Integrand size = 15, antiderivative size = 69 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^4} \, dx=-\frac {1}{2} \left (c^4-\frac {1}{x^4}\right ) x \text {csch}^{\frac {3}{2}}(2 \log (c x))+\frac {1}{2} c^6 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \csc ^{-1}\left (c^2 x^2\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x)) \]
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Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5671, 5669, 342, 281, 294, 222} \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^4} \, dx=\frac {1}{2} c^6 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \csc ^{-1}\left (c^2 x^2\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))-\frac {1}{2} x \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x)) \]
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Rule 222
Rule 281
Rule 294
Rule 342
Rule 5669
Rule 5671
Rubi steps \begin{align*} \text {integral}& = c^3 \text {Subst}\left (\int \frac {\text {csch}^{\frac {3}{2}}(2 \log (x))}{x^4} \, dx,x,c x\right ) \\ & = \left (c^6 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {1}{x^4}\right )^{3/2} x^7} \, dx,x,c x\right ) \\ & = -\left (\left (c^6 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))\right ) \text {Subst}\left (\int \frac {x^5}{\left (1-x^4\right )^{3/2}} \, dx,x,\frac {1}{c x}\right )\right ) \\ & = -\left (\frac {1}{2} \left (c^6 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))\right ) \text {Subst}\left (\int \frac {x^2}{\left (1-x^2\right )^{3/2}} \, dx,x,\frac {1}{c^2 x^2}\right )\right ) \\ & = -\frac {1}{2} \left (c^4-\frac {1}{x^4}\right ) x \text {csch}^{\frac {3}{2}}(2 \log (c x))+\frac {1}{2} \left (c^6 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\frac {1}{c^2 x^2}\right ) \\ & = -\frac {1}{2} \left (c^4-\frac {1}{x^4}\right ) x \text {csch}^{\frac {3}{2}}(2 \log (c x))+\frac {1}{2} c^6 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \csc ^{-1}\left (c^2 x^2\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x)) \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.77 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^4} \, dx=-\frac {\sqrt {2} c^2 \sqrt {\frac {c^2 x^2}{-1+c^4 x^4}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1-c^4 x^4\right )}{x} \]
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\[\int \frac {\operatorname {csch}\left (2 \ln \left (c x \right )\right )^{\frac {3}{2}}}{x^{4}}d x\]
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none
Time = 0.25 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.13 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^4} \, dx=-\frac {\sqrt {2} c^{3} x \arctan \left (\frac {{\left (c^{4} x^{4} - 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}}}{c x}\right ) + \sqrt {2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} c^{2}}{x} \]
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\[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^4} \, dx=\int \frac {\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}{x^{4}}\, dx \]
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\[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^4} \, dx=\int { \frac {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}{x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^4} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^4} \, dx=\int \frac {{\left (\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}}{x^4} \,d x \]
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