Integrand size = 15, antiderivative size = 108 \[ \int \frac {x^5}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\frac {2 x^2}{21 c^4 \sqrt {\text {sech}(2 \log (c x))}}+\frac {x^6}{7 \sqrt {\text {sech}(2 \log (c x))}}+\frac {\sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) \operatorname {EllipticF}\left (2 \cot ^{-1}(c x),\frac {1}{2}\right )}{21 c^5 \left (c^4+\frac {1}{x^4}\right ) x \sqrt {\text {sech}(2 \log (c x))}} \]
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Time = 0.06 (sec) , antiderivative size = 108, 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, 283, 331, 226} \[ \int \frac {x^5}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\frac {2 x^2}{21 c^4 \sqrt {\text {sech}(2 \log (c x))}}+\frac {\sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) \operatorname {EllipticF}\left (2 \cot ^{-1}(c x),\frac {1}{2}\right )}{21 c^5 x \left (c^4+\frac {1}{x^4}\right ) \sqrt {\text {sech}(2 \log (c x))}}+\frac {x^6}{7 \sqrt {\text {sech}(2 \log (c x))}} \]
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Rule 226
Rule 283
Rule 331
Rule 342
Rule 5668
Rule 5670
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^5}{\sqrt {\text {sech}(2 \log (x))}} \, dx,x,c x\right )}{c^6} \\ & = \frac {\text {Subst}\left (\int \sqrt {1+\frac {1}{x^4}} x^6 \, dx,x,c x\right )}{c^7 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}} \\ & = -\frac {\text {Subst}\left (\int \frac {\sqrt {1+x^4}}{x^8} \, dx,x,\frac {1}{c x}\right )}{c^7 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}} \\ & = \frac {x^6}{7 \sqrt {\text {sech}(2 \log (c x))}}-\frac {2 \text {Subst}\left (\int \frac {1}{x^4 \sqrt {1+x^4}} \, dx,x,\frac {1}{c x}\right )}{7 c^7 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}} \\ & = \frac {2 x^2}{21 c^4 \sqrt {\text {sech}(2 \log (c x))}}+\frac {x^6}{7 \sqrt {\text {sech}(2 \log (c x))}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\frac {1}{c x}\right )}{21 c^7 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}} \\ & = \frac {2 x^2}{21 c^4 \sqrt {\text {sech}(2 \log (c x))}}+\frac {x^6}{7 \sqrt {\text {sech}(2 \log (c x))}}+\frac {\sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) \operatorname {EllipticF}\left (2 \cot ^{-1}(c x),\frac {1}{2}\right )}{21 c^5 \left (c^4+\frac {1}{x^4}\right ) x \sqrt {\text {sech}(2 \log (c x))}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.71 \[ \int \frac {x^5}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\frac {\sqrt {1+c^4 x^4} \sqrt {\frac {c^2 x^2}{2+2 c^4 x^4}} \left (\left (1+c^4 x^4\right )^{3/2}-\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-c^4 x^4\right )\right )}{7 c^6} \]
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Result contains complex when optimal does not.
Time = 0.92 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.20
method | result | size |
risch | \(\frac {x^{2} \left (3 c^{4} x^{4}+2\right ) \sqrt {2}}{42 c^{4} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}-\frac {\sqrt {-i c^{2} x^{2}+1}\, \sqrt {i c^{2} x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {i c^{2}}, i\right ) \sqrt {2}\, x}{21 c^{4} \sqrt {i c^{2}}\, \left (c^{4} x^{4}+1\right ) \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}\) | \(130\) |
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none
Time = 0.08 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.75 \[ \int \frac {x^5}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=-\frac {2 \, \sqrt {2} \sqrt {c^{4}} c \left (-\frac {1}{c^{4}}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {1}{c^{4}}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - \sqrt {2} {\left (3 \, c^{8} x^{8} + 5 \, c^{4} x^{4} + 2\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}}}{42 \, c^{6}} \]
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\[ \int \frac {x^5}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\int \frac {x^{5}}{\sqrt {\operatorname {sech}{\left (2 \log {\left (c x \right )} \right )}}}\, dx \]
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\[ \int \frac {x^5}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\int { \frac {x^{5}}{\sqrt {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )}} \,d x } \]
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\[ \int \frac {x^5}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\int { \frac {x^{5}}{\sqrt {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )}} \,d x } \]
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Timed out. \[ \int \frac {x^5}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\int \frac {x^5}{\sqrt {\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}}} \,d x \]
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