Integrand size = 12, antiderivative size = 24 \[ \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x^{3/2}} \, dx=-\frac {2 \coth ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}}-\log (1-x)+\log (x) \]
[Out]
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6038, 36, 31, 29} \[ \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x^{3/2}} \, dx=-\log (1-x)+\log (x)-\frac {2 \coth ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}} \]
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 6038
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \coth ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}}+\int \frac {1}{(1-x) x} \, dx \\ & = -\frac {2 \coth ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}}+\int \frac {1}{1-x} \, dx+\int \frac {1}{x} \, dx \\ & = -\frac {2 \coth ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}}-\log (1-x)+\log (x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x^{3/2}} \, dx=-\frac {2 \coth ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}}-\log (1-x)+\log (x) \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21
method | result | size |
derivativedivides | \(-\frac {2 \,\operatorname {arccoth}\left (\sqrt {x}\right )}{\sqrt {x}}-\ln \left (\sqrt {x}-1\right )+\ln \left (x \right )-\ln \left (\sqrt {x}+1\right )\) | \(29\) |
default | \(-\frac {2 \,\operatorname {arccoth}\left (\sqrt {x}\right )}{\sqrt {x}}-\ln \left (\sqrt {x}-1\right )+\ln \left (x \right )-\ln \left (\sqrt {x}+1\right )\) | \(29\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x^{3/2}} \, dx=-\frac {x \log \left (x - 1\right ) - x \log \left (x\right ) + \sqrt {x} \log \left (\frac {x + 2 \, \sqrt {x} + 1}{x - 1}\right )}{x} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (20) = 40\).
Time = 0.40 (sec) , antiderivative size = 126, normalized size of antiderivative = 5.25 \[ \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x^{3/2}} \, dx=- \frac {2 x^{\frac {3}{2}} \operatorname {acoth}{\left (\sqrt {x} \right )}}{x^{2} - x} + \frac {2 \sqrt {x} \operatorname {acoth}{\left (\sqrt {x} \right )}}{x^{2} - x} + \frac {x^{2} \log {\left (x \right )}}{x^{2} - x} - \frac {2 x^{2} \log {\left (\sqrt {x} + 1 \right )}}{x^{2} - x} + \frac {2 x^{2} \operatorname {acoth}{\left (\sqrt {x} \right )}}{x^{2} - x} - \frac {x \log {\left (x \right )}}{x^{2} - x} + \frac {2 x \log {\left (\sqrt {x} + 1 \right )}}{x^{2} - x} - \frac {2 x \operatorname {acoth}{\left (\sqrt {x} \right )}}{x^{2} - x} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75 \[ \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x^{3/2}} \, dx=-\frac {2 \, \operatorname {arcoth}\left (\sqrt {x}\right )}{\sqrt {x}} - \log \left (x - 1\right ) + \log \left (x\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.92 \[ \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x^{3/2}} \, dx=\frac {2 \, \log \left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1}\right )}{\frac {\sqrt {x} + 1}{\sqrt {x} - 1} + 1} - 2 \, \log \left (\frac {\sqrt {x} + 1}{{\left | \sqrt {x} - 1 \right |}}\right ) + 2 \, \log \left ({\left | \frac {\sqrt {x} + 1}{\sqrt {x} - 1} + 1 \right |}\right ) \]
[In]
[Out]
Time = 3.94 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x^{3/2}} \, dx=2\,\ln \left (\sqrt {x}\right )-\ln \left (x-1\right )-\frac {2\,\mathrm {acoth}\left (\sqrt {x}\right )}{\sqrt {x}} \]
[In]
[Out]