Optimal. Leaf size=74 \[ c^3 x \sqrt {1-\frac {1}{c^4 x^4}} F\left (\left .\csc ^{-1}(c x)\right |-1\right ) \sqrt {\text {csch}(2 \log (c x))}-c^3 x \sqrt {1-\frac {1}{c^4 x^4}} E\left (\left .\csc ^{-1}(c x)\right |-1\right ) \sqrt {\text {csch}(2 \log (c x))} \]
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Rubi [A] time = 0.07, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {5552, 5550, 335, 307, 221, 1181, 424} \[ c^3 x \sqrt {1-\frac {1}{c^4 x^4}} F\left (\left .\csc ^{-1}(c x)\right |-1\right ) \sqrt {\text {csch}(2 \log (c x))}-c^3 x \sqrt {1-\frac {1}{c^4 x^4}} E\left (\left .\csc ^{-1}(c x)\right |-1\right ) \sqrt {\text {csch}(2 \log (c x))} \]
Antiderivative was successfully verified.
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Rule 221
Rule 307
Rule 335
Rule 424
Rule 1181
Rule 5550
Rule 5552
Rubi steps
\begin {align*} \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^3} \, dx &=c^2 \operatorname {Subst}\left (\int \frac {\sqrt {\text {csch}(2 \log (x))}}{x^3} \, dx,x,c x\right )\\ &=\left (c^3 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {1}{x^4}} x^4} \, dx,x,c x\right )\\ &=-\left (\left (c^3 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )\right )\\ &=\left (c^3 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )-\left (c^3 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )\\ &=c^3 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))} F\left (\left .\csc ^{-1}(c x)\right |-1\right )-\left (c^3 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx,x,\frac {1}{c x}\right )\\ &=-c^3 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))} E\left (\left .\csc ^{-1}(c x)\right |-1\right )+c^3 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))} F\left (\left .\csc ^{-1}(c x)\right |-1\right )\\ \end {align*}
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Mathematica [C] time = 0.10, size = 58, normalized size = 0.78 \[ -\frac {\sqrt {2-2 c^4 x^4} \sqrt {\frac {c^2 x^2}{c^4 x^4-1}} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};c^4 x^4\right )}{x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 126, normalized size = 1.70 \[ \frac {\left (c^{4} x^{4}-1\right ) \sqrt {2}\, \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}{x^{2}}-\frac {c^{2} \sqrt {c^{2} x^{2}+1}\, \sqrt {-c^{2} x^{2}+1}\, \left (\EllipticF \left (x \sqrt {-c^{2}}, i\right )-\EllipticE \left (x \sqrt {-c^{2}}, i\right )\right ) \sqrt {2}\, \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}{\sqrt {-c^{2}}\, x} \]
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 \[ \int \frac {\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}}{x^{3}}\,{d x} \]
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 \[ \int \frac {\sqrt {\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}}}{x^3} \,d x \]
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 \[ \int \frac {\sqrt {\operatorname {csch}{\left (2 \log {\left (c x \right )} \right )}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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