Optimal. Leaf size=34 \[ \frac {\text {sech}^5(x)}{5}+\frac {3}{8} \tan ^{-1}(\sinh (x))+\frac {1}{4} \tanh (x) \text {sech}^3(x)+\frac {3}{8} \tanh (x) \text {sech}(x) \]
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Rubi [A] time = 0.05, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3501, 3768, 3770} \[ \frac {\text {sech}^5(x)}{5}+\frac {3}{8} \tan ^{-1}(\sinh (x))+\frac {1}{4} \tanh (x) \text {sech}^3(x)+\frac {3}{8} \tanh (x) \text {sech}(x) \]
Antiderivative was successfully verified.
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Rule 3501
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {sech}^7(x)}{1+\tanh (x)} \, dx &=\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} \tan ^{-1}(\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*}
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Mathematica [A] time = 0.04, size = 34, normalized size = 1.00 \[ \frac {1}{40} \left (8 \text {sech}^5(x)+30 \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )+10 \tanh (x) \text {sech}^3(x)+15 \tanh (x) \text {sech}(x)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 670, normalized size = 19.71 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 45, normalized size = 1.32 \[ \frac {15 \, e^{\left (9 \, x\right )} + 70 \, e^{\left (7 \, x\right )} + 128 \, e^{\left (5 \, x\right )} - 70 \, e^{\left (3 \, x\right )} - 15 \, e^{x}}{20 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{5}} + \frac {3}{4} \, \arctan \left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 67, normalized size = 1.97 \[ \frac {-\frac {5 \left (\tanh ^{9}\left (\frac {x}{2}\right )\right )}{4}+2 \left (\tanh ^{8}\left (\frac {x}{2}\right )\right )-\frac {\left (\tanh ^{7}\left (\frac {x}{2}\right )\right )}{2}+4 \left (\tanh ^{4}\left (\frac {x}{2}\right )\right )+\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{2}+\frac {5 \tanh \left (\frac {x}{2}\right )}{4}+\frac {2}{5}}{\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{5}}+\frac {3 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 73, normalized size = 2.15 \[ \frac {15 \, e^{\left (-x\right )} + 70 \, e^{\left (-3 \, x\right )} + 128 \, e^{\left (-5 \, x\right )} - 70 \, e^{\left (-7 \, x\right )} - 15 \, e^{\left (-9 \, x\right )}}{20 \, {\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} - \frac {3}{4} \, \arctan \left (e^{\left (-x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 137, normalized size = 4.03 \[ \frac {3\,\mathrm {atan}\left ({\mathrm {e}}^x\right )}{4}-\frac {32\,{\mathrm {e}}^{3\,x}}{5\,\left (5\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}+5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}+1\right )}-\frac {12\,{\mathrm {e}}^x}{5\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}+\frac {3\,{\mathrm {e}}^x}{4\,\left ({\mathrm {e}}^{2\,x}+1\right )}+\frac {2\,{\mathrm {e}}^x}{5\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )}+\frac {{\mathrm {e}}^x}{2\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}^{7}{\relax (x )}}{\tanh {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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