Optimal. Leaf size=96 \[ -\frac{3 x}{16 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^4 x^4}}\right )}{16 c^8 x^3 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^5}{8 \text{csch}^{\frac{3}{2}}(2 \log (c x))} \]
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Rubi [A] time = 0.0673089, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5552, 5550, 266, 47, 63, 206} \[ -\frac{3 x}{16 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^4 x^4}}\right )}{16 c^8 x^3 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^5}{8 \text{csch}^{\frac{3}{2}}(2 \log (c x))} \]
Antiderivative was successfully verified.
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Rule 5552
Rule 5550
Rule 266
Rule 47
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{x^4}{\text{csch}^{\frac{3}{2}}(2 \log (c x))} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4}{\text{csch}^{\frac{3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^5}\\ &=\frac{\operatorname{Subst}\left (\int \left (1-\frac{1}{x^4}\right )^{3/2} x^7 \, dx,x,c x\right )}{c^8 \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{(1-x)^{3/2}}{x^3} \, dx,x,\frac{1}{c^4 x^4}\right )}{4 c^8 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{x^5}{8 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{1-x}}{x^2} \, dx,x,\frac{1}{c^4 x^4}\right )}{16 c^8 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=-\frac{3 x}{16 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^5}{8 \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x} \, dx,x,\frac{1}{c^4 x^4}\right )}{32 c^8 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=-\frac{3 x}{16 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^5}{8 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-\frac{1}{c^4 x^4}}\right )}{16 c^8 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=-\frac{3 x}{16 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^5}{8 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^4 x^4}}\right )}{16 c^8 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ \end{align*}
Mathematica [A] time = 0.175055, size = 87, normalized size = 0.91 \[ \frac{c^3 x^3 \sqrt{1-c^4 x^4} \left (2 c^4 x^4-5\right )-3 c x \sin ^{-1}\left (c^2 x^2\right )}{32 c^5 \sqrt{2-2 c^4 x^4} \sqrt{\frac{c^2 x^2}{c^4 x^4-1}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 113, normalized size = 1.2 \begin{align*}{\frac{{x}^{3} \left ( 2\,{c}^{4}{x}^{4}-5 \right ) \sqrt{2}}{64\,{c}^{2}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}-1}}}}}}+{\frac{3\,\sqrt{2}x}{64\,{c}^{2}}\ln \left ({{c}^{4}{x}^{2}{\frac{1}{\sqrt{{c}^{4}}}}}+\sqrt{{c}^{4}{x}^{4}-1} \right ){\frac{1}{\sqrt{{c}^{4}}}}{\frac{1}{\sqrt{{c}^{4}{x}^{4}-1}}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}-1}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{csch}\left (2 \, \log \left (c x\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57994, size = 219, normalized size = 2.28 \begin{align*} \frac{2 \, \sqrt{2}{\left (2 \, c^{9} x^{9} - 7 \, c^{5} x^{5} + 5 \, c x\right )} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} - 1}} + 3 \, \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 )}{128 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{csch}^{\frac{3}{2}}{\left (2 \log{\left (c x \right )} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{csch}\left (2 \, \log \left (c x\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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