Optimal. Leaf size=49 \[ \frac {2 x}{\sqrt {\sqrt {x^2+1}+1}}-\sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {\sqrt {x^2+1}+1}}\right ) \]
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Rubi [F] time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{\sqrt {1+\sqrt {1+x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+\sqrt {1+x^2}}} \, dx &=\int \frac {1}{\sqrt {1+\sqrt {1+x^2}}} \, dx\\ \end {align*}
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Mathematica [C] time = 0.22, size = 106, normalized size = 2.16 \begin {gather*} \frac {\sqrt {\sqrt {x^2+1}+1} \left (-2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {1}{2}-\frac {\sqrt {x^2+1}}{2}\right )+4 \sqrt {x^2+1}-\sqrt {2} \sqrt {\sqrt {x^2+1}-1} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}-1}}{\sqrt {2}}\right )-2\right )}{2 x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 49, normalized size = 1.00 \begin {gather*} \frac {2 x}{\sqrt {\sqrt {x^2+1}+1}}-\sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {\sqrt {x^2+1}+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.87, size = 51, normalized size = 1.04 \begin {gather*} \frac {\sqrt {2} x \arctan \left (\frac {\sqrt {2} \sqrt {\sqrt {x^{2} + 1} + 1}}{x}\right ) + 2 \, \sqrt {\sqrt {x^{2} + 1} + 1} {\left (\sqrt {x^{2} + 1} - 1\right )}}{x} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {\sqrt {x^{2} + 1} + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 20, normalized size = 0.41 \begin {gather*} \frac {\sqrt {2}\, x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}, \frac {3}{4}\right ], \left [\frac {3}{2}, \frac {3}{2}\right ], -x^{2}\right )}{2} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {\sqrt {x^{2} + 1} + 1}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {\sqrt {x^2+1}+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.73, size = 32, normalized size = 0.65 \begin {gather*} \frac {x \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2}, \frac {3}{4} \\ \frac {3}{2}, \frac {3}{2} \end {matrix}\middle | {x^{2} e^{i \pi }} \right )}}{2 \pi } \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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