Optimal. Leaf size=18 \[ \frac {2 x}{\sqrt {1+\sqrt {1+x^2}}} \]
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Rubi [F]
time = 0.06, 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+\sqrt {1+x^2}}}{\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+\sqrt {1+x^2}}}{\sqrt {1+x^2}} \, dx &=\int \frac {\sqrt {1+\sqrt {1+x^2}}}{\sqrt {1+x^2}} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 18, normalized size = 1.00 \begin {gather*} \frac {2 x}{\sqrt {1+\sqrt {1+x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.17, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {1+\sqrt {x^{2}+1}}}{\sqrt {x^{2}+1}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 25, normalized size = 1.39 \begin {gather*} \frac {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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (15) = 30\).
time = 0.45, size = 31, normalized size = 1.72 \begin {gather*} \frac {\sqrt {2} x \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{\pi \sqrt {\sqrt {x^{2} + 1} + 1}} \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*} \text {could not integrate} \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 = -0.06 \begin {gather*} \int \frac {\sqrt {\sqrt {x^2+1}+1}}{\sqrt {x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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