Integrand size = 15, antiderivative size = 21 \[ \int \frac {1}{x \sqrt {a^2+x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a^2+x^2}}{a}\right )}{a} \]
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Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {272, 65, 213} \[ \int \frac {1}{x \sqrt {a^2+x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a^2+x^2}}{a}\right )}{a} \]
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Rule 65
Rule 213
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {a^2+x}} \, dx,x,x^2\right ) \\ & = \text {Subst}\left (\int \frac {1}{-a^2+x^2} \, dx,x,\sqrt {a^2+x^2}\right ) \\ & = -\frac {\text {arctanh}\left (\frac {\sqrt {a^2+x^2}}{a}\right )}{a} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(49\) vs. \(2(21)=42\).
Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.33 \[ \int \frac {1}{x \sqrt {a^2+x^2}} \, dx=-\frac {\log \left (a+\sqrt {a^2+x^2}\right )}{2 a}+\frac {\log \left (-a^2+a \sqrt {a^2+x^2}\right )}{2 a} \]
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Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.67
method | result | size |
default | \(-\frac {\ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}+x^{2}}}{x}\right )}{\sqrt {a^{2}}}\) | \(35\) |
pseudoelliptic | \(\frac {\ln \left (-a +\sqrt {a^{2}+x^{2}}\right )-\ln \left (a +\sqrt {a^{2}+x^{2}}\right )}{2 a}\) | \(35\) |
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Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (19) = 38\).
Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.90 \[ \int \frac {1}{x \sqrt {a^2+x^2}} \, dx=-\frac {\log \left (a - x + \sqrt {a^{2} + x^{2}}\right ) - \log \left (-a - x + \sqrt {a^{2} + x^{2}}\right )}{a} \]
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Time = 0.51 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.33 \[ \int \frac {1}{x \sqrt {a^2+x^2}} \, dx=- \frac {\operatorname {asinh}{\left (\frac {a}{x} \right )}}{a} \]
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none
Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x \sqrt {a^2+x^2}} \, dx=-\frac {\operatorname {arsinh}\left (\frac {a}{{\left | x \right |}}\right )}{a} \]
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none
Time = 0.31 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \frac {1}{x \sqrt {a^2+x^2}} \, dx=-\frac {\log \left (a + \sqrt {a^{2} + x^{2}}\right )}{2 \, a} + \frac {\log \left (-a + \sqrt {a^{2} + x^{2}}\right )}{2 \, a} \]
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Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {1}{x \sqrt {a^2+x^2}} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {a^2+x^2}}{\sqrt {-a^2}}\right )}{\sqrt {-a^2}} \]
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