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

Optimal. Leaf size=19 \[ \sqrt{1-x^2} x+2 x+\sin ^{-1}(x) \]

[Out]

2*x + x*Sqrt[1 - x^2] + ArcSin[x]

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Rubi [A]  time = 0.0249363, antiderivative size = 19, 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 = {6742, 195, 216} \[ \sqrt{1-x^2} x+2 x+\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[1 - x] + Sqrt[1 + x])^2,x]

[Out]

2*x + x*Sqrt[1 - x^2] + ArcSin[x]

Rule 6742

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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

Mathematica [A]  time = 0.0148542, size = 18, normalized size = 0.95 \[ x \left (\sqrt{1-x^2}+2\right )+\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x*(2 + Sqrt[1 - x^2]) + ArcSin[x]

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Maple [B]  time = 0.005, size = 58, normalized size = 3.1 \begin{align*} 2\,x-\sqrt{1+x} \left ( 1-x \right ) ^{{\frac{3}{2}}}+\sqrt{1-x}\sqrt{1+x}+{\arcsin \left ( x \right ) \sqrt{ \left ( 1-x \right ) \left ( 1+x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.90388, size = 23, normalized size = 1.21 \begin{align*} \sqrt{-x^{2} + 1} x + 2 \, x + \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, algorithm="maxima")

[Out]

sqrt(-x^2 + 1)*x + 2*x + arcsin(x)

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Fricas [B]  time = 1.22892, size = 107, normalized size = 5.63 \begin{align*} \sqrt{x + 1} x \sqrt{-x + 1} + 2 \, x - 2 \, \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{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, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 24.3767, size = 44, normalized size = 2.32 \begin{align*} 2 x + 4 \left (\begin{cases} \frac{x \sqrt{1 - x} \sqrt{x + 1}}{4} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) + 2 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x + 4*Piecewise((x*sqrt(1 - x)*sqrt(x + 1)/4 + asin(sqrt(2)*sqrt(x + 1)/2)/2, (x >= -1) & (x < 1))) + 2

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Giac [A]  time = 1.16092, size = 43, normalized size = 2.26 \begin{align*} \sqrt{x + 1} x \sqrt{-x + 1} + 2 \, x + 2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) + 2 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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