Integrand size = 29, antiderivative size = 19 \[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a^2+x^2}} \, dx=2 \sqrt {x+\sqrt {a^2+x^2}} \]
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Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2147, 30} \[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a^2+x^2}} \, dx=2 \sqrt {\sqrt {a^2+x^2}+x} \]
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Rule 30
Rule 2147
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a^2+x^2}} \, dx=2 \sqrt {x+\sqrt {a^2+x^2}} \]
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Time = 0.44 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(2 \sqrt {x +\sqrt {a^{2}+x^{2}}}\) | \(16\) |
default | \(2 \sqrt {x +\sqrt {a^{2}+x^{2}}}\) | \(16\) |
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Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a^2+x^2}} \, dx=2 \, \sqrt {x + \sqrt {a^{2} + x^{2}}} \]
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Time = 0.09 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a^2+x^2}} \, dx=2 \sqrt {x + \sqrt {a^{2} + x^{2}}} \]
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\[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a^2+x^2}} \, dx=\int { \frac {\sqrt {x + \sqrt {a^{2} + x^{2}}}}{\sqrt {a^{2} + x^{2}}} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a^2+x^2}} \, dx=2 \, \sqrt {x + \sqrt {a^{2} + x^{2}}} \]
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Time = 0.48 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a^2+x^2}} \, dx=2\,\sqrt {x+\sqrt {a^2+x^2}} \]
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