Optimal. Leaf size=86 \[ -\frac {2}{7 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{7 c^7 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^4}{7 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]
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Rubi [A] time = 0.06, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5552, 5550, 335, 277, 221} \[ -\frac {2}{7 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{7 c^7 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^4}{7 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]
Antiderivative was successfully verified.
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Rule 221
Rule 277
Rule 335
Rule 5550
Rule 5552
Rubi steps
\begin {align*} \int \frac {x^3}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{\text {csch}^{\frac {3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^4}\\ &=\frac {\operatorname {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 {\operatorname {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 \operatorname {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 \operatorname {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 F\left (\left .\csc ^{-1}(c x)\right |-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*}
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Mathematica [C] time = 0.13, size = 65, normalized size = 0.76 \[ \frac {\sqrt {1-c^4 x^4} \sqrt {\frac {c^2 x^2}{c^4 x^4-1}} \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};c^4 x^4\right )}{2 \sqrt {2} c^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3}}{\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}, x\right ) \]
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 \[ \int \frac {x^{3}}{\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 124, normalized size = 1.44 \[ \frac {x^{2} \left (c^{4} x^{4}-3\right ) \sqrt {2}}{28 c^{2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}+\frac {\sqrt {c^{2} x^{2}+1}\, \sqrt {-c^{2} x^{2}+1}\, \EllipticF \left (x \sqrt {-c^{2}}, i\right ) \sqrt {2}\, x}{7 \sqrt {-c^{2}}\, \left (c^{4} x^{4}-1\right ) c^{2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}} \]
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 \[ \int \frac {x^{3}}{\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^3}{{\left (\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{3}}{\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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