Integrand size = 15, antiderivative size = 28 \[ \int \frac {x^4}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\frac {\left (c^4+\frac {1}{x^4}\right ) x^5}{6 c^4 \sqrt {\text {sech}(2 \log (c x))}} \]
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Time = 0.03 (sec) , antiderivative size = 28, 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 = {5670, 5668, 270} \[ \int \frac {x^4}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\frac {x^5 \left (c^4+\frac {1}{x^4}\right )}{6 c^4 \sqrt {\text {sech}(2 \log (c x))}} \]
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Rule 270
Rule 5668
Rule 5670
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^4}{\sqrt {\text {sech}(2 \log (x))}} \, dx,x,c x\right )}{c^5} \\ & = \frac {\text {Subst}\left (\int \sqrt {1+\frac {1}{x^4}} x^5 \, dx,x,c x\right )}{c^6 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}} \\ & = \frac {\left (c^4+\frac {1}{x^4}\right ) x^5}{6 c^4 \sqrt {\text {sech}(2 \log (c x))}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {x^4}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\frac {\left (1+c^4 x^4\right )^2 \sqrt {\frac {c^2 x^2}{2+2 c^4 x^4}}}{6 c^6 x} \]
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Time = 0.16 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39
method | result | size |
risch | \(\frac {\sqrt {2}\, x \left (c^{4} x^{4}+1\right )}{12 \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}\, c^{4}}\) | \(39\) |
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Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71 \[ \int \frac {x^4}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\frac {\sqrt {2} {\left (c^{8} x^{8} + 2 \, c^{4} x^{4} + 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}}}{12 \, c^{6} x} \]
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\[ \int \frac {x^4}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\int \frac {x^{4}}{\sqrt {\operatorname {sech}{\left (2 \log {\left (c x \right )} \right )}}}\, dx \]
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Time = 0.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {x^4}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\frac {{\left (\sqrt {2} c^{4} x^{4} + \sqrt {2}\right )} \sqrt {c^{4} x^{4} + 1}}{12 \, c^{5}} \]
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\[ \int \frac {x^4}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\int { \frac {x^{4}}{\sqrt {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )}} \,d x } \]
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Time = 2.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {x^4}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\frac {{\left (c^4\,x^4+1\right )}^2\,\sqrt {\frac {2\,c^2\,x^2}{c^4\,x^4+1}}}{12\,c^6\,x} \]
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