Integrand size = 15, antiderivative size = 118 \[ \int \frac {x^7}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {4}{77 c^4 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {6 x^4}{77 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^8}{11 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{77 c^{11} \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]
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Time = 0.06 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5671, 5669, 342, 283, 331, 227} \[ \int \frac {x^7}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=-\frac {6 x^4}{77 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {4}{77 c^4 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{77 c^{11} x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^8}{11 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]
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Rule 227
Rule 283
Rule 331
Rule 342
Rule 5669
Rule 5671
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^7}{\text {csch}^{\frac {3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^8} \\ & = \frac {\text {Subst}\left (\int \left (1-\frac {1}{x^4}\right )^{3/2} x^{10} \, dx,x,c x\right )}{c^{11} \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = -\frac {\text {Subst}\left (\int \frac {\left (1-x^4\right )^{3/2}}{x^{12}} \, dx,x,\frac {1}{c x}\right )}{c^{11} \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {x^8}{11 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {6 \text {Subst}\left (\int \frac {\sqrt {1-x^4}}{x^8} \, dx,x,\frac {1}{c x}\right )}{11 c^{11} \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = -\frac {6 x^4}{77 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^8}{11 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {12 \text {Subst}\left (\int \frac {1}{x^4 \sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )}{77 c^{11} \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {4}{77 c^4 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {6 x^4}{77 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^8}{11 \text {csch}^{\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 )}{77 c^{11} \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {4}{77 c^4 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {6 x^4}{77 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^8}{11 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{77 c^{11} \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\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 = 0.12 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.68 \[ \int \frac {x^7}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {x^2 \left (\left (1-c^4 x^4\right )^{5/2}-\operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},c^4 x^4\right )\right )}{22 c^6 \sqrt {2-2 c^4 x^4} \sqrt {\frac {c^2 x^2}{-1+c^4 x^4}}} \]
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Time = 0.57 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.13
method | result | size |
risch | \(\frac {x^{2} \left (7 c^{8} x^{8}-13 c^{4} x^{4}+4\right ) \sqrt {2}}{308 c^{6} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}+\frac {\sqrt {c^{2} x^{2}+1}\, \sqrt {-c^{2} x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {-c^{2}}, i\right ) \sqrt {2}\, x}{77 c^{6} \sqrt {-c^{2}}\, \left (c^{4} x^{4}-1\right ) \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}\) | \(133\) |
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Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.68 \[ \int \frac {x^7}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {\sqrt {2} {\left (7 \, c^{14} x^{12} - 20 \, c^{10} x^{8} + 17 \, c^{6} x^{4} - 4 \, c^{2}\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} - 4 \, \sqrt {2} \sqrt {c^{4}} F(\arcsin \left (\frac {1}{c x}\right )\,|\,-1)}{308 \, c^{10}} \]
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\[ \int \frac {x^7}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x^{7}}{\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}\, dx \]
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\[ \int \frac {x^7}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int { \frac {x^{7}}{\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^7}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {x^7}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x^7}{{\left (\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}} \,d x \]
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