Optimal. Leaf size=89 \[ -\frac{3 \tanh ^{-1}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}-\frac{\coth \left (a+b \log \left (c x^n\right )\right ) \text{csch}^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac{3 \coth \left (a+b \log \left (c x^n\right )\right ) \text{csch}\left (a+b \log \left (c x^n\right )\right )}{8 b n} \]
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Rubi [A] time = 0.0702584, antiderivative size = 89, normalized size of antiderivative = 1., 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} \[ -\frac{3 \tanh ^{-1}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}-\frac{\coth \left (a+b \log \left (c x^n\right )\right ) \text{csch}^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac{3 \coth \left (a+b \log \left (c x^n\right )\right ) \text{csch}\left (a+b \log \left (c x^n\right )\right )}{8 b n} \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \frac{\text{csch}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \text{csch}^5(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\coth \left (a+b \log \left (c x^n\right )\right ) \text{csch}^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}-\frac{3 \operatorname{Subst}\left (\int \text{csch}^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{4 n}\\ &=\frac{3 \coth \left (a+b \log \left (c x^n\right )\right ) \text{csch}\left (a+b \log \left (c x^n\right )\right )}{8 b n}-\frac{\coth \left (a+b \log \left (c x^n\right )\right ) \text{csch}^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac{3 \operatorname{Subst}\left (\int \text{csch}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{8 n}\\ &=-\frac{3 \tanh ^{-1}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}+\frac{3 \coth \left (a+b \log \left (c x^n\right )\right ) \text{csch}\left (a+b \log \left (c x^n\right )\right )}{8 b n}-\frac{\coth \left (a+b \log \left (c x^n\right )\right ) \text{csch}^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}\\ \end{align*}
Mathematica [A] time = 0.0581078, size = 135, normalized size = 1.52 \[ \frac{3 \log \left (\tanh \left (\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right )\right )}{8 b n}+\frac{\text{sech}^4\left (\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{64 b n}+\frac{3 \text{sech}^2\left (\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{32 b n}-\frac{\text{csch}^4\left (\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{64 b n}+\frac{3 \text{csch}^2\left (\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{32 b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 84, normalized size = 0.9 \begin{align*} -{\frac{{\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \right ) \left ({\rm csch} \left (a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{3}}{4\,bn}}+{\frac{3\,{\rm csch} \left (a+b\ln \left ( c{x}^{n} \right ) \right ){\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \right )}{8\,bn}}-{\frac{3\,{\it Artanh} \left ({{\rm e}^{a+b\ln \left ( c{x}^{n} \right ) }} \right ) }{4\,bn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.26848, size = 313, normalized size = 3.52 \begin{align*} \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 )}} - \frac{3 \, \log \left (\frac{{\left (c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )} + 1\right )} e^{\left (-a\right )}}{c^{b}}\right )}{8 \, b n} + \frac{3 \, \log \left (\frac{{\left (c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )} - 1\right )} e^{\left (-a\right )}}{c^{b}}\right )}{8 \, b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.23032, size = 5852, normalized size = 65.75 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}^{5}{\left (a + b \log{\left (c x^{n} \right )} \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23348, size = 329, normalized size = 3.7 \begin{align*} -\frac{1}{16} \, c^{5 \, b}{\left (\frac{3 \, c^{b} e^{\left (-5 \, a\right )} \log \left (2 \, x^{b n}{\left | c \right |}^{b} \cos \left (-\frac{1}{2} \, \pi b \mathrm{sgn}\left (c\right ) + \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^{6 \, b} n} - \frac{3 \, c^{b} e^{\left (-5 \, a\right )} \log \left (-2 \, x^{b n}{\left | c \right |}^{b} \cos \left (-\frac{1}{2} \, \pi b \mathrm{sgn}\left (c\right ) + \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^{6 \, b} n} - \frac{4 \,{\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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