Optimal. Leaf size=118 \[ -\frac {6 x^4}{77 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {4}{77 c^4 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{77 c^{11} x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^8}{11 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]
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Rubi [A] time = 0.08, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5552, 5550, 335, 277, 325, 221} \[ -\frac {6 x^4}{77 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {4}{77 c^4 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{77 c^{11} x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^8}{11 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]
Antiderivative was successfully verified.
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Rule 221
Rule 277
Rule 325
Rule 335
Rule 5550
Rule 5552
Rubi steps
\begin {align*} \int \frac {x^7}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^7}{\text {csch}^{\frac {3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^8}\\ &=\frac {\operatorname {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 {\operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 F\left (\left .\csc ^{-1}(c x)\right |-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*}
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Mathematica [C] time = 0.18, size = 80, normalized size = 0.68 \[ \frac {x^2 \left (\left (1-c^4 x^4\right )^{5/2}-\, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};c^4 x^4\right )\right )}{22 c^6 \sqrt {2-2 c^4 x^4} \sqrt {\frac {c^2 x^2}{c^4 x^4-1}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{7}}{\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}, x\right ) \]
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 \[ \int \frac {x^{7}}{\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 133, normalized size = 1.13 \[ \frac {x^{2} \left (7 c^{8} x^{8}-13 c^{4} x^{4}+4\right ) \sqrt {2}}{308 c^{6} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}+\frac {\sqrt {c^{2} x^{2}+1}\, \sqrt {-c^{2} x^{2}+1}\, \EllipticF \left (x \sqrt {-c^{2}}, i\right ) \sqrt {2}\, x}{77 c^{6} \sqrt {-c^{2}}\, \left (c^{4} x^{4}-1\right ) \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}} \]
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 \[ \int \frac {x^{7}}{\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^7}{{\left (\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{7}}{\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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