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

Optimal. Leaf size=45 \[ \frac{2 x}{\sqrt{\sqrt{1-x^2}+1}}-\frac{2 x^3}{3 \left (\sqrt{1-x^2}+1\right )^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.009672, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2129} \[ \frac{2 x}{\sqrt{\sqrt{1-x^2}+1}}-\frac{2 x^3}{3 \left (\sqrt{1-x^2}+1\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + Sqrt[1 - x^2]],x]

[Out]

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

Rule 2129

Int[Sqrt[(a_) + (b_.)*Sqrt[(c_) + (d_.)*(x_)^2]], x_Symbol] :> Simp[(2*b^2*d*x^3)/(3*(a + b*Sqrt[c + d*x^2])^(
3/2)), x] + Simp[(2*a*x)/Sqrt[a + b*Sqrt[c + d*x^2]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2*c, 0]

Rubi steps

\begin{align*} \int \sqrt{1+\sqrt{1-x^2}} \, dx &=-\frac{2 x^3}{3 \left (1+\sqrt{1-x^2}\right )^{3/2}}+\frac{2 x}{\sqrt{1+\sqrt{1-x^2}}}\\ \end{align*}

Mathematica [A]  time = 0.0659507, size = 35, normalized size = 0.78 \[ \frac{2 x \left (\sqrt{1-x^2}+2\right )}{3 \sqrt{\sqrt{1-x^2}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Sqrt[1 - x^2]],x]

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.027, size = 60, normalized size = 1.3 \begin{align*}{\frac{{\frac{i}{8}}}{\sqrt{\pi }} \left ({\frac{32\,i}{3}}\sqrt{\pi }\sqrt{2}{x}^{3}\cos \left ({\frac{3\,\arcsin \left ( x \right ) }{2}} \right ) -{8\,i\sqrt{\pi }\sqrt{2} \left ( -{\frac{4\,{x}^{4}}{3}}+{\frac{2\,{x}^{2}}{3}}+{\frac{2}{3}} \right ) \sin \left ({\frac{3\,\arcsin \left ( x \right ) }{2}} \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*I/Pi^(1/2)*(32/3*I*Pi^(1/2)*2^(1/2)*x^3*cos(3/2*arcsin(x))-8*I*Pi^(1/2)*2^(1/2)*(-4/3*x^4+2/3*x^2+2/3)*sin
(3/2*arcsin(x))/(-x^2+1)^(1/2))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.07101, size = 80, normalized size = 1.78 \begin{align*} \frac{2 \,{\left (x^{2} - \sqrt{-x^{2} + 1} + 1\right )} \sqrt{\sqrt{-x^{2} + 1} + 1}}{3 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 1.13682, size = 413, normalized size = 9.18 \begin{align*} \begin{cases} \frac{\sqrt{2} x^{3} \Gamma \left (- \frac{1}{4}\right ) \Gamma \left (\frac{1}{4}\right )}{12 i \pi \sqrt{x^{2} - 1} \sqrt{i \sqrt{x^{2} - 1} + 1} + 12 \pi \sqrt{i \sqrt{x^{2} - 1} + 1}} - \frac{3 \sqrt{2} i x \sqrt{x^{2} - 1} \Gamma \left (- \frac{1}{4}\right ) \Gamma \left (\frac{1}{4}\right )}{12 i \pi \sqrt{x^{2} - 1} \sqrt{i \sqrt{x^{2} - 1} + 1} + 12 \pi \sqrt{i \sqrt{x^{2} - 1} + 1}} - \frac{3 \sqrt{2} x \Gamma \left (- \frac{1}{4}\right ) \Gamma \left (\frac{1}{4}\right )}{12 i \pi \sqrt{x^{2} - 1} \sqrt{i \sqrt{x^{2} - 1} + 1} + 12 \pi \sqrt{i \sqrt{x^{2} - 1} + 1}} & \text{for}\: \left |{x^{2}}\right | > 1 \\\frac{\sqrt{2} x^{3} \Gamma \left (- \frac{1}{4}\right ) \Gamma \left (\frac{1}{4}\right )}{12 \pi \sqrt{1 - x^{2}} \sqrt{\sqrt{1 - x^{2}} + 1} + 12 \pi \sqrt{\sqrt{1 - x^{2}} + 1}} - \frac{3 \sqrt{2} x \sqrt{1 - x^{2}} \Gamma \left (- \frac{1}{4}\right ) \Gamma \left (\frac{1}{4}\right )}{12 \pi \sqrt{1 - x^{2}} \sqrt{\sqrt{1 - x^{2}} + 1} + 12 \pi \sqrt{\sqrt{1 - x^{2}} + 1}} - \frac{3 \sqrt{2} x \Gamma \left (- \frac{1}{4}\right ) \Gamma \left (\frac{1}{4}\right )}{12 \pi \sqrt{1 - x^{2}} \sqrt{\sqrt{1 - x^{2}} + 1} + 12 \pi \sqrt{\sqrt{1 - x^{2}} + 1}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((sqrt(2)*x**3*gamma(-1/4)*gamma(1/4)/(12*I*pi*sqrt(x**2 - 1)*sqrt(I*sqrt(x**2 - 1) + 1) + 12*pi*sqrt
(I*sqrt(x**2 - 1) + 1)) - 3*sqrt(2)*I*x*sqrt(x**2 - 1)*gamma(-1/4)*gamma(1/4)/(12*I*pi*sqrt(x**2 - 1)*sqrt(I*s
qrt(x**2 - 1) + 1) + 12*pi*sqrt(I*sqrt(x**2 - 1) + 1)) - 3*sqrt(2)*x*gamma(-1/4)*gamma(1/4)/(12*I*pi*sqrt(x**2
- 1)*sqrt(I*sqrt(x**2 - 1) + 1) + 12*pi*sqrt(I*sqrt(x**2 - 1) + 1)), Abs(x**2) > 1), (sqrt(2)*x**3*gamma(-1/4
)*gamma(1/4)/(12*pi*sqrt(1 - x**2)*sqrt(sqrt(1 - x**2) + 1) + 12*pi*sqrt(sqrt(1 - x**2) + 1)) - 3*sqrt(2)*x*sq
rt(1 - x**2)*gamma(-1/4)*gamma(1/4)/(12*pi*sqrt(1 - x**2)*sqrt(sqrt(1 - x**2) + 1) + 12*pi*sqrt(sqrt(1 - x**2)
+ 1)) - 3*sqrt(2)*x*gamma(-1/4)*gamma(1/4)/(12*pi*sqrt(1 - x**2)*sqrt(sqrt(1 - x**2) + 1) + 12*pi*sqrt(sqrt(1
- x**2) + 1)), True))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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