Optimal. Leaf size=55 \[ \frac {\tanh ^{-1}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}-\frac {\coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
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Rubi [A] time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3768, 3770} \[ \frac {\tanh ^{-1}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}-\frac {\coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \text {csch}^3(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}\left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac {\operatorname {Subst}\left (\int \text {csch}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{2 n}\\ &=\frac {\tanh ^{-1}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}-\frac {\coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{2 b n}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 81, normalized size = 1.47 \[ -\frac {\log \left (\tanh \left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )\right )}{2 b n}-\frac {\text {sech}^2\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}-\frac {\text {csch}^2\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.35, size = 643, normalized size = 11.69 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 210, normalized size = 3.82 \[ \frac {1}{2} \, c^{3 \, b} {\left (\frac {c^{b} e^{\left (-3 \, a\right )} \log \left (\sqrt {2 \, x^{b n} {\left | c \right |}^{b} \cos \left (-\frac {1}{2} \, \pi b \mathrm {sgn}\relax (c) + \frac {1}{2} \, \pi b\right ) e^{a} + x^{2 \, b n} {\left | c \right |}^{2 \, b} e^{\left (2 \, a\right )} + 1}\right )}{b c^{4 \, b} n} - \frac {c^{b} e^{\left (-3 \, a\right )} \log \left (\sqrt {-2 \, x^{b n} {\left | c \right |}^{b} \cos \left (-\frac {1}{2} \, \pi b \mathrm {sgn}\relax (c) + \frac {1}{2} \, \pi b\right ) e^{a} + x^{2 \, b n} {\left | c \right |}^{2 \, b} e^{\left (2 \, a\right )} + 1}\right )}{b c^{4 \, b} n} - \frac {2 \, {\left (c^{2 \, b} x^{3 \, b n} e^{\left (2 \, a\right )} + x^{b n}\right )} e^{\left (-2 \, a\right )}}{{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )}^{2} b c^{2 \, b} n}\right )} e^{\left (3 \, a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 51, normalized size = 0.93 \[ -\frac {\coth \left (a +b \ln \left (c \,x^{n}\right )\right ) \mathrm {csch}\left (a +b \ln \left (c \,x^{n}\right )\right )}{2 b n}+\frac {\arctanh \left ({\mathrm e}^{a +b \ln \left (c \,x^{n}\right )}\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 150, normalized size = 2.73 \[ -\frac {c^{3 \, b} e^{\left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right )} + c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )}}{b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n} + \frac {\log \left (\frac {{\left (c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )} + 1\right )} e^{\left (-a\right )}}{c^{b}}\right )}{2 \, b n} - \frac {\log \left (\frac {{\left (c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )} - 1\right )} e^{\left (-a\right )}}{c^{b}}\right )}{2 \, b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.49, size = 140, normalized size = 2.55 \[ \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,\sqrt {-b^2\,n^2}}{b\,n\,{\left (c\,x^n\right )}^b}\right )}{\sqrt {-b^2\,n^2}}+\frac {{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (b\,n-\frac {b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}\right )}-\frac {2\,{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (b\,n-\frac {2\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}\right )} \]
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 \[ \int \frac {\operatorname {csch}^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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