Integrand size = 15, antiderivative size = 111 \[ \int \frac {x^3}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {2}{7 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^4}{7 \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {2 \sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) \operatorname {EllipticF}\left (2 \cot ^{-1}(c x),\frac {1}{2}\right )}{7 c \left (c^4+\frac {1}{x^4}\right )^2 x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \]
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Time = 0.06 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5670, 5668, 342, 283, 226} \[ \int \frac {x^3}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {2}{7 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {2 \sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) \operatorname {EllipticF}\left (2 \cot ^{-1}(c x),\frac {1}{2}\right )}{7 c x^3 \left (c^4+\frac {1}{x^4}\right )^2 \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^4}{7 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \]
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Rule 226
Rule 283
Rule 342
Rule 5668
Rule 5670
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^3}{\text {sech}^{\frac {3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^4} \\ & = \frac {\text {Subst}\left (\int \left (1+\frac {1}{x^4}\right )^{3/2} x^6 \, dx,x,c x\right )}{c^7 \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^8} \, dx,x,\frac {1}{c x}\right )}{c^7 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {x^4}{7 \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {6 \text {Subst}\left (\int \frac {\sqrt {1+x^4}}{x^4} \, dx,x,\frac {1}{c x}\right )}{7 c^7 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {2}{7 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^4}{7 \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\frac {1}{c x}\right )}{7 c^7 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {2}{7 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^4}{7 \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {2 \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 )}{7 c \left (c^4+\frac {1}{x^4}\right )^2 x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.14 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.55 \[ \int \frac {x^3}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {\sqrt {1+c^4 x^4} \sqrt {\frac {c^2 x^2}{2+2 c^4 x^4}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},-c^4 x^4\right )}{2 c^4} \]
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Result contains complex when optimal does not.
Time = 0.59 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.16
method | result | size |
risch | \(\frac {x^{2} \left (c^{4} x^{4}+3\right ) \sqrt {2}}{28 c^{2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}+\frac {\sqrt {-i c^{2} x^{2}+1}\, \sqrt {i c^{2} x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {i c^{2}}, i\right ) \sqrt {2}\, x}{7 \sqrt {i c^{2}}\, \left (c^{4} x^{4}+1\right ) c^{2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}\) | \(129\) |
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none
Time = 0.08 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.71 \[ \int \frac {x^3}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {4 \, \sqrt {2} \sqrt {c^{4}} c \left (-\frac {1}{c^{4}}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {1}{c^{4}}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + \sqrt {2} {\left (c^{8} x^{8} + 4 \, c^{4} x^{4} + 3\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}}}{28 \, c^{4}} \]
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\[ \int \frac {x^3}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x^{3}}{\operatorname {sech}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}\, dx \]
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\[ \int \frac {x^3}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int { \frac {x^{3}}{\operatorname {sech}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^3}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {x^3}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x^3}{{\left (\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}} \,d x \]
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