Integrand size = 15, antiderivative size = 23 \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^4} \, dx=-\frac {1}{2} \left (c^4+\frac {1}{x^4}\right ) x \sqrt {\text {sech}(2 \log (c x))} \]
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Time = 0.03 (sec) , antiderivative size = 23, 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, 267} \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^4} \, dx=-\frac {1}{2} x \left (c^4+\frac {1}{x^4}\right ) \sqrt {\text {sech}(2 \log (c x))} \]
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Rule 267
Rule 5668
Rule 5670
Rubi steps \begin{align*} \text {integral}& = c^3 \text {Subst}\left (\int \frac {\sqrt {\text {sech}(2 \log (x))}}{x^4} \, dx,x,c x\right ) \\ & = \left (c^4 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {1}{x^4}} x^5} \, dx,x,c x\right ) \\ & = -\frac {1}{2} \left (c^4+\frac {1}{x^4}\right ) x \sqrt {\text {sech}(2 \log (c x))} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^4} \, dx=-\frac {c^2}{2 x \sqrt {\frac {c^2 x^2}{2+2 c^4 x^4}}} \]
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Time = 0.15 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.65
method | result | size |
risch | \(-\frac {\left (c^{4} x^{4}+1\right ) \sqrt {2}\, \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}{2 x^{3}}\) | \(38\) |
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none
Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.61 \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^4} \, dx=-\frac {\sqrt {2} {\left (c^{4} x^{4} + 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}}}{2 \, x^{3}} \]
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\[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^4} \, dx=\int \frac {\sqrt {\operatorname {sech}{\left (2 \log {\left (c x \right )} \right )}}}{x^{4}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (19) = 38\).
Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83 \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^4} \, dx=-\frac {1}{2} \, c^{3} {\left (\frac {\sqrt {2}}{\sqrt {\frac {1}{c^{4} x^{4}} + 1}} + \frac {\sqrt {2}}{c^{4} x^{4} \sqrt {\frac {1}{c^{4} x^{4}} + 1}}\right )} \]
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Timed out. \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^4} \, dx=\text {Timed out} \]
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Time = 2.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.52 \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^4} \, dx=-\frac {\sqrt {\frac {2\,c^2\,x^2}{c^4\,x^4+1}}}{2\,x^3}-\frac {c^4\,x\,\sqrt {\frac {2\,c^2\,x^2}{c^4\,x^4+1}}}{2} \]
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