Integrand size = 19, antiderivative size = 47 \[ \int \frac {1}{x \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {\arctan \left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\text {arctanh}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \]
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Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3557, 335, 218, 212, 209} \[ \int \frac {1}{x \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {\arctan \left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\text {arctanh}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \]
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Rule 209
Rule 212
Rule 218
Rule 335
Rule 3557
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\sqrt {\coth (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+x^2\right )} \, dx,x,\coth \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \\ & = -\frac {2 \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \\ & = \frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \\ & = \frac {\arctan \left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\text {arctanh}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {\arctan \left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\text {arctanh}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \]
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Time = 0.50 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {\operatorname {arctanh}\left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+\arctan \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )}{n b}\) | \(37\) |
default | \(\frac {\operatorname {arctanh}\left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+\arctan \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )}{n b}\) | \(37\) |
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Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (43) = 86\).
Time = 0.28 (sec) , antiderivative size = 303, normalized size of antiderivative = 6.45 \[ \int \frac {1}{x \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {2 \, \arctan \left (-\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1\right )} \sqrt {\frac {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}}\right ) + \log \left (-\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1\right )} \sqrt {\frac {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}}\right )}{2 \, b n} \]
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\[ \int \frac {1}{x \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int \frac {1}{x \sqrt {\coth {\left (a + b \log {\left (c x^{n} \right )} \right )}}}\, dx \]
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\[ \int \frac {1}{x \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int { \frac {1}{x \sqrt {\coth \left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{x \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}} \, dx=\text {Timed out} \]
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Time = 2.33 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {\mathrm {atan}\left (\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )+\mathrm {atanh}\left (\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n} \]
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