Integrand size = 15, antiderivative size = 20 \[ \int \frac {\text {csch}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\text {arctanh}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
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Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3855} \[ \int \frac {\text {csch}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\text {arctanh}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
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Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \text {csch}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {\text {arctanh}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(54\) vs. \(2(20)=40\).
Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.70 \[ \int \frac {\text {csch}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\log \left (\cosh \left (\frac {a}{2}+\frac {1}{2} b \log \left (c x^n\right )\right )\right )}{b n}+\frac {\log \left (\sinh \left (\frac {a}{2}+\frac {1}{2} b \log \left (c x^n\right )\right )\right )}{b n} \]
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Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15
method | result | size |
derivativedivides | \(\frac {\ln \left (\tanh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}{n b}\) | \(23\) |
default | \(\frac {\ln \left (\tanh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}{n b}\) | \(23\) |
parallelrisch | \(\frac {\ln \left (\tanh \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )\right )}{n b}\) | \(24\) |
risch | \(\frac {\ln \left (c^{b} \left (x^{n}\right )^{b} {\mathrm e}^{a} {\mathrm e}^{\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}}-1\right )}{b n}-\frac {\ln \left (c^{b} \left (x^{n}\right )^{b} {\mathrm e}^{a} {\mathrm e}^{\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}}+1\right )}{b n}\) | \(217\) |
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 65, normalized size of antiderivative = 3.25 \[ \int \frac {\text {csch}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\log \left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 1\right ) - \log \left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - 1\right )}{b n} \]
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Time = 0.44 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.10 \[ \int \frac {\text {csch}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=- \begin {cases} - \log {\left (x \right )} \operatorname {csch}{\left (a \right )} & \text {for}\: b = 0 \\- \log {\left (x \right )} \operatorname {csch}{\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\- \frac {\log {\left (\tanh {\left (\frac {a}{2} + \frac {b \log {\left (c x^{n} \right )}}{2} \right )} \right )}}{b n} & \text {otherwise} \end {cases} \]
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none
Time = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\text {csch}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\log \left (\tanh \left (\frac {1}{2} \, b \log \left (c x^{n}\right ) + \frac {1}{2} \, a\right )\right )}{b n} \]
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Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (20) = 40\).
Time = 0.31 (sec) , antiderivative size = 145, normalized size of antiderivative = 7.25 \[ \int \frac {\text {csch}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-c^{b} {\left (\frac {c^{b} e^{\left (-a\right )} \log \left (\sqrt {2 \, x^{b n} {\left | c \right |}^{b} \cos \left (-\frac {1}{2} \, \pi b \mathrm {sgn}\left (c\right ) + \frac {1}{2} \, \pi b\right ) e^{a} + x^{2 \, b n} {\left | c \right |}^{2 \, b} e^{\left (2 \, a\right )} + 1}\right )}{b c^{2 \, b} n} - \frac {c^{b} e^{\left (-a\right )} \log \left (\sqrt {-2 \, x^{b n} {\left | c \right |}^{b} \cos \left (-\frac {1}{2} \, \pi b \mathrm {sgn}\left (c\right ) + \frac {1}{2} \, \pi b\right ) e^{a} + x^{2 \, b n} {\left | c \right |}^{2 \, b} e^{\left (2 \, a\right )} + 1}\right )}{b c^{2 \, b} n}\right )} e^{a} \]
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Time = 2.41 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.15 \[ \int \frac {\text {csch}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,\sqrt {-b^2\,n^2}}{b\,n\,{\left (c\,x^n\right )}^b}\right )}{\sqrt {-b^2\,n^2}} \]
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