Integrand size = 13, antiderivative size = 130 \[ \int \frac {x}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=-\frac {6}{5 \left (c^4-\frac {1}{x^4}\right ) x^2 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^2}{5 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {12 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {12 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{5 c^5 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]
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Time = 0.05 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {5671, 5669, 342, 283, 313, 227, 1195, 435} \[ \int \frac {x}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=-\frac {6}{5 x^2 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {12 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{5 c^5 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {12 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^2}{5 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]
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Rule 227
Rule 283
Rule 313
Rule 342
Rule 435
Rule 1195
Rule 5669
Rule 5671
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x}{\text {csch}^{\frac {3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^2} \\ & = \frac {\text {Subst}\left (\int \left (1-\frac {1}{x^4}\right )^{3/2} x^4 \, dx,x,c x\right )}{c^5 \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^6} \, dx,x,\frac {1}{c x}\right )}{c^5 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {x^2}{5 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {6 \text {Subst}\left (\int \frac {\sqrt {1-x^4}}{x^2} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = -\frac {6}{5 \left (c^4-\frac {1}{x^4}\right ) x^2 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^2}{5 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {12 \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = -\frac {6}{5 \left (c^4-\frac {1}{x^4}\right ) x^2 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^2}{5 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {12 \text {Subst}\left (\int \frac {1}{\sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {12 \text {Subst}\left (\int \frac {1+x^2}{\sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = -\frac {6}{5 \left (c^4-\frac {1}{x^4}\right ) x^2 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^2}{5 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {12 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{5 c^5 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {12 \text {Subst}\left (\int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = -\frac {6}{5 \left (c^4-\frac {1}{x^4}\right ) x^2 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^2}{5 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {12 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {12 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{5 c^5 \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 = 60, normalized size of antiderivative = 0.46 \[ \int \frac {x}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {1}{4},\frac {3}{4},c^4 x^4\right )}{2 c^2 \sqrt {2-2 c^4 x^4} \sqrt {\frac {c^2 x^2}{-1+c^4 x^4}}} \]
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Time = 0.65 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.17
method | result | size |
risch | \(\frac {\left (c^{8} x^{8}+4 c^{4} x^{4}-5\right ) \sqrt {2}}{20 \left (c^{4} x^{4}-1\right ) c^{2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}-\frac {3 \sqrt {c^{2} x^{2}+1}\, \sqrt {-c^{2} x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {-c^{2}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {-c^{2}}, i\right )\right ) \sqrt {2}\, x}{5 \sqrt {-c^{2}}\, \left (c^{4} x^{4}-1\right ) \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}\) | \(152\) |
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\[ \int \frac {x}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int { \frac {x}{\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x}{\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}\, dx \]
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\[ \int \frac {x}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int { \frac {x}{\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {x}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x}{{\left (\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}} \,d x \]
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