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

Optimal. Leaf size=31 \[ -\frac{x^2}{2}+\frac{1}{2} \sqrt{1-x^2} x+x+\frac{1}{2} \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0461493, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6688, 195, 216} \[ -\frac{x^2}{2}+\frac{1}{2} \sqrt{1-x^2} x+x+\frac{1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x]*(Sqrt[1 - x] + Sqrt[1 + x]),x]

[Out]

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

Rule 6688

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

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

Mathematica [A]  time = 0.015452, size = 31, normalized size = 1. \[ -\frac{x^2}{2}+\frac{1}{2} \sqrt{1-x^2} x+x+\frac{1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.003, size = 63, normalized size = 2. \begin{align*} x-{\frac{{x}^{2}}{2}}-{\frac{1}{2}\sqrt{1+x} \left ( 1-x \right ) ^{{\frac{3}{2}}}}+{\frac{1}{2}\sqrt{1-x}\sqrt{1+x}}+{\frac{\arcsin \left ( x \right ) }{2}\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)+(1+x)^(1/2)),x)

[Out]

x-1/2*x^2-1/2*(1+x)^(1/2)*(1-x)^(3/2)+1/2*(1-x)^(1/2)*(1+x)^(1/2)+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 = 1.57816, size = 31, normalized size = 1. \begin{align*} -\frac{1}{2} \, x^{2} + \frac{1}{2} \, \sqrt{-x^{2} + 1} x + x + \frac{1}{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)+(1+x)^(1/2)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.29629, size = 122, normalized size = 3.94 \begin{align*} -\frac{1}{2} \, x^{2} + \frac{1}{2} \, \sqrt{x + 1} x \sqrt{-x + 1} + x - \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)+(1+x)^(1/2)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.49646, size = 48, normalized size = 1.55 \begin{align*} - \frac{\left (1 - x\right )^{2}}{2} - 2 \left (\begin{cases} - \frac{x \sqrt{1 - x} \sqrt{x + 1}}{4} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{1 - x}}{2} \right )}}{2} & \text{for}\: x \leq 1 \wedge x > -1 \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19635, size = 51, normalized size = 1.65 \begin{align*} -\frac{1}{2} \,{\left (x - 1\right )}^{2} + \frac{1}{2} \, \sqrt{x + 1} x \sqrt{-x + 1} - \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{-x + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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