Optimal. Leaf size=66 \[ \frac{1}{2} x \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))-\frac{1}{2} c^6 x^3 \left (\frac{1}{c^4 x^4}+1\right )^{3/2} \text{csch}^{-1}\left (c^2 x^2\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x)) \]
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Rubi [A] time = 0.0574587, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5551, 5549, 335, 275, 288, 215} \[ \frac{1}{2} x \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))-\frac{1}{2} c^6 x^3 \left (\frac{1}{c^4 x^4}+1\right )^{3/2} \text{csch}^{-1}\left (c^2 x^2\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x)) \]
Antiderivative was successfully verified.
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Rule 5551
Rule 5549
Rule 335
Rule 275
Rule 288
Rule 215
Rubi steps
\begin{align*} \int \frac{\text{sech}^{\frac{3}{2}}(2 \log (c x))}{x^4} \, dx &=c^3 \operatorname{Subst}\left (\int \frac{\text{sech}^{\frac{3}{2}}(2 \log (x))}{x^4} \, dx,x,c x\right )\\ &=\left (c^6 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{1}{x^4}\right )^{3/2} x^7} \, dx,x,c x\right )\\ &=-\left (\left (c^6 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))\right ) \operatorname{Subst}\left (\int \frac{x^5}{\left (1+x^4\right )^{3/2}} \, dx,x,\frac{1}{c x}\right )\right )\\ &=-\left (\frac{1}{2} \left (c^6 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (1+x^2\right )^{3/2}} \, dx,x,\frac{1}{c^2 x^2}\right )\right )\\ &=\frac{1}{2} \left (c^4+\frac{1}{x^4}\right ) x \text{sech}^{\frac{3}{2}}(2 \log (c x))-\frac{1}{2} \left (c^6 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,\frac{1}{c^2 x^2}\right )\\ &=\frac{1}{2} \left (c^4+\frac{1}{x^4}\right ) x \text{sech}^{\frac{3}{2}}(2 \log (c x))-\frac{1}{2} c^6 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{-1}\left (c^2 x^2\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))\\ \end{align*}
Mathematica [C] time = 0.105205, size = 51, normalized size = 0.77 \[ \frac{\sqrt{2} c^2 \sqrt{\frac{c^2 x^2}{c^4 x^4+1}} \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};c^4 x^4+1\right )}{x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \left ({\rm sech} \left (2\,\ln \left ( cx \right ) \right ) \right ) ^{{\frac{3}{2}}}}\, dx \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{\operatorname{sech}\left (2 \, \log \left (c x\right )\right )^{\frac{3}{2}}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.06589, size = 194, normalized size = 2.94 \begin{align*} \frac{\sqrt{2} c^{3} x \log \left (\frac{c^{5} x^{5} + 2 \, c x - 2 \,{\left (c^{4} x^{4} + 1\right )} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} + 1}}}{c x^{5}}\right ) + 2 \, \sqrt{2} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} + 1}} c^{2}}{2 \, x} \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{\operatorname{sech}^{\frac{3}{2}}{\left (2 \log{\left (c x \right )} \right )}}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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