Optimal. Leaf size=162 \[ -\frac {2 x^2}{15 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {4}{15 c^4 x^2 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{15 c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {4 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{15 c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^6}{9 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]
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Rubi [A] time = 0.10, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5552, 5550, 335, 277, 325, 307, 221, 1181, 424} \[ -\frac {2 x^2}{15 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {4}{15 c^4 x^2 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{15 c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {4 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{15 c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^6}{9 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]
Antiderivative was successfully verified.
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Rule 221
Rule 277
Rule 307
Rule 325
Rule 335
Rule 424
Rule 1181
Rule 5550
Rule 5552
Rubi steps
\begin {align*} \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^6}\\ &=\frac {\operatorname {Subst}\left (\int \left (1-\frac {1}{x^4}\right )^{3/2} x^8 \, dx,x,c x\right )}{c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^4\right )^{3/2}}{x^{10}} \, dx,x,\frac {1}{c x}\right )}{c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}\\ &=\frac {x^6}{9 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {2 \operatorname {Subst}\left (\int \frac {\sqrt {1-x^4}}{x^6} \, dx,x,\frac {1}{c x}\right )}{3 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}\\ &=-\frac {2 x^2}{15 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^6}{9 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )}{15 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}\\ &=\frac {4}{15 c^4 \left (c^4-\frac {1}{x^4}\right ) x^2 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {2 x^2}{15 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^6}{9 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {4 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )}{15 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}\\ &=\frac {4}{15 c^4 \left (c^4-\frac {1}{x^4}\right ) x^2 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {2 x^2}{15 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^6}{9 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )}{15 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {4 \operatorname {Subst}\left (\int \frac {1+x^2}{\sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )}{15 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}\\ &=\frac {4}{15 c^4 \left (c^4-\frac {1}{x^4}\right ) x^2 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {2 x^2}{15 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^6}{9 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{15 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {4 \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx,x,\frac {1}{c x}\right )}{15 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}\\ &=\frac {4}{15 c^4 \left (c^4-\frac {1}{x^4}\right ) x^2 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {2 x^2}{15 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^6}{9 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {4 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{15 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{15 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}\\ \end {align*}
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Mathematica [C] time = 0.13, size = 63, normalized size = 0.39 \[ -\frac {x^4 \, _2F_1\left (-\frac {3}{2},\frac {3}{4};\frac {7}{4};c^4 x^4\right )}{6 c^2 \sqrt {2-2 c^4 x^4} \sqrt {\frac {c^2 x^2}{c^4 x^4-1}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{5}}{\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 140, normalized size = 0.86 \[ \frac {x^{4} \left (5 c^{4} x^{4}-11\right ) \sqrt {2}}{180 c^{2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}+\frac {\sqrt {c^{2} x^{2}+1}\, \sqrt {-c^{2} x^{2}+1}\, \left (\EllipticF \left (x \sqrt {-c^{2}}, i\right )-\EllipticE \left (x \sqrt {-c^{2}}, i\right )\right ) \sqrt {2}\, x}{15 \sqrt {-c^{2}}\, \left (c^{4} x^{4}-1\right ) c^{4} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^5}{{\left (\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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