Optimal. Leaf size=119 \[ -\frac {2}{5 c^4 \sqrt {\text {csch}(2 \log (c x))}}+\frac {2 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}-\frac {2 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}+\frac {x^4}{5 \sqrt {\text {csch}(2 \log (c x))}} \]
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Rubi [A] time = 0.09, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \frac {2 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}-\frac {2 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}-\frac {2}{5 c^4 \sqrt {\text {csch}(2 \log (c x))}}+\frac {x^4}{5 \sqrt {\text {csch}(2 \log (c x))}} \]
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^3}{\sqrt {\text {csch}(2 \log (c x))}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{\sqrt {\text {csch}(2 \log (x))}} \, dx,x,c x\right )}{c^4}\\ &=\frac {\operatorname {Subst}\left (\int \sqrt {1-\frac {1}{x^4}} x^4 \, dx,x,c x\right )}{c^5 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {1-x^4}}{x^6} \, dx,x,\frac {1}{c x}\right )}{c^5 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}}\\ &=\frac {x^4}{5 \sqrt {\text {csch}(2 \log (c x))}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}}\\ &=-\frac {2}{5 c^4 \sqrt {\text {csch}(2 \log (c x))}}+\frac {x^4}{5 \sqrt {\text {csch}(2 \log (c x))}}-\frac {2 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}}\\ &=-\frac {2}{5 c^4 \sqrt {\text {csch}(2 \log (c x))}}+\frac {x^4}{5 \sqrt {\text {csch}(2 \log (c x))}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}}-\frac {2 \operatorname {Subst}\left (\int \frac {1+x^2}{\sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}}\\ &=-\frac {2}{5 c^4 \sqrt {\text {csch}(2 \log (c x))}}+\frac {x^4}{5 \sqrt {\text {csch}(2 \log (c x))}}+\frac {2 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}}-\frac {2 \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}}\\ &=-\frac {2}{5 c^4 \sqrt {\text {csch}(2 \log (c x))}}+\frac {x^4}{5 \sqrt {\text {csch}(2 \log (c x))}}-\frac {2 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}}+\frac {2 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}}\\ \end {align*}
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Mathematica [C] time = 0.12, size = 60, normalized size = 0.50 \[ \frac {x^4 \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {7}{4};c^4 x^4\right )}{3 \sqrt {2-2 c^4 x^4} \sqrt {\frac {c^2 x^2}{c^4 x^4-1}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3}}{\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}}, x\right ) \]
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 \[ \int \frac {x^{3}}{\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 127, normalized size = 1.07 \[ \frac {x^{4} \sqrt {2}}{10 \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}-\frac {\sqrt {c^{2} x^{2}+1}\, \sqrt {-c^{2} x^{2}+1}\, \left (\EllipticF \left (x \sqrt {-c^{2}}, i\right )-\EllipticE \left (x \sqrt {-c^{2}}, i\right )\right ) \sqrt {2}\, x}{5 \sqrt {-c^{2}}\, \left (c^{4} x^{4}-1\right ) c^{2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}} \]
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 \[ \int \frac {x^{3}}{\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}}\,{d x} \]
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 \[ \int \frac {x^3}{\sqrt {\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}}} \,d x \]
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 \[ \int \frac {x^{3}}{\sqrt {\operatorname {csch}{\left (2 \log {\left (c x \right )} \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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