Optimal. Leaf size=19 \[ 2 \sqrt{\sqrt{a^2+x^2}+x} \]
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Rubi [A] time = 0.0651093, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2122, 30} \[ 2 \sqrt{\sqrt{a^2+x^2}+x} \]
Antiderivative was successfully verified.
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Rule 2122
Rule 30
Rubi steps
\begin{align*} \int \frac{\sqrt{x+\sqrt{a^2+x^2}}}{\sqrt{a^2+x^2}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{x}} \, dx,x,x+\sqrt{a^2+x^2}\right )\\ &=2 \sqrt{x+\sqrt{a^2+x^2}}\\ \end{align*}
Mathematica [A] time = 0.0100317, size = 19, normalized size = 1. \[ 2 \sqrt{\sqrt{a^2+x^2}+x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.027, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{x+\sqrt{{a}^{2}+{x}^{2}}}{\frac{1}{\sqrt{{a}^{2}+{x}^{2}}}}}\, dx \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 \frac{\sqrt{x + \sqrt{a^{2} + x^{2}}}}{\sqrt{a^{2} + x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02462, size = 39, normalized size = 2.05 \begin{align*} 2 \, \sqrt{x + \sqrt{a^{2} + x^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.206048, size = 15, normalized size = 0.79 \begin{align*} 2 \sqrt{x + \sqrt{a^{2} + x^{2}}} \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 \frac{\sqrt{x + \sqrt{a^{2} + x^{2}}}}{\sqrt{a^{2} + x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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