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

Optimal. Leaf size=22 \[ -\sqrt{1-x^2} x-2 x-\sin ^{-1}(x) \]

[Out]

-2*x - x*Sqrt[1 - x^2] - ArcSin[x]

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Rubi [A]  time = 0.0545297, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {6688, 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])*(Sqrt[1 - x] + Sqrt[1 + x]),x]

[Out]

-2*x - x*Sqrt[1 - x^2] - ArcSin[x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, 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 ) \left (\sqrt{1-x}+\sqrt{1+x}\right ) \, dx &=-\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 = 0.0154404, size = 21, 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])*(Sqrt[1 - x] + Sqrt[1 + x]),x]

[Out]

-(x*(2 + Sqrt[1 - x^2])) - ArcSin[x]

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Maple [B]  time = 0.003, size = 59, normalized size = 2.7 \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))*((1-x)^(1/2)+(1+x)^(1/2)),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 = 1.53986, size = 27, normalized size = 1.23 \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))*((1-x)^(1/2)+(1+x)^(1/2)),x, algorithm="maxima")

[Out]

-sqrt(-x^2 + 1)*x - 2*x - arcsin(x)

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Fricas [B]  time = 1.01104, size = 108, normalized size = 4.91 \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))*((1-x)^(1/2)+(1+x)^(1/2)),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 = 30.6975, size = 46, normalized size = 2.09 \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))*((1-x)**(1/2)+(1+x)**(1/2)),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 = 1.14039, size = 45, normalized size = 2.05 \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))*((1-x)^(1/2)+(1+x)^(1/2)),x, algorithm="giac")

[Out]

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