Optimal. Leaf size=79 \[ \frac {2 \sqrt {x^2+1} x}{3 \sqrt {\sqrt {x^2+1}+1}}-\frac {2 x}{3 \sqrt {\sqrt {x^2+1}+1}}+\sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {\sqrt {x^2+1}+1}}\right ) \]
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Rubi [F] time = 0.10, 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 {\sqrt {1+x^2}}{\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 {\sqrt {1+x^2}}{\sqrt {1+\sqrt {1+x^2}}} \, dx &=\int \frac {\sqrt {1+x^2}}{\sqrt {1+\sqrt {1+x^2}}} \, dx\\ \end {align*}
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Mathematica [C] time = 0.18, size = 111, normalized size = 1.41 \begin {gather*} \frac {\sqrt {\sqrt {x^2+1}+1} \left (6 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {1}{2}-\frac {\sqrt {x^2+1}}{2}\right )+4 x^2-8 \sqrt {x^2+1}+3 \sqrt {2} \sqrt {\sqrt {x^2+1}-1} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}-1}}{\sqrt {2}}\right )+2\right )}{6 x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 79, normalized size = 1.00 \begin {gather*} -\frac {2 x}{3 \sqrt {1+\sqrt {1+x^2}}}+\frac {2 x \sqrt {1+x^2}}{3 \sqrt {1+\sqrt {1+x^2}}}+\sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {1+\sqrt {1+x^2}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.14, size = 58, normalized size = 0.73 \begin {gather*} -\frac {3 \, \sqrt {2} x \arctan \left (\frac {\sqrt {2} \sqrt {\sqrt {x^{2} + 1} + 1}}{x}\right ) - 2 \, {\left (x^{2} - 2 \, \sqrt {x^{2} + 1} + 2\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{3 \, 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 {\sqrt {x^{2} + 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 [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}+1}}{\sqrt {1+\sqrt {x^{2}+1}}}\, dx\]
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 {\sqrt {x^{2} + 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 = -0.01 \begin {gather*} \int \frac {\sqrt {x^2+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 [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} + 1}}{\sqrt {\sqrt {x^{2} + 1} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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