Integrand size = 10, antiderivative size = 25 \[ \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=-\frac {1}{\sqrt {x}}-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{x}+\text {arctanh}\left (\sqrt {x}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6038, 53, 65, 212} \[ \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=\text {arctanh}\left (\sqrt {x}\right )-\frac {1}{\sqrt {x}}-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{x} \]
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Rule 53
Rule 65
Rule 212
Rule 6038
Rubi steps \begin{align*} \text {integral}& = -\frac {\coth ^{-1}\left (\sqrt {x}\right )}{x}+\frac {1}{2} \int \frac {1}{(1-x) x^{3/2}} \, dx \\ & = -\frac {1}{\sqrt {x}}-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{x}+\frac {1}{2} \int \frac {1}{(1-x) \sqrt {x}} \, dx \\ & = -\frac {1}{\sqrt {x}}-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{x}+\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {1}{\sqrt {x}}-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{x}+\text {arctanh}\left (\sqrt {x}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.80 \[ \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=-\frac {1}{\sqrt {x}}-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{x}-\frac {1}{2} \log \left (1-\sqrt {x}\right )+\frac {1}{2} \log \left (1+\sqrt {x}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28
method | result | size |
derivativedivides | \(-\frac {\operatorname {arccoth}\left (\sqrt {x}\right )}{x}+\frac {\ln \left (\sqrt {x}+1\right )}{2}-\frac {1}{\sqrt {x}}-\frac {\ln \left (\sqrt {x}-1\right )}{2}\) | \(32\) |
default | \(-\frac {\operatorname {arccoth}\left (\sqrt {x}\right )}{x}+\frac {\ln \left (\sqrt {x}+1\right )}{2}-\frac {1}{\sqrt {x}}-\frac {\ln \left (\sqrt {x}-1\right )}{2}\) | \(32\) |
parts | \(-\frac {\operatorname {arccoth}\left (\sqrt {x}\right )}{x}+\frac {\ln \left (\sqrt {x}+1\right )}{2}-\frac {1}{\sqrt {x}}-\frac {\ln \left (\sqrt {x}-1\right )}{2}\) | \(32\) |
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Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=\frac {{\left (x - 1\right )} \log \left (\frac {x + 2 \, \sqrt {x} + 1}{x - 1}\right ) - 2 \, \sqrt {x}}{2 \, x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (20) = 40\).
Time = 0.55 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.68 \[ \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=\frac {x^{\frac {5}{2}} \operatorname {acoth}{\left (\sqrt {x} \right )}}{x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {2 x^{\frac {3}{2}} \operatorname {acoth}{\left (\sqrt {x} \right )}}{x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {\sqrt {x} \operatorname {acoth}{\left (\sqrt {x} \right )}}{x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {x^{2}}{x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {x}{x^{\frac {5}{2}} - x^{\frac {3}{2}}} \]
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Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=-\frac {\operatorname {arcoth}\left (\sqrt {x}\right )}{x} - \frac {1}{\sqrt {x}} + \frac {1}{2} \, \log \left (\sqrt {x} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {x} - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (19) = 38\).
Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.60 \[ \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=\frac {2}{\frac {\sqrt {x} + 1}{\sqrt {x} - 1} + 1} + \frac {2 \, {\left (\sqrt {x} + 1\right )} \log \left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1}\right )}{{\left (\sqrt {x} - 1\right )} {\left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1} + 1\right )}^{2}} \]
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Time = 4.68 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x^2} \, dx=\mathrm {atanh}\left (\sqrt {x}\right )-\frac {\mathrm {acoth}\left (\sqrt {x}\right )+\sqrt {x}}{x} \]
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