Optimal. Leaf size=22 \[ -\sqrt{1-x^2} x-2 x-\sin ^{-1}(x) \]
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Rubi [A] time = 0.0925122, 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 \[ -\sqrt{1-x^2} x-2 x-\sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[(-Sqrt[1 - x] - Sqrt[1 + x])*(Sqrt[1 - x] + Sqrt[1 + x]),x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - 2 \int ^{\sqrt{x + 1}} x \left (x + \sqrt{- x^{2} + 2}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-(1-x)**(1/2)-(1+x)**(1/2))*((1-x)**(1/2)+(1+x)**(1/2)),x)
[Out]
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Mathematica [A] time = 0.0226164, size = 34, normalized size = 1.55 \[ -x \left (\sqrt{1-x^2}+2\right )-2 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )-2 \]
Antiderivative was successfully verified.
[In] Integrate[(-Sqrt[1 - x] - Sqrt[1 + x])*(Sqrt[1 - x] + Sqrt[1 + x]),x]
[Out]
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Maple [B] time = 0.002, size = 59, normalized size = 2.7 \[ -2\,x-\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}+\sqrt{1-x}\sqrt{1+x}-{\arcsin \left ( x \right ) \sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]
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]
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Maxima [A] time = 0.768891, size = 27, normalized size = 1.23 \[ -\sqrt{-x^{2} + 1} x - 2 \, x - \arcsin \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(sqrt(x + 1) + sqrt(-x + 1))^2,x, algorithm="maxima")
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Fricas [A] time = 0.276811, size = 119, normalized size = 5.41 \[ -\frac{{\left (x^{3} + 2 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 2 \,{\left (x^{2} + 2 \, \sqrt{x + 1} \sqrt{-x + 1} - 2\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 2 \, x}{x^{2} + 2 \, \sqrt{x + 1} \sqrt{-x + 1} - 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(sqrt(x + 1) + sqrt(-x + 1))^2,x, algorithm="fricas")
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Sympy [A] time = 39.2566, size = 46, normalized size = 2.09 \[ - 2 x - 4 \left (\begin{cases} \frac{x \sqrt{- x + 1} \sqrt{x + 1}}{4} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) - 2 \]
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]
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GIAC/XCAS [A] time = 0.292131, size = 45, normalized size = 2.05 \[ -\sqrt{x + 1} x \sqrt{-x + 1} - 2 \, x - 2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) - 2 \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(sqrt(x + 1) + sqrt(-x + 1))^2,x, algorithm="giac")
[Out]