Optimal. Leaf size=41 \[ \frac{2 x^3}{3 \left (\sqrt{x^2+1}+1\right )^{3/2}}+\frac{2 x}{\sqrt{\sqrt{x^2+1}+1}} \]
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Rubi [A] time = 0.0073905, antiderivative size = 41, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0572284, 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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Maple [C] time = 0.017, size = 55, normalized size = 1.3 \begin{align*} -{\frac{1}{8\,\sqrt{\pi }} \left ( -{\frac{32\,\sqrt{\pi }\sqrt{2}{x}^{3}}{3}\cosh \left ({\frac{3\,{\it Arcsinh} \left ( x \right ) }{2}} \right ) }-8\,{\frac{\sqrt{\pi }\sqrt{2} \left ( -4/3\,{x}^{4}-2/3\,{x}^{2}+2/3 \right ) \sinh \left ( 3/2\,{\it Arcsinh} \left ( x \right ) \right ) }{\sqrt{{x}^{2}+1}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sqrt{x^{2} + 1} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.08139, size = 77, normalized size = 1.88 \begin{align*} \frac{2 \,{\left (x^{2} + \sqrt{x^{2} + 1} - 1\right )} \sqrt{\sqrt{x^{2} + 1} + 1}}{3 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.08258, size = 197, normalized size = 4.8 \begin{align*} - \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sqrt{x^{2} + 1} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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