Integrand size = 11, antiderivative size = 24 \[ \int \frac {\sqrt {4+x}}{x} \, dx=2 \sqrt {4+x}-4 \text {arctanh}\left (\frac {\sqrt {4+x}}{2}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {52, 65, 213} \[ \int \frac {\sqrt {4+x}}{x} \, dx=2 \sqrt {x+4}-4 \text {arctanh}\left (\frac {\sqrt {x+4}}{2}\right ) \]
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Rule 52
Rule 65
Rule 213
Rubi steps \begin{align*} \text {integral}& = 2 \sqrt {4+x}+4 \int \frac {1}{x \sqrt {4+x}} \, dx \\ & = 2 \sqrt {4+x}+8 \text {Subst}\left (\int \frac {1}{-4+x^2} \, dx,x,\sqrt {4+x}\right ) \\ & = 2 \sqrt {4+x}-4 \text {arctanh}\left (\frac {\sqrt {4+x}}{2}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {4+x}}{x} \, dx=2 \sqrt {4+x}-4 \text {arctanh}\left (\frac {\sqrt {4+x}}{2}\right ) \]
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Time = 0.14 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17
method | result | size |
trager | \(2 \sqrt {4+x}+2 \ln \left (\frac {-8-x +4 \sqrt {4+x}}{x}\right )\) | \(28\) |
derivativedivides | \(2 \sqrt {4+x}-2 \ln \left (\sqrt {4+x}+2\right )+2 \ln \left (\sqrt {4+x}-2\right )\) | \(29\) |
default | \(2 \sqrt {4+x}-2 \ln \left (\sqrt {4+x}+2\right )+2 \ln \left (\sqrt {4+x}-2\right )\) | \(29\) |
meijerg | \(-\frac {-2 \left (2-4 \ln \left (2\right )+\ln \left (x \right )\right ) \sqrt {\pi }+4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {1+\frac {x}{4}}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+\frac {x}{4}}}{2}\right )}{\sqrt {\pi }}\) | \(54\) |
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Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {4+x}}{x} \, dx=2 \, \sqrt {x + 4} - 2 \, \log \left (\sqrt {x + 4} + 2\right ) + 2 \, \log \left (\sqrt {x + 4} - 2\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (19) = 38\).
Time = 0.51 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {\sqrt {4+x}}{x} \, dx=\begin {cases} 2 \sqrt {x + 4} - 4 \operatorname {acoth}{\left (\frac {\sqrt {x + 4}}{2} \right )} & \text {for}\: \left |{x + 4}\right | > 4 \\2 \sqrt {x + 4} - 4 \operatorname {atanh}{\left (\frac {\sqrt {x + 4}}{2} \right )} & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {4+x}}{x} \, dx=2 \, \sqrt {x + 4} - 2 \, \log \left (\sqrt {x + 4} + 2\right ) + 2 \, \log \left (\sqrt {x + 4} - 2\right ) \]
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Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt {4+x}}{x} \, dx=2 \, \sqrt {x + 4} - 2 \, \log \left (\sqrt {x + 4} + 2\right ) + 2 \, \log \left ({\left | \sqrt {x + 4} - 2 \right |}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {4+x}}{x} \, dx=2\,\sqrt {x+4}-4\,\mathrm {atanh}\left (\frac {\sqrt {x+4}}{2}\right ) \]
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