Optimal. Leaf size=20 \[ -\frac {\tanh ^{-1}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
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Rubi [A] time = 0.02, antiderivative size = 20, 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} \[ -\frac {\tanh ^{-1}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {csch}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \text {csch}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\tanh ^{-1}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ \end {align*}
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Mathematica [B] time = 0.06, size = 54, normalized size = 2.70 \[ \frac {\log \left (\sinh \left (\frac {a}{2}+\frac {1}{2} b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\log \left (\cosh \left (\frac {a}{2}+\frac {1}{2} b \log \left (c x^n\right )\right )\right )}{b n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 65, normalized size = 3.25 \[ -\frac {\log \left (\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + 1\right ) - \log \left (\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) - 1\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.58, size = 145, normalized size = 7.25 \[ -c^{b} {\left (\frac {c^{b} e^{\left (-a\right )} \log \left (\sqrt {2 \, x^{b n} {\left | c \right |}^{b} \cos \left (-\frac {1}{2} \, \pi b \mathrm {sgn}\relax (c) + \frac {1}{2} \, \pi b\right ) e^{a} + x^{2 \, b n} {\left | c \right |}^{2 \, b} e^{\left (2 \, a\right )} + 1}\right )}{b c^{2 \, b} n} - \frac {c^{b} e^{\left (-a\right )} \log \left (\sqrt {-2 \, x^{b n} {\left | c \right |}^{b} \cos \left (-\frac {1}{2} \, \pi b \mathrm {sgn}\relax (c) + \frac {1}{2} \, \pi b\right ) e^{a} + x^{2 \, b n} {\left | c \right |}^{2 \, b} e^{\left (2 \, a\right )} + 1}\right )}{b c^{2 \, b} n}\right )} e^{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 23, normalized size = 1.15 \[ \frac {\ln \left (\tanh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}{n b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 22, normalized size = 1.10 \[ \frac {\log \left (\tanh \left (\frac {1}{2} \, b \log \left (c x^{n}\right ) + \frac {1}{2} \, a\right )\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.62, size = 43, normalized size = 2.15 \[ -\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,\sqrt {-b^2\,n^2}}{b\,n\,{\left (c\,x^n\right )}^b}\right )}{\sqrt {-b^2\,n^2}} \]
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 \[ \int \frac {\operatorname {csch}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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