Optimal. Leaf size=41 \[ \frac {2 x^3}{3 \left (1+\sqrt {1+x^2}\right )^{3/2}}+\frac {2 x}{\sqrt {1+\sqrt {1+x^2}}} \]
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Rubi [A]
time = 0.00, 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 = {2154}
\begin {gather*} \frac {2 x}{\sqrt {\sqrt {x^2+1}+1}}+\frac {2 x^3}{3 \left (\sqrt {x^2+1}+1\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2154
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 = 31, normalized size = 0.76 \begin {gather*} \frac {2 x \left (2+\sqrt {1+x^2}\right )}{3 \sqrt {1+\sqrt {1+x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order
2.
time = 0.02, size = 55, normalized size = 1.34
method | result | size |
meijerg | \(-\frac {-\frac {32 \sqrt {\pi }\, \sqrt {2}\, x^{3} \cosh \left (\frac {3 \arcsinh \left (x \right )}{2}\right )}{3}-\frac {8 \sqrt {\pi }\, \sqrt {2}\, \left (-\frac {4}{3} x^{4}-\frac {2}{3} x^{2}+\frac {2}{3}\right ) \sinh \left (\frac {3 \arcsinh \left (x \right )}{2}\right )}{\sqrt {x^{2}+1}}}{8 \sqrt {\pi }}\) | \(55\) |
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 = 0.38, size = 28, normalized size = 0.68 \begin {gather*} \frac {2 \, {\left (x^{2} + \sqrt {x^{2} + 1} - 1\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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 197 vs.
\(2 (36) = 72\).
time = 0.58, size = 197, normalized size = 4.80 \begin {gather*} - \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}} \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 = -0.02 \begin {gather*} \int \sqrt {\sqrt {x^2+1}+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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