Optimal. Leaf size=56 \[ \sinh (2 \log (c x)) \sqrt{\text{sech}(2 \log (c x))}+i \sqrt{\text{sech}(2 \log (c x))} \sqrt{\cosh (2 \log (c x))} E(i \log (c x)|2) \]
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Rubi [A] time = 0.035545, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3768, 3771, 2639} \[ \sinh (2 \log (c x)) \sqrt{\text{sech}(2 \log (c x))}+i \sqrt{\text{sech}(2 \log (c x))} \sqrt{\cosh (2 \log (c x))} E(i \log (c x)|2) \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{\text{sech}^{\frac{3}{2}}(2 \log (c x))}{x} \, dx &=\operatorname{Subst}\left (\int \text{sech}^{\frac{3}{2}}(2 x) \, dx,x,\log (c x)\right )\\ &=\sqrt{\text{sech}(2 \log (c x))} \sinh (2 \log (c x))-\operatorname{Subst}\left (\int \frac{1}{\sqrt{\text{sech}(2 x)}} \, dx,x,\log (c x)\right )\\ &=\sqrt{\text{sech}(2 \log (c x))} \sinh (2 \log (c x))-\left (\sqrt{\cosh (2 \log (c x))} \sqrt{\text{sech}(2 \log (c x))}\right ) \operatorname{Subst}\left (\int \sqrt{\cosh (2 x)} \, dx,x,\log (c x)\right )\\ &=i \sqrt{\cosh (2 \log (c x))} E(i \log (c x)|2) \sqrt{\text{sech}(2 \log (c x))}+\sqrt{\text{sech}(2 \log (c x))} \sinh (2 \log (c x))\\ \end{align*}
Mathematica [A] time = 0.100691, size = 45, normalized size = 0.8 \[ \frac{\tanh (2 \log (c x))+\frac{i E(i \log (c x)|2)}{\sqrt{\cosh (2 \log (c x))}}}{\sqrt{\text{sech}(2 \log (c x))}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.314, size = 127, normalized size = 2.3 \begin{align*}{ \left ( \sqrt{-2\, \left ( 1/2\,cx-1/2\,{\frac{1}{cx}} \right ) ^{2}-1}\sqrt{- \left ({\frac{cx}{2}}-{\frac{1}{2\,cx}} \right ) ^{2}}{\it EllipticE} \left ({\frac{cx}{2}}+{\frac{1}{2\,cx}},\sqrt{2} \right ) +2\, \left ( 1/2\,cx+1/2\,{\frac{1}{cx}} \right ) \left ( 1/2\,cx-1/2\,{\frac{1}{cx}} \right ) ^{2} \right ) \left ({\frac{cx}{2}}-{\frac{1}{2\,cx}} \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( 1/2\,cx+1/2\,{\frac{1}{cx}} \right ) ^{2}-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{\operatorname{sech}\left (2 \, \log \left (c x\right )\right )^{\frac{3}{2}}}{x}\,{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{\operatorname{sech}\left (2 \, \log \left (c x\right )\right )^{\frac{3}{2}}}{x}, 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{\operatorname{sech}^{\frac{3}{2}}{\left (2 \log{\left (c x \right )} \right )}}{x}\, 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{\operatorname{sech}\left (2 \, \log \left (c x\right )\right )^{\frac{3}{2}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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