Integrand size = 15, antiderivative size = 86 \[ \int \frac {x^3}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=-\frac {2}{7 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^4}{7 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{7 c^7 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]
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Time = 0.05 (sec) , antiderivative size = 86, 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 = {5671, 5669, 342, 283, 227} \[ \int \frac {x^3}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=-\frac {2}{7 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{7 c^7 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^4}{7 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]
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Rule 227
Rule 283
Rule 342
Rule 5669
Rule 5671
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^3}{\text {csch}^{\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 {csch}^{\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 {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {x^4}{7 \text {csch}^{\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 {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = -\frac {2}{7 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^4}{7 \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 )}{7 c^7 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = -\frac {2}{7 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^4}{7 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{7 c^7 \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 = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.76 \[ \int \frac {x^3}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {\sqrt {1-c^4 x^4} \sqrt {\frac {c^2 x^2}{-1+c^4 x^4}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},c^4 x^4\right )}{2 \sqrt {2} c^4} \]
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Time = 0.55 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.44
method | result | size |
risch | \(\frac {x^{2} \left (c^{4} x^{4}-3\right ) \sqrt {2}}{28 c^{2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}+\frac {\sqrt {c^{2} x^{2}+1}\, \sqrt {-c^{2} x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {-c^{2}}, i\right ) \sqrt {2}\, x}{7 \sqrt {-c^{2}}\, \left (c^{4} x^{4}-1\right ) c^{2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}\) | \(124\) |
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Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.83 \[ \int \frac {x^3}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {\sqrt {2} {\left (c^{10} x^{8} - 4 \, c^{6} x^{4} + 3 \, c^{2}\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} - 4 \, \sqrt {2} \sqrt {c^{4}} F(\arcsin \left (\frac {1}{c x}\right )\,|\,-1)}{28 \, c^{6}} \]
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\[ \int \frac {x^3}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x^{3}}{\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}\, dx \]
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\[ \int \frac {x^3}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int { \frac {x^{3}}{\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {x^3}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x^3}{{\left (\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}} \,d x \]
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