Integrand size = 15, antiderivative size = 92 \[ \int \frac {x^4}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {3 x}{16 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^5}{8 \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {3 \text {arctanh}\left (\sqrt {1+\frac {1}{c^4 x^4}}\right )}{16 c^8 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \]
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Time = 0.04 (sec) , antiderivative size = 92, 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, 272, 43, 65, 213} \[ \int \frac {x^4}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {3 \text {arctanh}\left (\sqrt {\frac {1}{c^4 x^4}+1}\right )}{16 c^8 x^3 \left (\frac {1}{c^4 x^4}+1\right )^{3/2} \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {3 x}{16 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^5}{8 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \]
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Rule 43
Rule 65
Rule 213
Rule 272
Rule 5668
Rule 5670
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^4}{\text {sech}^{\frac {3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^5} \\ & = \frac {\text {Subst}\left (\int \left (1+\frac {1}{x^4}\right )^{3/2} x^7 \, dx,x,c x\right )}{c^8 \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 {(1+x)^{3/2}}{x^3} \, dx,x,\frac {1}{c^4 x^4}\right )}{4 c^8 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {x^5}{8 \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {3 \text {Subst}\left (\int \frac {\sqrt {1+x}}{x^2} \, dx,x,\frac {1}{c^4 x^4}\right )}{16 c^8 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {3 x}{16 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^5}{8 \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {3 \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,\frac {1}{c^4 x^4}\right )}{32 c^8 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {3 x}{16 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^5}{8 \text {sech}^{\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 )}{16 c^8 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {3 x}{16 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^5}{8 \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {3 \text {arctanh}\left (\sqrt {1+\frac {1}{c^4 x^4}}\right )}{16 c^8 \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 = 90, normalized size of antiderivative = 0.98 \[ \int \frac {x^4}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {c^3 x^3 \sqrt {1+c^4 x^4} \left (5+2 c^4 x^4\right )+3 c x \text {arcsinh}\left (c^2 x^2\right )}{32 \sqrt {2} c^5 \sqrt {\frac {c^2 x^2}{1+c^4 x^4}} \sqrt {1+c^4 x^4}} \]
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Time = 0.22 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.23
method | result | size |
risch | \(\frac {x^{3} \left (2 c^{4} x^{4}+5\right ) \sqrt {2}}{64 c^{2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}+\frac {3 \ln \left (\frac {c^{4} x^{2}}{\sqrt {c^{4}}}+\sqrt {c^{4} x^{4}+1}\right ) \sqrt {2}\, x}{64 \sqrt {c^{4}}\, c^{2} \sqrt {c^{4} x^{4}+1}\, \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}\) | \(113\) |
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Time = 0.29 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.10 \[ \int \frac {x^4}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {2 \, \sqrt {2} {\left (2 \, c^{9} x^{9} + 7 \, c^{5} x^{5} + 5 \, c x\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}} + 3 \, \sqrt {2} \log \left (-2 \, c^{4} x^{4} - 2 \, {\left (c^{5} x^{5} + c x\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}} - 1\right )}{128 \, c^{5}} \]
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\[ \int \frac {x^4}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x^{4}}{\operatorname {sech}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}\, dx \]
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\[ \int \frac {x^4}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int { \frac {x^{4}}{\operatorname {sech}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^4}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {x^4}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x^4}{{\left (\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}} \,d x \]
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