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

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 = 0.0225083, 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, Rules used = {6689} \[ \frac{1}{3} (x+1)^{3/2}-\frac{1}{3} (x-1)^{3/2} \]

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

Rule 6689

Int[(u_.)*((e_.)*Sqrt[(a_.) + (b_.)*(x_)^(n_.)] + (f_.)*Sqrt[(c_.) + (d_.)*(x_)^(n_.)])^(m_), x_Symbol] :> Dis
t[(a*e^2 - c*f^2)^m, Int[ExpandIntegrand[u/(e*Sqrt[a + b*x^n] - f*Sqrt[c + d*x^n])^m, x], x], x] /; FreeQ[{a,
b, c, d, e, f, n}, x] && ILtQ[m, 0] && EqQ[b*e^2 - d*f^2, 0]

Rubi steps

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

Mathematica [A]  time = 0.0202374, size = 23, normalized size = 1. \[ \frac{1}{3} (x+1)^{3/2}-\frac{1}{3} (x-1)^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-1 + x)^(3/2)/3 + (1 + x)^(3/2)/3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)^(1/2)+(1+x)^(1/2)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.22195, size = 53, normalized size = 2.3 \begin{align*} \frac{1}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} - \frac{1}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)^(1/2)+(1+x)^(1/2)),x, algorithm="fricas")

[Out]

1/3*(x + 1)^(3/2) - 1/3*(x - 1)^(3/2)

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Sympy [B]  time = 0.3948, size = 51, normalized size = 2.22 \begin{align*} \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}} \end{align*}

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 [A]  time = 1.1578, size = 20, normalized size = 0.87 \begin{align*} \frac{1}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} - \frac{1}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)^(1/2)+(1+x)^(1/2)),x, algorithm="giac")

[Out]

1/3*(x + 1)^(3/2) - 1/3*(x - 1)^(3/2)