Integrand size = 17, antiderivative size = 43 \[ \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
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Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3554, 3556} \[ \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
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Rule 3554
Rule 3556
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \tanh ^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {\tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\text {Subst}\left (\int \tanh (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
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Time = 0.17 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.09
method | result | size |
parallelrisch | \(-\frac {2 \ln \left (x \right ) b n +{\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}+2 \ln \left (1-\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{2 b n}\) | \(47\) |
derivativedivides | \(\frac {-\frac {{\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{2}-\frac {\ln \left (\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )-1\right )}{2}-\frac {\ln \left (\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2}}{n b}\) | \(56\) |
default | \(\frac {-\frac {{\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{2}-\frac {\ln \left (\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )-1\right )}{2}-\frac {\ln \left (\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2}}{n b}\) | \(56\) |
risch | \(\ln \left (x \right )-\frac {2 a}{b n}-\frac {2 \ln \left (c \right )}{n}-\frac {2 \ln \left (x^{n}\right )}{n}-\frac {i \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{n}+\frac {i \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{n}+\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{n}-\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{n}+\frac {2 \left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{2 a} {\mathrm e}^{i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}}{b n {\left (\left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{2 a} {\mathrm e}^{i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}+1\right )}^{2}}+\frac {\ln \left (\left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{2 a} {\mathrm e}^{i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}+1\right )}{b n}\) | \(449\) |
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Leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (41) = 82\).
Time = 0.26 (sec) , antiderivative size = 566, normalized size of antiderivative = 13.16 \[ \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} \log \left (x\right ) + 4 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \log \left (x\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + b n \log \left (x\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 2 \, {\left (b n \log \left (x\right ) - 1\right )} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + b n \log \left (x\right ) + 2 \, {\left (3 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} \log \left (x\right ) + b n \log \left (x\right ) - 1\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 4 \, \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 )^{3} + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 1\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 4 \, {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 1\right )} \log \left (\frac {2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\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 )}\right ) + 4 \, {\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} \log \left (x\right ) + {\left (b n \log \left (x\right ) - 1\right )} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 4 \, b n \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 )^{3} + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 2 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, {\left (3 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + b n\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + b n + 4 \, {\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \]
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Time = 0.75 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.58 \[ \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\begin {cases} \log {\left (x \right )} \tanh ^{3}{\left (a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \tanh ^{3}{\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\frac {\log {\left (c x^{n} \right )}}{n} - \frac {\log {\left (\tanh {\left (a + b \log {\left (c x^{n} \right )} \right )} + 1 \right )}}{b n} - \frac {\tanh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{2 b n} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (41) = 82\).
Time = 0.27 (sec) , antiderivative size = 304, normalized size of antiderivative = 7.07 \[ \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {4 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 3}{4 \, {\left (b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} - \frac {2 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 3}{4 \, {\left (b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} + \frac {3 \, {\left (2 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 1\right )}}{4 \, {\left (b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} - \frac {3}{4 \, {\left (b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} + \frac {\log \left (\frac {{\left (c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 1\right )} e^{\left (-2 \, a\right )}}{c^{2 \, b}}\right )}{b n} - \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (41) = 82\).
Time = 0.35 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.95 \[ \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\log \left (\sqrt {2 \, x^{2 \, b n} {\left | c \right |}^{2 \, b} \cos \left (\pi b \mathrm {sgn}\left (c\right ) - \pi b\right ) e^{\left (2 \, a\right )} + x^{4 \, b n} {\left | c \right |}^{4 \, b} e^{\left (4 \, a\right )} + 1}\right )}{b n} - \frac {3 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} + 2 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 3}{2 \, {\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1\right )}^{2} b n} - \log \left (x\right ) \]
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Time = 1.83 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.19 \[ \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2}{b\,n+b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}}-\ln \left (x\right )-\frac {2}{b\,n+2\,b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+b\,n\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}}+\frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+1\right )}{b\,n} \]
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