Integrand size = 15, antiderivative size = 66 \[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x^4} \, dx=\frac {1}{2} \left (c^4+\frac {1}{x^4}\right ) x \text {sech}^{\frac {3}{2}}(2 \log (c x))-\frac {1}{2} c^6 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{-1}\left (c^2 x^2\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x)) \]
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Time = 0.04 (sec) , antiderivative size = 66, 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 = {5670, 5668, 342, 281, 294, 221} \[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x^4} \, dx=\frac {1}{2} x \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))-\frac {1}{2} c^6 x^3 \left (\frac {1}{c^4 x^4}+1\right )^{3/2} \text {csch}^{-1}\left (c^2 x^2\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x)) \]
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Rule 221
Rule 281
Rule 294
Rule 342
Rule 5668
Rule 5670
Rubi steps \begin{align*} \text {integral}& = c^3 \text {Subst}\left (\int \frac {\text {sech}^{\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 {sech}^{\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 {sech}^{\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 {sech}^{\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 {sech}^{\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 {sech}^{\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 {sech}^{\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 \text {csch}^{-1}\left (c^2 x^2\right ) \text {sech}^{\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 = 0.14 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.77 \[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x^4} \, dx=\frac {\sqrt {2} c^2 \sqrt {\frac {c^2 x^2}{1+c^4 x^4}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+c^4 x^4\right )}{x} \]
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\[\int \frac {\operatorname {sech}\left (2 \ln \left (c x \right )\right )^{\frac {3}{2}}}{x^{4}}d x\]
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none
Time = 0.25 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.41 \[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x^4} \, dx=\frac {\sqrt {2} c^{3} x \log \left (\frac {c^{5} x^{5} + 2 \, c x - 2 \, {\left (c^{4} x^{4} + 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}}}{c x^{5}}\right ) + 2 \, \sqrt {2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}} c^{2}}{2 \, x} \]
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\[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x^4} \, dx=\int \frac {\operatorname {sech}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}{x^{4}}\, dx \]
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\[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x^4} \, dx=\int { \frac {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}{x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x^4} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x^4} \, dx=\int \frac {{\left (\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}}{x^4} \,d x \]
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