Integrand size = 11, antiderivative size = 25 \[ \int \frac {\text {sech}^6(x)}{1+\tanh (x)} \, dx=-\frac {2}{3} (1-\tanh (x))^3+\frac {1}{4} (1-\tanh (x))^4 \]
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Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3568, 45} \[ \int \frac {\text {sech}^6(x)}{1+\tanh (x)} \, dx=\frac {1}{4} (1-\tanh (x))^4-\frac {2}{3} (1-\tanh (x))^3 \]
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Rule 45
Rule 3568
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int (1-x)^2 (1+x) \, dx,x,\tanh (x)\right ) \\ & = \text {Subst}\left (\int \left (2 (1-x)^2-(1-x)^3\right ) \, dx,x,\tanh (x)\right ) \\ & = -\frac {2}{3} (1-\tanh (x))^3+\frac {1}{4} (1-\tanh (x))^4 \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {\text {sech}^6(x)}{1+\tanh (x)} \, dx=\frac {1}{12} \tanh (x) \left (12-6 \tanh (x)-4 \tanh ^2(x)+3 \tanh ^3(x)\right ) \]
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Time = 4.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76
method | result | size |
risch | \(-\frac {4 \left (4 \,{\mathrm e}^{2 x}+1\right )}{3 \left (1+{\mathrm e}^{2 x}\right )^{4}}\) | \(19\) |
parallelrisch | \(-\frac {39}{100}+\frac {2 \tanh \left (x \right )}{3}+\frac {\tanh \left (x \right ) \operatorname {sech}\left (x \right )^{2}}{3}+\frac {\operatorname {sech}\left (x \right )^{4}}{4}\) | \(21\) |
derivativedivides | \(\frac {\tanh \left (x \right )^{4}}{4}-\frac {\tanh \left (x \right )^{3}}{3}-\frac {\tanh \left (x \right )^{2}}{2}+\tanh \left (x \right )\) | \(22\) |
default | \(\frac {\tanh \left (x \right )^{4}}{4}-\frac {\tanh \left (x \right )^{3}}{3}-\frac {\tanh \left (x \right )^{2}}{2}+\tanh \left (x \right )\) | \(22\) |
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Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (17) = 34\).
Time = 0.25 (sec) , antiderivative size = 140, normalized size of antiderivative = 5.60 \[ \int \frac {\text {sech}^6(x)}{1+\tanh (x)} \, dx=-\frac {4 \, {\left (5 \, \cosh \left (x\right ) + 3 \, \sinh \left (x\right )\right )}}{3 \, {\left (\cosh \left (x\right )^{7} + 7 \, \cosh \left (x\right ) \sinh \left (x\right )^{6} + \sinh \left (x\right )^{7} + {\left (21 \, \cosh \left (x\right )^{2} + 4\right )} \sinh \left (x\right )^{5} + 4 \, \cosh \left (x\right )^{5} + 5 \, {\left (7 \, \cosh \left (x\right )^{3} + 4 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} + {\left (35 \, \cosh \left (x\right )^{4} + 40 \, \cosh \left (x\right )^{2} + 6\right )} \sinh \left (x\right )^{3} + 6 \, \cosh \left (x\right )^{3} + {\left (21 \, \cosh \left (x\right )^{5} + 40 \, \cosh \left (x\right )^{3} + 18 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + {\left (7 \, \cosh \left (x\right )^{6} + 20 \, \cosh \left (x\right )^{4} + 18 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right ) + 5 \, \cosh \left (x\right )\right )}} \]
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\[ \int \frac {\text {sech}^6(x)}{1+\tanh (x)} \, dx=\int \frac {\operatorname {sech}^{6}{\left (x \right )}}{\tanh {\left (x \right )} + 1}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (17) = 34\).
Time = 0.19 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.72 \[ \int \frac {\text {sech}^6(x)}{1+\tanh (x)} \, dx=\frac {16 \, e^{\left (-2 \, x\right )}}{3 \, {\left (4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} + \frac {8 \, e^{\left (-4 \, x\right )}}{4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1} + \frac {4}{3 \, {\left (4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} \]
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none
Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \frac {\text {sech}^6(x)}{1+\tanh (x)} \, dx=-\frac {4 \, {\left (4 \, e^{\left (2 \, x\right )} + 1\right )}}{3 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{4}} \]
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Time = 0.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \frac {\text {sech}^6(x)}{1+\tanh (x)} \, dx=-\frac {4\,\left (4\,{\mathrm {e}}^{2\,x}+1\right )}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^4} \]
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