Integrand size = 15, antiderivative size = 26 \[ \int \text {csch}^3\left (a+2 \log \left (\frac {c}{\sqrt {x}}\right )\right ) \, dx=\frac {2 c^2 e^{-3 a}}{\left (e^{-2 a}-\frac {c^4}{x^2}\right )^2} \]
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Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {5665, 5667, 269, 267} \[ \int \text {csch}^3\left (a+2 \log \left (\frac {c}{\sqrt {x}}\right )\right ) \, dx=\frac {2 e^{-3 a} c^2}{\left (e^{-2 a}-\frac {c^4}{x^2}\right )^2} \]
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Rule 267
Rule 269
Rule 5665
Rule 5667
Rubi steps \begin{align*} \text {integral}& = -\left (\left (2 c^2\right ) \text {Subst}\left (\int \frac {\text {csch}^3(a+2 \log (x))}{x^3} \, dx,x,\frac {c}{\sqrt {x}}\right )\right ) \\ & = -\left (\left (16 c^2 e^{-3 a}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {e^{-2 a}}{x^4}\right )^3 x^9} \, dx,x,\frac {c}{\sqrt {x}}\right )\right ) \\ & = -\left (\left (16 c^2 e^{-3 a}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-e^{-2 a}+x^4\right )^3} \, dx,x,\frac {c}{\sqrt {x}}\right )\right ) \\ & = \frac {2 c^2 e^{-3 a}}{\left (e^{-2 a}-\frac {c^4}{x^2}\right )^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(65\) vs. \(2(26)=52\).
Time = 0.07 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.50 \[ \int \text {csch}^3\left (a+2 \log \left (\frac {c}{\sqrt {x}}\right )\right ) \, dx=-\frac {2 c^6 \left (\left (c^4-2 x^2\right ) \cosh (a)+\left (c^4+2 x^2\right ) \sinh (a)\right ) (\cosh (2 a)+\sinh (2 a))}{\left (\left (-c^4+x^2\right ) \cosh (a)-\left (c^4+x^2\right ) \sinh (a)\right )^2} \]
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\[\int \operatorname {csch}\left (a +2 \ln \left (\frac {c}{\sqrt {x}}\right )\right )^{3}d x\]
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none
Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.88 \[ \int \text {csch}^3\left (a+2 \log \left (\frac {c}{\sqrt {x}}\right )\right ) \, dx=-\frac {2 \, {\left (c^{10} e^{\left (5 \, a\right )} - 2 \, c^{6} x^{2} e^{\left (3 \, a\right )}\right )}}{c^{8} e^{\left (4 \, a\right )} - 2 \, c^{4} x^{2} e^{\left (2 \, a\right )} + x^{4}} \]
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\[ \int \text {csch}^3\left (a+2 \log \left (\frac {c}{\sqrt {x}}\right )\right ) \, dx=\int \operatorname {csch}^{3}{\left (a + 2 \log {\left (\frac {c}{\sqrt {x}} \right )} \right )}\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.88 \[ \int \text {csch}^3\left (a+2 \log \left (\frac {c}{\sqrt {x}}\right )\right ) \, dx=-\frac {2 \, {\left (c^{10} e^{\left (5 \, a\right )} - 2 \, c^{6} x^{2} e^{\left (3 \, a\right )}\right )}}{c^{8} e^{\left (4 \, a\right )} - 2 \, c^{4} x^{2} e^{\left (2 \, a\right )} + x^{4}} \]
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none
Time = 0.30 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \text {csch}^3\left (a+2 \log \left (\frac {c}{\sqrt {x}}\right )\right ) \, dx=-\frac {2 \, {\left (c^{10} e^{\left (5 \, a\right )} - 2 \, c^{6} x^{2} e^{\left (3 \, a\right )}\right )}}{{\left (c^{4} e^{\left (2 \, a\right )} - x^{2}\right )}^{2}} \]
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Time = 2.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \text {csch}^3\left (a+2 \log \left (\frac {c}{\sqrt {x}}\right )\right ) \, dx=\frac {2\,c^2\,x^4\,{\mathrm {e}}^a}{{\mathrm {e}}^{4\,a}\,c^8-2\,{\mathrm {e}}^{2\,a}\,c^4\,x^2+x^4} \]
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