Optimal. Leaf size=19 \[ \frac {\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
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Rubi [A] time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3770} \[ \frac {\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {sech}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \text {sech}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 19, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 34, normalized size = 1.79 \[ \frac {2 \, \arctan \left (\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 27, normalized size = 1.42 \[ \frac {2 \, \arctan \left (\frac {c^{2 \, b} x^{b n} e^{a}}{c^{b}}\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 20, normalized size = 1.05 \[ \frac {\arctan \left (\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 19, normalized size = 1.00 \[ \frac {\arctan \left (\sinh \left (b \log \left (c x^{n}\right ) + a\right )\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.41, size = 41, normalized size = 2.16 \[ -\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,\sqrt {b^2\,n^2}}{b\,n\,{\left (c\,x^n\right )}^b}\right )}{\sqrt {b^2\,n^2}} \]
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 \[ \int \frac {\operatorname {sech}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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