+x}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 19, normalized size = 1.00 \[ -\frac {2}{\sqrt {x+1}}-\frac {2}{\sqrt {x-1}} \]

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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fricas [A]  time = 0.99, size = 28, normalized size = 1.47 \[ -\frac {2 \, {\left ({\left (x + 1\right )} \sqrt {x - 1} + \sqrt {x + 1} {\left (x - 1\right )}\right )}}{x^{2} - 1} \]

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, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.15, size = 15, normalized size = 0.79 \[ -\frac {2}{\sqrt {x + 1}} - \frac {2}{\sqrt {x - 1}} \]

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, algorithm="giac")

[Out]

-2/sqrt(x + 1) - 2/sqrt(x - 1)

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maple [A]  time = 0.00, size = 16, normalized size = 0.84 \[ -\frac {2}{\sqrt {x -1}}-\frac {2}{\sqrt {x +1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.63, size = 15, normalized size = 0.79 \[ -\frac {2}{\sqrt {x + 1}} - \frac {2}{\sqrt {x - 1}} \]

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, algorithm="maxima")

[Out]

-2/sqrt(x + 1) - 2/sqrt(x - 1)

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mupad [B]  time = 0.40, size = 15, normalized size = 0.79 \[ -\frac {2}{\sqrt {x-1}}-\frac {2}{\sqrt {x+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- 2/(x - 1)^(1/2) - 2/(x + 1)^(1/2)

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sympy [B]  time = 6.63, 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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