Optimal. Leaf size=45 \[ \frac {2 x}{\sqrt {\sqrt {1-x^2}+1}}-\frac {2 x^3}{3 \left (\sqrt {1-x^2}+1\right )^{3/2}} \]
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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.
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Rule 2129
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.
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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.
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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.
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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.
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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.
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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.
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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.
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