Integrand size = 19, antiderivative size = 70 \[ \int \frac {\coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\arctan \left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\text {arctanh}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}-\frac {2 \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{b n} \]
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Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3554, 3557, 335, 218, 212, 209} \[ \int \frac {\coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\arctan \left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\text {arctanh}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}-\frac {2 \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{b n} \]
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Rule 209
Rule 212
Rule 218
Rule 335
Rule 3554
Rule 3557
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \coth ^{\frac {3}{2}}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {2 \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{b n}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {\coth (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {2 \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{b 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 \sqrt {\coth \left (a+b \log \left (c x^n\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 {2 \sqrt {\coth \left (a+b \log \left (c x^n\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}-\frac {2 \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{b n} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.81 \[ \int \frac {\coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\arctan \left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )+\text {arctanh}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )-2 \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{b n} \]
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Time = 0.49 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(\frac {-2 \sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}-\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}-1\right )}{2}+\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}+1\right )}{2}+\arctan \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )}{n b}\) | \(74\) |
default | \(\frac {-2 \sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}-\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}-1\right )}{2}+\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}+1\right )}{2}+\arctan \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )}{n b}\) | \(74\) |
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Leaf count of result is larger than twice the leaf count of optimal. 334 vs. \(2 (64) = 128\).
Time = 0.26 (sec) , antiderivative size = 334, normalized size of antiderivative = 4.77 \[ \int \frac {\coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {4 \, \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 )}} + 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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Timed out. \[ \int \frac {\coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]
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\[ \int \frac {\coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\coth \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}{x} \,d x } \]
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Timed out. \[ \int \frac {\coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]
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Time = 2.61 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.73 \[ \int \frac {\coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\mathrm {atan}\left (\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )+\mathrm {atanh}\left (\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )-2\,\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}}{b\,n} \]
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