Integrand size = 15, antiderivative size = 28 \[ \int \frac {x^6}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {\left (c^4+\frac {1}{x^4}\right ) x^7}{10 c^4 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \]
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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} \[ \int \frac {x^6}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {x^7 \left (c^4+\frac {1}{x^4}\right )}{10 c^4 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \]
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Rule 270
Rule 5668
Rule 5670
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^6}{\text {sech}^{\frac {3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^7} \\ & = \frac {\text {Subst}\left (\int \left (1+\frac {1}{x^4}\right )^{3/2} x^9 \, dx,x,c x\right )}{c^{10} \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {\left (c^4+\frac {1}{x^4}\right ) x^7}{10 c^4 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {x^6}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {\left (1+c^4 x^4\right )^3 \sqrt {\frac {c^2 x^2}{2+2 c^4 x^4}}}{20 c^8 x} \]
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Time = 0.16 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68
method | result | size |
risch | \(\frac {\sqrt {2}\, x \left (c^{8} x^{8}+2 c^{4} x^{4}+1\right )}{40 c^{6} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}\) | \(47\) |
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (24) = 48\).
Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00 \[ \int \frac {x^6}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {\sqrt {2} {\left (c^{12} x^{12} + 3 \, c^{8} x^{8} + 3 \, c^{4} x^{4} + 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}}}{40 \, c^{8} x} \]
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\[ \int \frac {x^6}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x^{6}}{\operatorname {sech}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}\, dx \]
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none
Time = 0.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {x^6}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {{\left (\sqrt {2} c^{4} x^{4} + \sqrt {2}\right )} {\left (c^{4} x^{4} + 1\right )}^{\frac {3}{2}}}{40 \, c^{7}} \]
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Timed out. \[ \int \frac {x^6}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\text {Timed out} \]
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Time = 2.16 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {x^6}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {{\left (c^4\,x^4+1\right )}^3\,\sqrt {\frac {2\,c^2\,x^2}{c^4\,x^4+1}}}{40\,c^8\,x} \]
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