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

Optimal. Leaf size=32 \[ \frac{\sqrt{1-x^2}}{2 x}-\frac{1}{2 x}+\frac{1}{2} \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.156235, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {6690, 277, 216} \[ \frac{\sqrt{1-x^2}}{2 x}-\frac{1}{2 x}+\frac{1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6690

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

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{1}{\left (\sqrt{1-x}+\sqrt{1+x}\right )^2} \, dx &=\frac{1}{4} \int \left (\frac{2}{x^2}-\frac{2 \sqrt{1-x^2}}{x^2}\right ) \, dx\\ &=-\frac{1}{2 x}-\frac{1}{2} \int \frac{\sqrt{1-x^2}}{x^2} \, dx\\ &=-\frac{1}{2 x}+\frac{\sqrt{1-x^2}}{2 x}+\frac{1}{2} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\frac{1}{2 x}+\frac{\sqrt{1-x^2}}{2 x}+\frac{1}{2} \sin ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0171405, size = 24, normalized size = 0.75 \[ \frac{\sqrt{1-x^2}+x \sin ^{-1}(x)-1}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.009, size = 50, normalized size = 1.6 \begin{align*} -{\frac{1}{2\,x}}-{\frac{1}{2\,x} \left ( -\arcsin \left ( x \right ) x-\sqrt{-{x}^{2}+1} \right ) \sqrt{1-x}\sqrt{1+x}{\frac{1}{\sqrt{-{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((sqrt(x + 1) + sqrt(-x + 1))^(-2), x)

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Fricas [A]  time = 1.11214, size = 116, normalized size = 3.62 \begin{align*} -\frac{2 \, x \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - \sqrt{x + 1} \sqrt{-x + 1} + 1}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(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{1}{\left (\sqrt{1 - x} + \sqrt{x + 1}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.14937, size = 198, normalized size = 6.19 \begin{align*} \frac{1}{2} \, \pi + \frac{2 \,{\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{1}{2 \, x} + \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/((1-x)^(1/2)+(1+x)^(1/2))^2,x, algorithm="giac")

[Out]

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