Optimal. Leaf size=86 \[ -\frac{4 \text{EllipticF}\left (\csc ^{-1}(c x),-1\right )}{7 c^7 x^3 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{2}{7 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^4}{7 \text{csch}^{\frac{3}{2}}(2 \log (c x))} \]
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Rubi [A] time = 0.0633788, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5552, 5550, 335, 277, 221} \[ -\frac{2}{7 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{4 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{7 c^7 x^3 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^4}{7 \text{csch}^{\frac{3}{2}}(2 \log (c x))} \]
Antiderivative was successfully verified.
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Rule 5552
Rule 5550
Rule 335
Rule 277
Rule 221
Rubi steps
\begin{align*} \int \frac{x^3}{\text{csch}^{\frac{3}{2}}(2 \log (c x))} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{\text{csch}^{\frac{3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (1-\frac{1}{x^4}\right )^{3/2} x^6 \, dx,x,c x\right )}{c^7 \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{\left (1-x^4\right )^{3/2}}{x^8} \, dx,x,\frac{1}{c x}\right )}{c^7 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{x^4}{7 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{6 \operatorname{Subst}\left (\int \frac{\sqrt{1-x^4}}{x^4} \, dx,x,\frac{1}{c x}\right )}{7 c^7 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=-\frac{2}{7 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^4}{7 \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{4 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^4}} \, dx,x,\frac{1}{c x}\right )}{7 c^7 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=-\frac{2}{7 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^4}{7 \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{4 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{7 c^7 \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 [C] time = 0.12084, size = 65, normalized size = 0.76 \[ \frac{\sqrt{1-c^4 x^4} \sqrt{\frac{c^2 x^2}{c^4 x^4-1}} \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};c^4 x^4\right )}{2 \sqrt{2} c^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 124, normalized size = 1.4 \begin{align*}{\frac{{x}^{2} \left ({c}^{4}{x}^{4}-3 \right ) \sqrt{2}}{28\,{c}^{2}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}-1}}}}}}+{\frac{\sqrt{2}x}{ \left ( 7\,{c}^{4}{x}^{4}-7 \right ){c}^{2}}\sqrt{{c}^{2}{x}^{2}+1}\sqrt{-{c}^{2}{x}^{2}+1}{\it EllipticF} \left ( x\sqrt{-{c}^{2}},i \right ){\frac{1}{\sqrt{-{c}^{2}}}}{\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^{3}}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3}}{\operatorname{csch}\left (2 \, \log \left (c x\right )\right )^{\frac{3}{2}}}, x\right ) \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^{3}}{\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^{3}}{\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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