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]

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

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Rubi [A]  time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.10, 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]])

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fricas [A]  time = 0.46, size = 34, normalized size = 0.76 \[ \frac {2 \, {\left (x^{2} - \sqrt {-x^{2} + 1} + 1\right )} \sqrt {\sqrt {-x^{2} + 1} + 1}}{3 \, x} \]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sqrt {-x^{2} + 1} + 1}\,{d x} \]

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)

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maple [C]  time = 0.04, size = 60, normalized size = 1.33 \[ \frac {i \left (\frac {32 i \sqrt {\pi }\, \sqrt {2}\, x^{3} \cos \left (\frac {3 \arcsin \relax (x )}{2}\right )}{3}-\frac {8 i \sqrt {\pi }\, \sqrt {2}\, \left (-\frac {4}{3} x^{4}+\frac {2}{3} x^{2}+\frac {2}{3}\right ) \sin \left (\frac {3 \arcsin \relax (x )}{2}\right )}{\sqrt {-x^{2}+1}}\right )}{8 \sqrt {\pi }} \]

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))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sqrt {-x^{2} + 1} + 1}\,{d x} \]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {\sqrt {1-x^2}+1} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - x^2)^(1/2) + 1)^(1/2),x)

[Out]

int(((1 - x^2)^(1/2) + 1)^(1/2), x)

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sympy [B]  time = 1.28, size = 418, normalized size = 9.29 \[ \begin {cases} - \frac {\sqrt {2} x^{3} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{- 12 i \pi \sqrt {x^{2} - 1} \sqrt {i \sqrt {x^{2} - 1} + 1} - 12 \pi \sqrt {i \sqrt {x^{2} - 1} + 1}} + \frac {3 \sqrt {2} i x \sqrt {x^{2} - 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{- 12 i \pi \sqrt {x^{2} - 1} \sqrt {i \sqrt {x^{2} - 1} + 1} - 12 \pi \sqrt {i \sqrt {x^{2} - 1} + 1}} + \frac {3 \sqrt {2} x \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{- 12 i \pi \sqrt {x^{2} - 1} \sqrt {i \sqrt {x^{2} - 1} + 1} - 12 \pi \sqrt {i \sqrt {x^{2} - 1} + 1}} & \text {for}\: \left |{x^{2}}\right | > 1 \\\frac {\sqrt {2} x^{3} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi \sqrt {1 - x^{2}} \sqrt {\sqrt {1 - x^{2}} + 1} + 12 \pi \sqrt {\sqrt {1 - x^{2}} + 1}} - \frac {3 \sqrt {2} x \sqrt {1 - x^{2}} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi \sqrt {1 - x^{2}} \sqrt {\sqrt {1 - x^{2}} + 1} + 12 \pi \sqrt {\sqrt {1 - x^{2}} + 1}} - \frac {3 \sqrt {2} x \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi \sqrt {1 - x^{2}} \sqrt {\sqrt {1 - x^{2}} + 1} + 12 \pi \sqrt {\sqrt {1 - x^{2}} + 1}} & \text {otherwise} \end {cases} \]

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*sq
rt(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*sqrt(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*sqrt(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(sq
rt(1 - x**2) + 1)), True))

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