Optimal. Leaf size=41 \[ \frac {2 x}{\sqrt {\sqrt {x^2+1}+1}}+\frac {2 x^3}{3 \left (\sqrt {x^2+1}+1\right )^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2129} \[ \frac {2 x^3}{3 \left (\sqrt {x^2+1}+1\right )^{3/2}}+\frac {2 x}{\sqrt {\sqrt {x^2+1}+1}} \]
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 = 0.06, size = 44, normalized size = 1.07 \[ \frac {2 \left (\sqrt {x^2+1}-1\right ) \sqrt {\sqrt {x^2+1}+1} \left (\sqrt {x^2+1}+2\right )}{3 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 28, normalized size = 0.68 \[ \frac {2 \, {\left (x^{2} + \sqrt {x^{2} + 1} - 1\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{3 \, x} \]
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 = 0.03, size = 55, normalized size = 1.34 \[ -\frac {-\frac {32 \sqrt {\pi }\, \sqrt {2}\, x^{3} \cosh \left (\frac {3 \arcsinh \relax (x )}{2}\right )}{3}-\frac {8 \sqrt {\pi }\, \sqrt {2}\, \left (-\frac {4}{3} x^{4}-\frac {2}{3} x^{2}+\frac {2}{3}\right ) \sinh \left (\frac {3 \arcsinh \relax (x )}{2}\right )}{\sqrt {x^{2}+1}}}{8 \sqrt {\pi }} \]
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 \[ \int \sqrt {\sqrt {x^2+1}+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.18, size = 197, normalized size = 4.80 \[ - \frac {\sqrt {2} x^{3} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi \sqrt {x^{2} + 1} \sqrt {\sqrt {x^{2} + 1} + 1} + 12 \pi \sqrt {\sqrt {x^{2} + 1} + 1}} - \frac {3 \sqrt {2} x \sqrt {x^{2} + 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi \sqrt {x^{2} + 1} \sqrt {\sqrt {x^{2} + 1} + 1} + 12 \pi \sqrt {\sqrt {x^{2} + 1} + 1}} - \frac {3 \sqrt {2} x \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi \sqrt {x^{2} + 1} \sqrt {\sqrt {x^{2} + 1} + 1} + 12 \pi \sqrt {\sqrt {x^{2} + 1} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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