Optimal. Leaf size=69 \[ \frac {x^3}{4 \sqrt {\text {csch}(2 \log (c x))}}-\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{c^4 x^4}}\right )}{4 c^4 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}} \]
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Rubi [A] time = 0.06, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5552, 5550, 266, 47, 63, 206} \[ \frac {x^3}{4 \sqrt {\text {csch}(2 \log (c x))}}-\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{c^4 x^4}}\right )}{4 c^4 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 206
Rule 266
Rule 5550
Rule 5552
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {\text {csch}(2 \log (c x))}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{\sqrt {\text {csch}(2 \log (x))}} \, dx,x,c x\right )}{c^3}\\ &=\frac {\operatorname {Subst}\left (\int \sqrt {1-\frac {1}{x^4}} x^3 \, dx,x,c x\right )}{c^4 \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}}{x^2} \, dx,x,\frac {1}{c^4 x^4}\right )}{4 c^4 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}}\\ &=\frac {x^3}{4 \sqrt {\text {csch}(2 \log (c x))}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,\frac {1}{c^4 x^4}\right )}{8 c^4 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}}\\ &=\frac {x^3}{4 \sqrt {\text {csch}(2 \log (c x))}}-\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-\frac {1}{c^4 x^4}}\right )}{4 c^4 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}}\\ &=\frac {x^3}{4 \sqrt {\text {csch}(2 \log (c x))}}-\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{c^4 x^4}}\right )}{4 c^4 \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 = 0.14, size = 74, normalized size = 1.07 \[ \frac {x \left (\sin ^{-1}\left (c^2 x^2\right )+c^2 x^2 \sqrt {1-c^4 x^4}\right )}{4 c^2 \sqrt {2-2 c^4 x^4} \sqrt {\frac {c^2 x^2}{c^4 x^4-1}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.66, size = 92, normalized size = 1.33 \[ \frac {2 \, \sqrt {2} {\left (c^{5} x^{5} - c x\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} + \sqrt {2} \log \left (2 \, c^{4} x^{4} - 2 \, {\left (c^{5} x^{5} - c x\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} - 1\right )}{16 \, c^{3}} \]
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 \[ \int \frac {x^{2}}{\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 97, normalized size = 1.41 \[ \frac {x^{3} \sqrt {2}}{8 \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}-\frac {\ln \left (\frac {c^{4} x^{2}}{\sqrt {c^{4}}}+\sqrt {c^{4} x^{4}-1}\right ) \sqrt {2}\, x}{8 \sqrt {c^{4}}\, \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}\, \sqrt {c^{4} x^{4}-1}} \]
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 \[ \int \frac {x^{2}}{\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}}\,{d x} \]
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 \[ \int \frac {x^2}{\sqrt {\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}}} \,d x \]
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 \[ \int \frac {x^{2}}{\sqrt {\operatorname {csch}{\left (2 \log {\left (c x \right )} \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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