Integrand size = 19, antiderivative size = 73 \[ \int \frac {\coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\arctan \left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\text {arctanh}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}-\frac {2 \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]
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Time = 0.04 (sec) , antiderivative size = 73, 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, 304, 209, 212} \[ \int \frac {\coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\arctan \left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\text {arctanh}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}-\frac {2 \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]
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Rule 209
Rule 212
Rule 304
Rule 335
Rule 3554
Rule 3557
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \coth ^{\frac {5}{2}}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {2 \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\text {Subst}\left (\int \sqrt {\coth (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {2 \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac {\text {Subst}\left (\int \frac {\sqrt {x}}{-1+x^2} \, dx,x,\coth \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \\ & = -\frac {2 \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac {2 \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \\ & = -\frac {2 \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 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 \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.85 \[ \int \frac {\coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\arctan \left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )-\text {arctanh}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )+\frac {2}{3} \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{b n} \]
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Time = 1.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {-\frac {2 {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}}}{3}-\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}-1\right )}{2}+\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}+1\right )}{2}-\arctan \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )}{n b}\) | \(76\) |
default | \(\frac {-\frac {2 {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}}}{3}-\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}-1\right )}{2}+\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}+1\right )}{2}-\arctan \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )}{n b}\) | \(76\) |
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Leaf count of result is larger than twice the leaf count of optimal. 626 vs. \(2 (65) = 130\).
Time = 0.27 (sec) , antiderivative size = 626, normalized size of antiderivative = 8.58 \[ \int \frac {\coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]
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\[ \int \frac {\coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\coth \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}{x} \,d x } \]
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Timed out. \[ \int \frac {\coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]
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Time = 2.98 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.89 \[ \int \frac {\coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\mathrm {atanh}\left (\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n}-\frac {\mathrm {atan}\left (\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n}-\frac {2\,{\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2}}{3\,b\,n} \]
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