3.424 \(\int \frac{(\sqrt{1-x}+\sqrt{1+x})^2}{x^2} \, dx\)

Optimal. Leaf size=26 \[ -\frac{2 \sqrt{1-x^2}}{x}-\frac{2}{x}-2 \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.080425, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {6742, 277, 216} \[ -\frac{2 \sqrt{1-x^2}}{x}-\frac{2}{x}-2 \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[1 - x] + Sqrt[1 + x])^2/x^2,x]

[Out]

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

Rule 6742

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

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

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

Mathematica [A]  time = 0.0267812, size = 22, normalized size = 0.85 \[ -\frac{2 \left (\sqrt{1-x^2}+x \sin ^{-1}(x)+1\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[1 - x] + Sqrt[1 + x])^2/x^2,x]

[Out]

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

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Maple [B]  time = 0.012, size = 50, normalized size = 1.9 \begin{align*} -2\,{x}^{-1}+2\,{\frac{ \left ( -\arcsin \left ( x \right ) x-\sqrt{-{x}^{2}+1} \right ) \sqrt{1+x}\sqrt{1-x}}{x\sqrt{-{x}^{2}+1}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.51806, size = 32, normalized size = 1.23 \begin{align*} -\frac{2 \, \sqrt{-x^{2} + 1}}{x} - \frac{2}{x} - 2 \, \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.12653, size = 112, normalized size = 4.31 \begin{align*} \frac{2 \,{\left (2 \, x \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - \sqrt{x + 1} \sqrt{-x + 1} - 1\right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\sqrt{1 - x} + \sqrt{x + 1}\right )^{2}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1-x)**(1/2)+(1+x)**(1/2))**2/x**2,x)

[Out]

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

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Giac [B]  time = 1.15333, size = 201, normalized size = 7.73 \begin{align*} -2 \, \pi - \frac{8 \,{\left (\frac{\sqrt{2} - \sqrt{-x + 1}}{\sqrt{x + 1}} - \frac{\sqrt{x + 1}}{\sqrt{2} - \sqrt{-x + 1}}\right )}}{{\left (\frac{\sqrt{2} - \sqrt{-x + 1}}{\sqrt{x + 1}} - \frac{\sqrt{x + 1}}{\sqrt{2} - \sqrt{-x + 1}}\right )}^{2} - 4} - \frac{2}{x} - 4 \, \arctan \left (\frac{\sqrt{x + 1}{\left (\frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{2 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*pi - 8*((sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)))/(((sqrt(2) - sqrt(-x +
1))/sqrt(x + 1) - sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)))^2 - 4) - 2/x - 4*arctan(1/2*sqrt(x + 1)*((sqrt(2) - s
qrt(-x + 1))^2/(x + 1) - 1)/(sqrt(2) - sqrt(-x + 1)))