Optimal. Leaf size=60 \[ \frac {\csc ^{-1}\left (c^2 x^2\right )}{2 c^2 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}+\frac {x}{2 \sqrt {\text {csch}(2 \log (c x))}} \]
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Rubi [A] time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {5546, 5544, 335, 275, 277, 216} \[ \frac {\csc ^{-1}\left (c^2 x^2\right )}{2 c^2 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}+\frac {x}{2 \sqrt {\text {csch}(2 \log (c x))}} \]
Antiderivative was successfully verified.
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Rule 216
Rule 275
Rule 277
Rule 335
Rule 5544
Rule 5546
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\text {csch}(2 \log (c x))}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {\text {csch}(2 \log (x))}} \, dx,x,c x\right )}{c}\\ &=\frac {\operatorname {Subst}\left (\int \sqrt {1-\frac {1}{x^4}} x \, dx,x,c x\right )}{c^2 \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^3} \, dx,x,\frac {1}{c x}\right )}{c^2 \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^2}}{x^2} \, dx,x,\frac {1}{c^2 x^2}\right )}{2 c^2 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}}\\ &=\frac {x}{2 \sqrt {\text {csch}(2 \log (c x))}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\frac {1}{c^2 x^2}\right )}{2 c^2 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}}\\ &=\frac {x}{2 \sqrt {\text {csch}(2 \log (c x))}}+\frac {\csc ^{-1}\left (c^2 x^2\right )}{2 c^2 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 77, normalized size = 1.28 \[ \frac {x \left (2 \sqrt {c^4 x^4-1}-2 \tan ^{-1}\left (\sqrt {c^4 x^4-1}\right )\right )}{4 \sqrt {2} \sqrt {\frac {c^2 x^2}{c^4 x^4-1}} \sqrt {c^4 x^4-1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 86, normalized size = 1.43 \[ -\frac {\sqrt {2} c x \arctan \left (\frac {{\left (c^{4} x^{4} - 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}}}{c x}\right ) - \sqrt {2} {\left (c^{4} x^{4} - 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}}}{4 \, c^{2} x} \]
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 [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\mathrm {csch}\left (2 \ln \left (c x \right )\right )}}\, dx \]
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 \[ \int \frac {1}{\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}}} \,d x \]
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 \[ \int \frac {1}{\sqrt {\operatorname {csch}{\left (2 \log {\left (c x \right )} \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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