Optimal. Leaf size=19 \[ 2 \sqrt {x+\sqrt {a^2+x^2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 = {2147, 30}
\begin {gather*} 2 \sqrt {\sqrt {a^2+x^2}+x} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2147
Rubi steps
\begin {align*} \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a^2+x^2}} \, dx &=\text {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*}
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Mathematica [A]
time = 0.03, size = 19, normalized size = 1.00 \begin {gather*} 2 \sqrt {x+\sqrt {a^2+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.76, size = 15, normalized size = 0.79 \begin {gather*} 2 \sqrt {x+\sqrt {a^2+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x +\sqrt {a^{2}+x^{2}}}}{\sqrt {a^{2}+x^{2}}}\, dx\]
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 = 0.31, size = 15, normalized size = 0.79 \begin {gather*} 2 \, \sqrt {x + \sqrt {a^{2} + x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 15, normalized size = 0.79 \begin {gather*} 2 \sqrt {x + \sqrt {a^{2} + x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 19, normalized size = 1.00 \begin {gather*} 2 \sqrt {x+\sqrt {a^{2}+x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.42, size = 15, normalized size = 0.79 \begin {gather*} 2\,\sqrt {x+\sqrt {a^2+x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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