Integrand size = 17, antiderivative size = 89 \[ \int \frac {\text {csch}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {3 \text {arctanh}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}+\frac {3 \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{8 b n}-\frac {\coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{4 b n} \]
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Time = 0.05 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3853, 3855} \[ \int \frac {\text {csch}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {3 \text {arctanh}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}-\frac {\coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{8 b n} \]
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Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\text {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 \text {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 \text {Subst}\left (\int \text {csch}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{8 n} \\ & = -\frac {3 \text {arctanh}\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*}
Time = 0.05 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.81 \[ \int \frac {\text {csch}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3 \text {csch}^2\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{32 b n}-\frac {\text {csch}^4\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{64 b n}-\frac {3 \log \left (\cosh \left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )\right )}{8 b n}+\frac {3 \log \left (\sinh \left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )\right )}{8 b n}+\frac {3 \text {sech}^2\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{32 b n}+\frac {\text {sech}^4\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{64 b n} \]
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Time = 21.53 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.72
method | result | size |
derivativedivides | \(\frac {\left (-\frac {{\operatorname {csch}\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}}{4}+\frac {3 \,\operatorname {csch}\left (a +b \ln \left (c \,x^{n}\right )\right )}{8}\right ) \coth \left (a +b \ln \left (c \,x^{n}\right )\right )-\frac {3 \,\operatorname {arctanh}\left ({\mathrm e}^{a +b \ln \left (c \,x^{n}\right )}\right )}{4}}{n b}\) | \(64\) |
default | \(\frac {\left (-\frac {{\operatorname {csch}\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}}{4}+\frac {3 \,\operatorname {csch}\left (a +b \ln \left (c \,x^{n}\right )\right )}{8}\right ) \coth \left (a +b \ln \left (c \,x^{n}\right )\right )-\frac {3 \,\operatorname {arctanh}\left ({\mathrm e}^{a +b \ln \left (c \,x^{n}\right )}\right )}{4}}{n b}\) | \(64\) |
parallelrisch | \(\frac {-{\coth \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{4}+{\tanh \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{4}-8 {\tanh \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{2}+24 \ln \left (\tanh \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )\right )+8 {\coth \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{2}}{64 b n}\) | \(102\) |
risch | \(\text {Expression too large to display}\) | \(744\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1806 vs. \(2 (83) = 166\).
Time = 0.27 (sec) , antiderivative size = 1806, normalized size of antiderivative = 20.29 \[ \int \frac {\text {csch}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \]
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\[ \int \frac {\text {csch}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\operatorname {csch}^{5}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (83) = 166\).
Time = 0.21 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.61 \[ \int \frac {\text {csch}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3 \, c^{7 \, b} e^{\left (7 \, b \log \left (x^{n}\right ) + 7 \, a\right )} - 11 \, c^{5 \, b} e^{\left (5 \, b \log \left (x^{n}\right ) + 5 \, a\right )} - 11 \, c^{3 \, b} e^{\left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right )} + 3 \, c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )}}{4 \, {\left (b c^{8 \, b} n e^{\left (8 \, b \log \left (x^{n}\right ) + 8 \, a\right )} - 4 \, b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} + 6 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 4 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} - \frac {3 \, \log \left (\frac {{\left (c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )} + 1\right )} e^{\left (-a\right )}}{c^{b}}\right )}{8 \, b n} + \frac {3 \, \log \left (\frac {{\left (c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )} - 1\right )} e^{\left (-a\right )}}{c^{b}}\right )}{8 \, b n} \]
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Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (83) = 166\).
Time = 0.35 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.79 \[ \int \frac {\text {csch}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {1}{8} \, c^{5 \, b} {\left (\frac {3 \, c^{b} e^{\left (-5 \, a\right )} \log \left (\sqrt {2 \, x^{b n} {\left | c \right |}^{b} \cos \left (-\frac {1}{2} \, \pi b \mathrm {sgn}\left (c\right ) + \frac {1}{2} \, \pi b\right ) e^{a} + x^{2 \, b n} {\left | c \right |}^{2 \, b} e^{\left (2 \, a\right )} + 1}\right )}{b c^{6 \, b} n} - \frac {3 \, c^{b} e^{\left (-5 \, a\right )} \log \left (\sqrt {-2 \, x^{b n} {\left | c \right |}^{b} \cos \left (-\frac {1}{2} \, \pi b \mathrm {sgn}\left (c\right ) + \frac {1}{2} \, \pi b\right ) e^{a} + x^{2 \, b n} {\left | c \right |}^{2 \, b} e^{\left (2 \, a\right )} + 1}\right )}{b c^{6 \, b} n} - \frac {2 \, {\left (3 \, c^{6 \, b} x^{7 \, b n} e^{\left (6 \, a\right )} - 11 \, c^{4 \, b} x^{5 \, b n} e^{\left (4 \, a\right )} - 11 \, c^{2 \, b} x^{3 \, b n} e^{\left (2 \, a\right )} + 3 \, x^{b n}\right )} e^{\left (-4 \, a\right )}}{{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )}^{4} b c^{4 \, b} n}\right )} e^{\left (5 \, a\right )} \]
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Time = 2.23 (sec) , antiderivative size = 318, normalized size of antiderivative = 3.57 \[ \int \frac {\text {csch}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2\,{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (b\,n-\frac {3\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {3\,b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}-\frac {b\,n\,{\mathrm {e}}^{-6\,a}}{{\left (c\,x^n\right )}^{6\,b}}\right )}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,\sqrt {-b^2\,n^2}}{b\,n\,{\left (c\,x^n\right )}^b}\right )}{4\,\sqrt {-b^2\,n^2}}-\frac {3\,{\mathrm {e}}^{-a}}{4\,{\left (c\,x^n\right )}^b\,\left (b\,n-\frac {b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}\right )}-\frac {4\,{\mathrm {e}}^{-3\,a}}{{\left (c\,x^n\right )}^{3\,b}\,\left (b\,n-\frac {4\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {6\,b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}-\frac {4\,b\,n\,{\mathrm {e}}^{-6\,a}}{{\left (c\,x^n\right )}^{6\,b}}+\frac {b\,n\,{\mathrm {e}}^{-8\,a}}{{\left (c\,x^n\right )}^{8\,b}}\right )}-\frac {{\mathrm {e}}^{-a}}{2\,{\left (c\,x^n\right )}^b\,\left (b\,n-\frac {2\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}\right )} \]
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