Integrand size = 17, antiderivative size = 55 \[ \int \frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\arctan \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac {\text {sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
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Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3853, 3855} \[ \int \frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\arctan \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac {\tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
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Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\text {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 {\text {Subst}\left (\int \text {sech}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{2 n} \\ & = \frac {\arctan \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*}
Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\arctan \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac {\text {sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
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Time = 19.64 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {sech}\left (a +b \ln \left (c \,x^{n}\right )\right ) \tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}+\arctan \left ({\mathrm e}^{a +b \ln \left (c \,x^{n}\right )}\right )}{n b}\) | \(45\) |
default | \(\frac {\frac {\operatorname {sech}\left (a +b \ln \left (c \,x^{n}\right )\right ) \tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}+\arctan \left ({\mathrm e}^{a +b \ln \left (c \,x^{n}\right )}\right )}{n b}\) | \(45\) |
parallelrisch | \(\frac {i \left (-1-\cosh \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )\right ) \ln \left (\tanh \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )-i\right )+i \left (\cosh \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )+1\right ) \ln \left (\tanh \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )+i\right )+2 \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}{2 b n \left (\cosh \left (4 b \ln \left (\sqrt {c \,x^{n}}\right )+2 a \right )+1\right )}\) | \(121\) |
risch | \(\frac {c^{b} \left (x^{n}\right )^{b} \left (\left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{3 a} {\mathrm e}^{\frac {3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{-\frac {3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}}-{\mathrm e}^{a} {\mathrm e}^{\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}}\right )}{b n {\left (\left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{2 a} {\mathrm e}^{i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}+1\right )}^{2}}+\frac {i \ln \left (c^{b} \left (x^{n}\right )^{b} {\mathrm e}^{a} {\mathrm e}^{\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}}+i\right )}{2 b n}-\frac {i \ln \left (c^{b} \left (x^{n}\right )^{b} {\mathrm e}^{a} {\mathrm e}^{\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}}-i\right )}{2 b n}\) | \(538\) |
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Leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (51) = 102\).
Time = 0.27 (sec) , antiderivative size = 452, normalized size of antiderivative = 8.22 \[ \int \frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 4 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 1\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 4 \, {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 1\right )} \arctan \left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right ) + {\left (3 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 4 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 2 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, {\left (3 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + b n\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + b n + 4 \, {\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \]
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\[ \int \frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\operatorname {sech}^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
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\[ \int \frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{3}}{x} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (51) = 102\).
Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.09 \[ \int \frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=c^{3 \, b} {\left (\frac {\arctan \left (\frac {c^{2 \, b} x^{b n} e^{a}}{c^{b}}\right ) e^{\left (-3 \, a\right )}}{b c^{2 \, b} c^{b} n} + \frac {{\left (c^{2 \, b} x^{3 \, b n} e^{\left (2 \, a\right )} - x^{b n}\right )} e^{\left (-2 \, a\right )}}{{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1\right )}^{2} b c^{2 \, b} n}\right )} e^{\left (3 \, a\right )} \]
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Time = 2.06 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.53 \[ \int \frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2\,{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (b\,n+\frac {2\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}\right )}-\frac {{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (b\,n+\frac {b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}\right )}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,\sqrt {b^2\,n^2}}{b\,n\,{\left (c\,x^n\right )}^b}\right )}{\sqrt {b^2\,n^2}} \]
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