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

Optimal. Leaf size=63 \[ \frac{1}{4} \sqrt{-x^2+\sqrt{x^2-1} x+1} \left (\sqrt{x^2-1}+3 x\right )+\frac{3 \sin ^{-1}\left (x-\sqrt{x^2-1}\right )}{4 \sqrt{2}} \]

[Out]

((3*x + Sqrt[-1 + x^2])*Sqrt[1 - x^2 + x*Sqrt[-1 + x^2]])/4 + (3*ArcSin[x - Sqrt[-1 + x^2]])/(4*Sqrt[2])

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Rubi [F]  time = 0.0266277, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \sqrt{1-x^2+x \sqrt{-1+x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[1 - x^2 + x*Sqrt[-1 + x^2]],x]

[Out]

Defer[Int][Sqrt[1 - x^2 + x*Sqrt[-1 + x^2]], x]

Rubi steps

\begin{align*} \int \sqrt{1-x^2+x \sqrt{-1+x^2}} \, dx &=\int \sqrt{1-x^2+x \sqrt{-1+x^2}} \, dx\\ \end{align*}

Mathematica [F]  time = 0.0257292, size = 0, normalized size = 0. \[ \int \sqrt{1-x^2+x \sqrt{-1+x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[1 - x^2 + x*Sqrt[-1 + x^2]],x]

[Out]

Integrate[Sqrt[1 - x^2 + x*Sqrt[-1 + x^2]], x]

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Maple [F]  time = 0.009, size = 0, normalized size = 0. \begin{align*} \int \sqrt{1-{x}^{2}+x\sqrt{{x}^{2}-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-x^{2} + \sqrt{x^{2} - 1} x + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-x^2 + sqrt(x^2 - 1)*x + 1), x)

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Fricas [A]  time = 5.35895, size = 190, normalized size = 3.02 \begin{align*} \frac{1}{4} \, \sqrt{-x^{2} + \sqrt{x^{2} - 1} x + 1}{\left (3 \, x + \sqrt{x^{2} - 1}\right )} + \frac{3}{8} \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-x^{2} + \sqrt{x^{2} - 1} x + 1}}{2 \, \sqrt{x^{2} - 1}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(-x^2 + sqrt(x^2 - 1)*x + 1)*(3*x + sqrt(x^2 - 1)) + 3/8*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-x^2 + sqrt(x
^2 - 1)*x + 1)/sqrt(x^2 - 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- x^{2} + x \sqrt{x^{2} - 1} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-x**2 + x*sqrt(x**2 - 1) + 1), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-x^{2} + \sqrt{x^{2} - 1} x + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-x^2 + sqrt(x^2 - 1)*x + 1), x)