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 = 0.270052, 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, 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*}

Mathematica [A]  time = 0.0174567, 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 = 0.002, size = 16, normalized size = 0.8 \begin{align*} -2\,{\frac{1}{\sqrt{-1+x}}}-2\,{\frac{1}{\sqrt{1+x}}} \end{align*}

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 = 0.933505, size = 20, normalized size = 1.05 \begin{align*} -\frac{2}{\sqrt{x + 1}} - \frac{2}{\sqrt{x - 1}} \end{align*}

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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Fricas [A]  time = 1.97151, size = 78, normalized size = 4.11 \begin{align*} -\frac{2 \,{\left ({\left (x + 1\right )} \sqrt{x - 1} + \sqrt{x + 1}{\left (x - 1\right )}\right )}}{x^{2} - 1} \end{align*}

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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Sympy [B]  time = 6.48919, size = 56, normalized size = 2.95 \begin{align*} - \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} \end{align*}

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 [A]  time = 1.09706, size = 20, normalized size = 1.05 \begin{align*} -\frac{2}{\sqrt{x + 1}} - \frac{2}{\sqrt{x - 1}} \end{align*}

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)