Integrand size = 17, antiderivative size = 45 \[ \int \sqrt {1+\sqrt {1-x^2}} \, dx=-\frac {2 x^3}{3 \left (1+\sqrt {1-x^2}\right )^{3/2}}+\frac {2 x}{\sqrt {1+\sqrt {1-x^2}}} \]
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Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2154} \[ \int \sqrt {1+\sqrt {1-x^2}} \, dx=\frac {2 x}{\sqrt {\sqrt {1-x^2}+1}}-\frac {2 x^3}{3 \left (\sqrt {1-x^2}+1\right )^{3/2}} \]
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Rule 2154
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.06 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78 \[ \int \sqrt {1+\sqrt {1-x^2}} \, dx=\frac {2 x \left (2+\sqrt {1-x^2}\right )}{3 \sqrt {1+\sqrt {1-x^2}}} \]
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Result contains higher order function than in optimal. Order 3 vs. order 2.
Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.33
method | result | size |
meijerg | \(\frac {i \left (\frac {32 i \sqrt {\pi }\, \sqrt {2}\, x^{3} \cos \left (\frac {3 \arcsin \left (x \right )}{2}\right )}{3}-\frac {8 i \sqrt {\pi }\, \sqrt {2}\, \left (-\frac {4}{3} x^{4}+\frac {2}{3} x^{2}+\frac {2}{3}\right ) \sin \left (\frac {3 \arcsin \left (x \right )}{2}\right )}{\sqrt {-x^{2}+1}}\right )}{8 \sqrt {\pi }}\) | \(60\) |
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none
Time = 0.34 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.76 \[ \int \sqrt {1+\sqrt {1-x^2}} \, dx=\frac {2 \, {\left (x^{2} - \sqrt {-x^{2} + 1} + 1\right )} \sqrt {\sqrt {-x^{2} + 1} + 1}}{3 \, x} \]
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Result contains complex when optimal does not.
Time = 0.71 (sec) , antiderivative size = 415, normalized size of antiderivative = 9.22 \[ \int \sqrt {1+\sqrt {1-x^2}} \, dx=\begin {cases} - \frac {\sqrt {2} i x^{3} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi \sqrt {x^{2} - 1} \sqrt {i \sqrt {x^{2} - 1} + 1} - 12 i \pi \sqrt {i \sqrt {x^{2} - 1} + 1}} - \frac {3 \sqrt {2} x \sqrt {x^{2} - 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi \sqrt {x^{2} - 1} \sqrt {i \sqrt {x^{2} - 1} + 1} - 12 i \pi \sqrt {i \sqrt {x^{2} - 1} + 1}} + \frac {3 \sqrt {2} i x \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi \sqrt {x^{2} - 1} \sqrt {i \sqrt {x^{2} - 1} + 1} - 12 i \pi \sqrt {i \sqrt {x^{2} - 1} + 1}} & \text {for}\: \left |{x^{2}}\right | > 1 \\\frac {\sqrt {2} x^{3} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi \sqrt {1 - x^{2}} \sqrt {\sqrt {1 - x^{2}} + 1} + 12 \pi \sqrt {\sqrt {1 - x^{2}} + 1}} - \frac {3 \sqrt {2} x \sqrt {1 - x^{2}} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi \sqrt {1 - x^{2}} \sqrt {\sqrt {1 - x^{2}} + 1} + 12 \pi \sqrt {\sqrt {1 - x^{2}} + 1}} - \frac {3 \sqrt {2} x \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi \sqrt {1 - x^{2}} \sqrt {\sqrt {1 - x^{2}} + 1} + 12 \pi \sqrt {\sqrt {1 - x^{2}} + 1}} & \text {otherwise} \end {cases} \]
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\[ \int \sqrt {1+\sqrt {1-x^2}} \, dx=\int { \sqrt {\sqrt {-x^{2} + 1} + 1} \,d x } \]
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\[ \int \sqrt {1+\sqrt {1-x^2}} \, dx=\int { \sqrt {\sqrt {-x^{2} + 1} + 1} \,d x } \]
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Timed out. \[ \int \sqrt {1+\sqrt {1-x^2}} \, dx=\int \sqrt {\sqrt {1-x^2}+1} \,d x \]
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