Integrand size = 9, antiderivative size = 2 \[ \int \frac {1}{\sqrt {1+x^2}} \, dx=\text {arcsinh}(x) \]
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Time = 0.00 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {221} \[ \int \frac {1}{\sqrt {1+x^2}} \, dx=\text {arcsinh}(x) \]
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Rule 221
Rubi steps \begin{align*} \text {integral}& = \text {arcsinh}(x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(16\) vs. \(2(2)=4\).
Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 8.00 \[ \int \frac {1}{\sqrt {1+x^2}} \, dx=-\log \left (-x+\sqrt {1+x^2}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.50
method | result | size |
default | \(\operatorname {arcsinh}\left (x \right )\) | \(3\) |
meijerg | \(\operatorname {arcsinh}\left (x \right )\) | \(3\) |
trager | \(\ln \left (x +\sqrt {x^{2}+1}\right )\) | \(11\) |
pseudoelliptic | \(\operatorname {arctanh}\left (\frac {\sqrt {x^{2}+1}}{x}\right )\) | \(13\) |
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Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (2) = 4\).
Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 7.00 \[ \int \frac {1}{\sqrt {1+x^2}} \, dx=-\log \left (-x + \sqrt {x^{2} + 1}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1+x^2}} \, dx=\operatorname {asinh}{\left (x \right )} \]
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none
Time = 0.29 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1+x^2}} \, dx=\operatorname {arsinh}\left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (2) = 4\).
Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 12.50 \[ \int \frac {1}{\sqrt {1+x^2}} \, dx=\frac {1}{2} \, \sqrt {x^{2} + 1} x - \frac {1}{2} \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \]
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Time = 0.00 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1+x^2}} \, dx=\mathrm {asinh}\left (x\right ) \]
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