Optimal. Leaf size=89 \[ \frac {3 \tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}+\frac {\tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )}{8 b n} \]
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Rubi [A] time = 0.06, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3768, 3770} \[ \frac {3 \tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}+\frac {\tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )}{8 b n} \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {sech}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \text {sech}^5(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \operatorname {Subst}\left (\int \text {sech}^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{4 n}\\ &=\frac {3 \text {sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \operatorname {Subst}\left (\int \text {sech}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{8 n}\\ &=\frac {3 \tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}+\frac {3 \text {sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{4 b n}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 75, normalized size = 0.84 \[ \frac {3 \tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )+2 \tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}^3\left (a+b \log \left (c x^n\right )\right )+3 \tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )}{8 b n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 1326, normalized size = 14.90 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 152, normalized size = 1.71 \[ \frac {1}{4} \, c^{5 \, b} {\left (\frac {3 \, \arctan \left (\frac {c^{2 \, b} x^{b n} e^{a}}{c^{b}}\right ) e^{\left (-5 \, a\right )}}{b c^{4 \, b} c^{b} n} + \frac {{\left (3 \, c^{6 \, b} x^{7 \, b n} e^{\left (6 \, a\right )} + 11 \, c^{4 \, b} x^{5 \, b n} e^{\left (4 \, a\right )} - 11 \, c^{2 \, b} x^{3 \, b n} e^{\left (2 \, a\right )} - 3 \, x^{b n}\right )} e^{\left (-4 \, a\right )}}{{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1\right )}^{4} b c^{4 \, b} n}\right )} e^{\left (5 \, a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 84, normalized size = 0.94 \[ \frac {\mathrm {sech}\left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}{4 b n}+\frac {3 \,\mathrm {sech}\left (a +b \ln \left (c \,x^{n}\right )\right ) \tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}{8 b n}+\frac {3 \arctan \left ({\mathrm e}^{a +b \ln \left (c \,x^{n}\right )}\right )}{4 b n} \]
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 \[ 96 \, c^{b} \int \frac {e^{\left (b \log \left (x^{n}\right ) + a\right )}}{128 \, {\left (c^{2 \, b} x e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + x\right )}}\,{d x} + \frac {3 \, c^{7 \, b} e^{\left (7 \, b \log \left (x^{n}\right ) + 7 \, a\right )} + 11 \, c^{5 \, b} e^{\left (5 \, b \log \left (x^{n}\right ) + 5 \, a\right )} - 11 \, c^{3 \, b} e^{\left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right )} - 3 \, c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )}}{4 \, {\left (b c^{8 \, b} n e^{\left (8 \, b \log \left (x^{n}\right ) + 8 \, a\right )} + 4 \, b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} + 6 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 4 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 314, normalized size = 3.53 \[ \frac {2\,{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (b\,n+\frac {3\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {3\,b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}+\frac {b\,n\,{\mathrm {e}}^{-6\,a}}{{\left (c\,x^n\right )}^{6\,b}}\right )}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,\sqrt {b^2\,n^2}}{b\,n\,{\left (c\,x^n\right )}^b}\right )}{4\,\sqrt {b^2\,n^2}}-\frac {3\,{\mathrm {e}}^{-a}}{4\,{\left (c\,x^n\right )}^b\,\left (b\,n+\frac {b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}\right )}+\frac {4\,{\mathrm {e}}^{-3\,a}}{{\left (c\,x^n\right )}^{3\,b}\,\left (b\,n+\frac {4\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {6\,b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}+\frac {4\,b\,n\,{\mathrm {e}}^{-6\,a}}{{\left (c\,x^n\right )}^{6\,b}}+\frac {b\,n\,{\mathrm {e}}^{-8\,a}}{{\left (c\,x^n\right )}^{8\,b}}\right )}-\frac {{\mathrm {e}}^{-a}}{2\,{\left (c\,x^n\right )}^b\,\left (b\,n+\frac {2\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}\right )} \]
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 \[ \int \frac {\operatorname {sech}^{5}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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