Integrand size = 11, antiderivative size = 34 \[ \int \frac {\text {sech}^7(x)}{1+\tanh (x)} \, dx=\frac {3}{8} \arctan (\sinh (x))+\frac {\text {sech}^5(x)}{5}+\frac {3}{8} \text {sech}(x) \tanh (x)+\frac {1}{4} \text {sech}^3(x) \tanh (x) \]
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Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3582, 3853, 3855} \[ \int \frac {\text {sech}^7(x)}{1+\tanh (x)} \, dx=\frac {3}{8} \arctan (\sinh (x))+\frac {\text {sech}^5(x)}{5}+\frac {1}{4} \tanh (x) \text {sech}^3(x)+\frac {3}{8} \tanh (x) \text {sech}(x) \]
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Rule 3582
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\text {sech}^5(x)}{5}+\int \text {sech}^5(x) \, dx \\ & = \frac {\text {sech}^5(x)}{5}+\frac {1}{4} \text {sech}^3(x) \tanh (x)+\frac {3}{4} \int \text {sech}^3(x) \, dx \\ & = \frac {\text {sech}^5(x)}{5}+\frac {3}{8} \text {sech}(x) \tanh (x)+\frac {1}{4} \text {sech}^3(x) \tanh (x)+\frac {3}{8} \int \text {sech}(x) \, dx \\ & = \frac {3}{8} \arctan (\sinh (x))+\frac {\text {sech}^5(x)}{5}+\frac {3}{8} \text {sech}(x) \tanh (x)+\frac {1}{4} \text {sech}^3(x) \tanh (x) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {\text {sech}^7(x)}{1+\tanh (x)} \, dx=\frac {1}{40} \left (30 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )+8 \text {sech}^5(x)+15 \text {sech}(x) \tanh (x)+10 \text {sech}^3(x) \tanh (x)\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(66\) vs. \(2(26)=52\).
Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.97
\[\frac {-\frac {5 \tanh \left (\frac {x}{2}\right )^{9}}{4}+2 \tanh \left (\frac {x}{2}\right )^{8}-\frac {\tanh \left (\frac {x}{2}\right )^{7}}{2}+4 \tanh \left (\frac {x}{2}\right )^{4}+\frac {\tanh \left (\frac {x}{2}\right )^{3}}{2}+\frac {5 \tanh \left (\frac {x}{2}\right )}{4}+\frac {2}{5}}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{5}}+\frac {3 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{4}\]
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Leaf count of result is larger than twice the leaf count of optimal. 670 vs. \(2 (26) = 52\).
Time = 0.26 (sec) , antiderivative size = 670, normalized size of antiderivative = 19.71 \[ \int \frac {\text {sech}^7(x)}{1+\tanh (x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {\text {sech}^7(x)}{1+\tanh (x)} \, dx=\int \frac {\operatorname {sech}^{7}{\left (x \right )}}{\tanh {\left (x \right )} + 1}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (26) = 52\).
Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.15 \[ \int \frac {\text {sech}^7(x)}{1+\tanh (x)} \, dx=\frac {15 \, e^{\left (-x\right )} + 70 \, e^{\left (-3 \, x\right )} + 128 \, e^{\left (-5 \, x\right )} - 70 \, e^{\left (-7 \, x\right )} - 15 \, e^{\left (-9 \, x\right )}}{20 \, {\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} - \frac {3}{4} \, \arctan \left (e^{\left (-x\right )}\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.32 \[ \int \frac {\text {sech}^7(x)}{1+\tanh (x)} \, dx=\frac {15 \, e^{\left (9 \, x\right )} + 70 \, e^{\left (7 \, x\right )} + 128 \, e^{\left (5 \, x\right )} - 70 \, e^{\left (3 \, x\right )} - 15 \, e^{x}}{20 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{5}} + \frac {3}{4} \, \arctan \left (e^{x}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 137, normalized size of antiderivative = 4.03 \[ \int \frac {\text {sech}^7(x)}{1+\tanh (x)} \, dx=\frac {3\,\mathrm {atan}\left ({\mathrm {e}}^x\right )}{4}-\frac {32\,{\mathrm {e}}^{3\,x}}{5\,\left (5\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}+5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}+1\right )}-\frac {12\,{\mathrm {e}}^x}{5\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}+\frac {3\,{\mathrm {e}}^x}{4\,\left ({\mathrm {e}}^{2\,x}+1\right )}+\frac {2\,{\mathrm {e}}^x}{5\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )}+\frac {{\mathrm {e}}^x}{2\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )} \]
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