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 = 0.04, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 2639
Rule 3768
Rule 3771
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*}
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Mathematica [A] time = 0.11, size = 45, normalized size = 0.80 \[ \frac {\tanh (2 \log (c x))+\frac {i E(i \log (c x)|2)}{\sqrt {\cosh (2 \log (c x))}}}{\sqrt {\text {sech}(2 \log (c x))}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.62, size = 127, normalized size = 2.27 \[ \frac {\sqrt {-2 \left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}-1}\, \sqrt {-\left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}}\, \EllipticE \left (\frac {c x}{2}+\frac {1}{2 c x}, \sqrt {2}\right )+2 \left (\frac {c x}{2}+\frac {1}{2 c x}\right ) \left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}}{\left (\frac {c x}{2}-\frac {1}{2 c x}\right ) \sqrt {2 \left (\frac {c x}{2}+\frac {1}{2 c x}\right )^{2}-1}} \]
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 \[ \int \frac {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}}{x} \,d x \]
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 \[ \int \frac {\operatorname {sech}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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