Integrand size = 15, antiderivative size = 27 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^2} \, dx=-\frac {1}{2} \left (c^4-\frac {1}{x^4}\right ) x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x)) \]
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Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5671, 5669, 267} \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^2} \, dx=-\frac {1}{2} x^3 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x)) \]
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Rule 267
Rule 5669
Rule 5671
Rubi steps \begin{align*} \text {integral}& = c \text {Subst}\left (\int \frac {\text {csch}^{\frac {3}{2}}(2 \log (x))}{x^2} \, dx,x,c x\right ) \\ & = \left (c^4 \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^5} \, dx,x,c x\right ) \\ & = -\frac {1}{2} \left (c^4-\frac {1}{x^4}\right ) x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x)) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^2} \, dx=-\sqrt {2} c^2 x \sqrt {\frac {c^2 x^2}{-1+c^4 x^4}} \]
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\[\int \frac {\operatorname {csch}\left (2 \ln \left (c x \right )\right )^{\frac {3}{2}}}{x^{2}}d x\]
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none
Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^2} \, dx=-\sqrt {2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} c^{2} x \]
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\[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^2} \, dx=\int \frac {\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}{x^{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (23) = 46\).
Time = 0.29 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.22 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^2} \, dx=-c {\left (\frac {\sqrt {2}}{{\left (\frac {1}{c x} + 1\right )}^{\frac {3}{2}} {\left (-\frac {1}{c x} + 1\right )}^{\frac {3}{2}} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}}} - \frac {\sqrt {2}}{c^{4} x^{4} {\left (\frac {1}{c x} + 1\right )}^{\frac {3}{2}} {\left (-\frac {1}{c x} + 1\right )}^{\frac {3}{2}} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}}}\right )} \]
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Timed out. \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^2} \, dx=\text {Timed out} \]
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Time = 2.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^2} \, dx=-c^2\,x\,\sqrt {\frac {2\,c^2\,x^2}{c^4\,x^4-1}} \]
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