Integrand size = 15, antiderivative size = 88 \[ \int \frac {x^2}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {1}{2 \left (c^4+\frac {1}{x^4}\right ) x \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^3}{6 \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {\text {csch}^{-1}\left (c^2 x^2\right )}{2 c^6 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \]
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Time = 0.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5670, 5668, 342, 281, 283, 221} \[ \int \frac {x^2}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {1}{2 x \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {\text {csch}^{-1}\left (c^2 x^2\right )}{2 c^6 x^3 \left (\frac {1}{c^4 x^4}+1\right )^{3/2} \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^3}{6 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \]
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Rule 221
Rule 281
Rule 283
Rule 342
Rule 5668
Rule 5670
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^2}{\text {sech}^{\frac {3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^3} \\ & = \frac {\text {Subst}\left (\int \left (1+\frac {1}{x^4}\right )^{3/2} x^5 \, dx,x,c x\right )}{c^6 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \\ & = -\frac {\text {Subst}\left (\int \frac {\left (1+x^4\right )^{3/2}}{x^7} \, dx,x,\frac {1}{c x}\right )}{c^6 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \\ & = -\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^{3/2}}{x^4} \, dx,x,\frac {1}{c^2 x^2}\right )}{2 c^6 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {x^3}{6 \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {\text {Subst}\left (\int \frac {\sqrt {1+x^2}}{x^2} \, dx,x,\frac {1}{c^2 x^2}\right )}{2 c^6 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {1}{2 \left (c^4+\frac {1}{x^4}\right ) x \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^3}{6 \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\frac {1}{c^2 x^2}\right )}{2 c^6 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {1}{2 \left (c^4+\frac {1}{x^4}\right ) x \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^3}{6 \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {\text {csch}^{-1}\left (c^2 x^2\right )}{2 c^6 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {x \left (\sqrt {1+c^4 x^4} \left (4+c^4 x^4\right )-3 \text {arctanh}\left (\sqrt {1+c^4 x^4}\right )\right )}{12 \sqrt {2} c^2 \sqrt {\frac {c^2 x^2}{1+c^4 x^4}} \sqrt {1+c^4 x^4}} \]
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\[\int \frac {x^{2}}{\operatorname {sech}\left (2 \ln \left (c x \right )\right )^{\frac {3}{2}}}d x\]
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none
Time = 0.25 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.24 \[ \int \frac {x^2}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {3 \, \sqrt {2} c x \log \left (\frac {c^{5} x^{5} + 2 \, c x - 2 \, {\left (c^{4} x^{4} + 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}}}{c x^{5}}\right ) + 2 \, \sqrt {2} {\left (c^{8} x^{8} + 5 \, c^{4} x^{4} + 4\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}}}{48 \, c^{4} x} \]
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\[ \int \frac {x^2}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x^{2}}{\operatorname {sech}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}\, dx \]
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\[ \int \frac {x^2}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int { \frac {x^{2}}{\operatorname {sech}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {x^2}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x^2}{{\left (\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}} \,d x \]
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