Integrand size = 11, antiderivative size = 96 \[ \int \frac {1}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {3}{4 \left (c^4-\frac {1}{x^4}\right ) x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x}{4 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{c^4 x^4}}\right )}{4 c^4 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]
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Time = 0.03 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {5665, 5663, 272, 43, 52, 65, 212} \[ \int \frac {1}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=-\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{c^4 x^4}}\right )}{4 c^4 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {3}{4 x^3 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x}{4 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]
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Rule 43
Rule 52
Rule 65
Rule 212
Rule 272
Rule 5663
Rule 5665
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\text {csch}^{\frac {3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c} \\ & = \frac {\text {Subst}\left (\int \left (1-\frac {1}{x^4}\right )^{3/2} x^3 \, dx,x,c x\right )}{c^4 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = -\frac {\text {Subst}\left (\int \frac {(1-x)^{3/2}}{x^2} \, dx,x,\frac {1}{c^4 x^4}\right )}{4 c^4 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {x}{4 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {3 \text {Subst}\left (\int \frac {\sqrt {1-x}}{x} \, dx,x,\frac {1}{c^4 x^4}\right )}{8 c^4 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {3}{4 \left (c^4-\frac {1}{x^4}\right ) x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x}{4 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,\frac {1}{c^4 x^4}\right )}{8 c^4 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {3}{4 \left (c^4-\frac {1}{x^4}\right ) x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x}{4 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {3 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-\frac {1}{c^4 x^4}}\right )}{4 c^4 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {3}{4 \left (c^4-\frac {1}{x^4}\right ) x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x}{4 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{c^4 x^4}}\right )}{4 c^4 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \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 = 63, normalized size of antiderivative = 0.66 \[ \int \frac {1}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {1}{2},\frac {1}{2},c^4 x^4\right )}{4 c^2 x \sqrt {2-2 c^4 x^4} \sqrt {\frac {c^2 x^2}{-1+c^4 x^4}}} \]
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Time = 0.19 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.35
method | result | size |
risch | \(\frac {\left (c^{8} x^{8}+c^{4} x^{4}-2\right ) \sqrt {2}}{16 x \left (c^{4} x^{4}-1\right ) c^{2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}-\frac {3 c^{2} \ln \left (\frac {c^{4} x^{2}}{\sqrt {c^{4}}}+\sqrt {c^{4} x^{4}-1}\right ) \sqrt {2}\, x}{16 \sqrt {c^{4}}\, \sqrt {c^{4} x^{4}-1}\, \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}\) | \(130\) |
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Time = 0.26 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {3 \, \sqrt {2} c^{3} x^{3} \log \left (2 \, c^{4} x^{4} - 2 \, {\left (c^{5} x^{5} - c x\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} - 1\right ) + 2 \, \sqrt {2} {\left (c^{8} x^{8} + c^{4} x^{4} - 2\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}}}{32 \, c^{4} x^{3}} \]
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\[ \int \frac {1}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {1}{\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}\, dx \]
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\[ \int \frac {1}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int { \frac {1}{\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {1}{{\left (\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}} \,d x \]
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