Optimal. Leaf size=19 \[ \frac {\text {ArcTan}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
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Rubi [A]
time = 0.01, 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 = {3855}
\begin {gather*} \frac {\text {ArcTan}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \end {gather*}
Antiderivative was successfully verified.
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Rule 3855
Rubi steps
\begin {align*} \int \frac {\text {sech}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\text {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 \begin {gather*} \frac {\text {ArcTan}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.60, size = 20, normalized size = 1.05
method | result | size |
derivativedivides | \(\frac {\arctan \left (\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{b n}\) | \(20\) |
default | \(\frac {\arctan \left (\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{b n}\) | \(20\) |
risch | \(\frac {i \ln \left (c^{b} \left (x^{n}\right )^{b} {\mathrm e}^{a} {\mathrm e}^{-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right ) \pi }{2}} {\mathrm e}^{\frac {i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}} {\mathrm e}^{-\frac {i b \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}}+i\right )}{b n}-\frac {i \ln \left (c^{b} \left (x^{n}\right )^{b} {\mathrm e}^{a} {\mathrm e}^{-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right ) \pi }{2}} {\mathrm e}^{\frac {i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}} {\mathrm e}^{-\frac {i b \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}}-i\right )}{b n}\) | \(222\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 19, normalized size = 1.00 \begin {gather*} \frac {\arctan \left (\sinh \left (b \log \left (c x^{n}\right ) + a\right )\right )}{b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 34, normalized size = 1.79 \begin {gather*} \frac {2 \, \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 )}{b n} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {sech}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 27, normalized size = 1.42 \begin {gather*} \frac {2 \, \arctan \left (\frac {c^{2 \, b} x^{b n} e^{a}}{c^{b}}\right )}{b n} \end {gather*}
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 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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