Integrand size = 7, antiderivative size = 18 \[ \int \sinh ^4(x) \tanh (x) \, dx=-\cosh ^2(x)+\frac {\cosh ^4(x)}{4}+\log (\cosh (x)) \]
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Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2670, 272, 45} \[ \int \sinh ^4(x) \tanh (x) \, dx=\frac {\cosh ^4(x)}{4}-\cosh ^2(x)+\log (\cosh (x)) \]
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Rule 45
Rule 272
Rule 2670
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x} \, dx,x,\cosh (x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {(1-x)^2}{x} \, dx,x,\cosh ^2(x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-2+\frac {1}{x}+x\right ) \, dx,x,\cosh ^2(x)\right ) \\ & = -\cosh ^2(x)+\frac {\cosh ^4(x)}{4}+\log (\cosh (x)) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \sinh ^4(x) \tanh (x) \, dx=-\cosh ^2(x)+\frac {\cosh ^4(x)}{4}+\log (\cosh (x)) \]
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Time = 6.59 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\left (\sinh ^{4}\left (x \right )\right )}{4}-\frac {\left (\sinh ^{2}\left (x \right )\right )}{2}+\ln \left (\cosh \left (x \right )\right )\) | \(17\) |
risch | \(-x +\frac {{\mathrm e}^{4 x}}{64}-\frac {3 \,{\mathrm e}^{2 x}}{16}-\frac {3 \,{\mathrm e}^{-2 x}}{16}+\frac {{\mathrm e}^{-4 x}}{64}+\ln \left (1+{\mathrm e}^{2 x}\right )\) | \(36\) |
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Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (16) = 32\).
Time = 0.25 (sec) , antiderivative size = 257, normalized size of antiderivative = 14.28 \[ \int \sinh ^4(x) \tanh (x) \, dx=\frac {\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{6} - 12 \, \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} - 64 \, x \cosh \left (x\right )^{4} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} - 90 \, \cosh \left (x\right )^{2} - 32 \, x\right )} \sinh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} - 30 \, \cosh \left (x\right )^{3} - 32 \, x \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} - 45 \, \cosh \left (x\right )^{4} - 96 \, x \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{2} - 12 \, \cosh \left (x\right )^{2} + 64 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}\right )} \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 8 \, {\left (\cosh \left (x\right )^{7} - 9 \, \cosh \left (x\right )^{5} - 32 \, x \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}{64 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}\right )}} \]
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\[ \int \sinh ^4(x) \tanh (x) \, dx=\int \frac {\tanh ^{5}{\left (x \right )}}{\operatorname {sech}^{4}{\left (x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (16) = 32\).
Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.94 \[ \int \sinh ^4(x) \tanh (x) \, dx=-\frac {1}{64} \, {\left (12 \, e^{\left (-2 \, x\right )} - 1\right )} e^{\left (4 \, x\right )} + x - \frac {3}{16} \, e^{\left (-2 \, x\right )} + \frac {1}{64} \, e^{\left (-4 \, x\right )} + \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (16) = 32\).
Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.39 \[ \int \sinh ^4(x) \tanh (x) \, dx=\frac {1}{64} \, {\left (48 \, e^{\left (4 \, x\right )} - 12 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-4 \, x\right )} - x + \frac {1}{64} \, e^{\left (4 \, x\right )} - \frac {3}{16} \, e^{\left (2 \, x\right )} + \log \left (e^{\left (2 \, x\right )} + 1\right ) \]
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Time = 0.44 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.94 \[ \int \sinh ^4(x) \tanh (x) \, dx=\ln \left ({\mathrm {e}}^{2\,x}+1\right )-x-\frac {3\,{\mathrm {e}}^{-2\,x}}{16}-\frac {3\,{\mathrm {e}}^{2\,x}}{16}+\frac {{\mathrm {e}}^{-4\,x}}{64}+\frac {{\mathrm {e}}^{4\,x}}{64} \]
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