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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, 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 = {6822, 283, 222} \begin {gather*} \frac {\sqrt {1-x^2}}{2 x}-\frac {1}{2 x}+\frac {1}{2} \sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 222

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]

Rule 283

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 6822

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]

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 = 0.09, size = 41, normalized size = 1.28 \begin {gather*} \frac {-1+\sqrt {1-x^2}+2 x \tan ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\right )}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded in comparison} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

cought exception: maximum recursion depth exceeded in comparison

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(49\) vs. \(2(24)=48\).
time = 0.01, size = 50, normalized size = 1.56

method	result	size
default	\(-\frac {1}{2 x}-\frac {\left (-\arcsin \left (x \right ) x -\sqrt {-x^{2}+1}\right ) \sqrt {1+x}\, \sqrt {1-x}}{2 x \sqrt {-x^{2}+1}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1-x)^(1/2)+(1+x)^(1/2))^2,x,method=_RETURNVERBOSE)

[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)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 44, normalized size = 1.38 \begin {gather*} -\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 {gather*}

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (\sqrt {1 - x} + \sqrt {x + 1}\right )^{2}}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (24) = 48\).
time = 0.03, size = 208, normalized size = 6.50 \begin {gather*} -\frac {\pi }{2}-\frac {2 \left (\frac {2 \sqrt {-x+1}}{-2 \sqrt {x+1}+2 \sqrt {2}}-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )}{-\left (\frac {2 \sqrt {-x+1}}{-2 \sqrt {x+1}+2 \sqrt {2}}-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{2}+4}-\arctan \left (\frac {\sqrt {-x+1} \left (\left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{2}-1\right )}{-2 \sqrt {x+1}+2 \sqrt {2}}\right )-\frac {1}{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)))

________________________________________________________________________________________

Mupad [B]
time = 0.62, size = 49, normalized size = 1.53 \begin {gather*} \frac {\left (\frac {x}{2}+\frac {1}{2}\right )\,\sqrt {1-x}}{x\,\sqrt {x+1}}-\frac {1}{2\,x}-2\,\mathrm {atan}\left (\frac {\sqrt {1-x}-1}{\sqrt {x+1}-1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________