Optimal. Leaf size=55 \[ \frac {\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac {\tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
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Rubi [A] time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3768, 3770} \[ \frac {\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac {\tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \text {sech}^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\text {sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\operatorname {Subst}\left (\int \text {sech}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{2 n}\\ &=\frac {\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac {\text {sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{2 b n}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 55, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac {\tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 452, normalized size = 8.22 \[ \frac {\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} + 3 \, \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 )^{2} + \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} + {\left (\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} + 4 \, \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 )^{3} + \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + 1\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + 2 \, \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + 4 \, {\left (\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} + \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + 1\right )} \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 ) + {\left (3 \, \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} - 1\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) - \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} + 4 \, b n \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 )^{3} + b n \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} + 2 \, b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + 2 \, {\left (3 \, b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + b n\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + b n + 4 \, {\left (b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} + b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 115, normalized size = 2.09 \[ c^{3 \, b} {\left (\frac {\arctan \left (\frac {c^{2 \, b} x^{b n} e^{a}}{c^{b}}\right ) e^{\left (-3 \, a\right )}}{b c^{2 \, b} c^{b} n} + \frac {{\left (c^{2 \, b} x^{3 \, b n} e^{\left (2 \, a\right )} - x^{b n}\right )} e^{\left (-2 \, a\right )}}{{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1\right )}^{2} b c^{2 \, b} n}\right )} e^{\left (3 \, a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 51, normalized size = 0.93 \[ \frac {\mathrm {sech}\left (a +b \ln \left (c \,x^{n}\right )\right ) \tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}{2 b n}+\frac {\arctan \left ({\mathrm e}^{a +b \ln \left (c \,x^{n}\right )}\right )}{b n} \]
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 \[ 8 \, c^{b} \int \frac {e^{\left (b \log \left (x^{n}\right ) + a\right )}}{8 \, {\left (c^{2 \, b} x e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + x\right )}}\,{d x} + \frac {c^{3 \, b} e^{\left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right )} - c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )}}{b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.40, size = 139, normalized size = 2.53 \[ \frac {2\,{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (b\,n+\frac {2\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}\right )}-\frac {{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (b\,n+\frac {b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}\right )}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,\sqrt {b^2\,n^2}}{b\,n\,{\left (c\,x^n\right )}^b}\right )}{\sqrt {b^2\,n^2}} \]
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 \[ \int \frac {\operatorname {sech}^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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