Optimal. Leaf size=25 \[ \frac {1}{4} (1-\tanh (x))^4-\frac {2}{3} (1-\tanh (x))^3 \]
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Rubi [A] time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3487, 43} \[ \frac {1}{4} (1-\tanh (x))^4-\frac {2}{3} (1-\tanh (x))^3 \]
Antiderivative was successfully verified.
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Rule 43
Rule 3487
Rubi steps
\begin {align*} \int \frac {\text {sech}^6(x)}{1+\tanh (x)} \, dx &=\operatorname {Subst}\left (\int (1-x)^2 (1+x) \, dx,x,\tanh (x)\right )\\ &=\operatorname {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*}
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Mathematica [A] time = 0.03, size = 20, normalized size = 0.80 \[ \frac {1}{12} (4 \sinh (2 x)+\sinh (4 x)+3) \text {sech}^4(x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 140, normalized size = 5.60 \[ -\frac {4 \, {\left (5 \, \cosh \relax (x) + 3 \, \sinh \relax (x)\right )}}{3 \, {\left (\cosh \relax (x)^{7} + 7 \, \cosh \relax (x) \sinh \relax (x)^{6} + \sinh \relax (x)^{7} + {\left (21 \, \cosh \relax (x)^{2} + 4\right )} \sinh \relax (x)^{5} + 4 \, \cosh \relax (x)^{5} + 5 \, {\left (7 \, \cosh \relax (x)^{3} + 4 \, \cosh \relax (x)\right )} \sinh \relax (x)^{4} + {\left (35 \, \cosh \relax (x)^{4} + 40 \, \cosh \relax (x)^{2} + 6\right )} \sinh \relax (x)^{3} + 6 \, \cosh \relax (x)^{3} + {\left (21 \, \cosh \relax (x)^{5} + 40 \, \cosh \relax (x)^{3} + 18 \, \cosh \relax (x)\right )} \sinh \relax (x)^{2} + {\left (7 \, \cosh \relax (x)^{6} + 20 \, \cosh \relax (x)^{4} + 18 \, \cosh \relax (x)^{2} + 3\right )} \sinh \relax (x) + 5 \, \cosh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 18, normalized size = 0.72 \[ -\frac {4 \, {\left (4 \, e^{\left (2 \, x\right )} + 1\right )}}{3 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 56, normalized size = 2.24 \[ -\frac {2 \left (-\left (\tanh ^{7}\left (\frac {x}{2}\right )\right )+\tanh ^{6}\left (\frac {x}{2}\right )-\frac {5 \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{3}-\frac {5 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3}+\tanh ^{2}\left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right )\right )}{\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 93, normalized size = 3.72 \[ \frac {16 \, e^{\left (-2 \, x\right )}}{3 \, {\left (4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} + \frac {8 \, e^{\left (-4 \, x\right )}}{4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1} + \frac {4}{3 \, {\left (4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 18, normalized size = 0.72 \[ -\frac {4\,\left (4\,{\mathrm {e}}^{2\,x}+1\right )}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^4} \]
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 \[ \int \frac {\operatorname {sech}^{6}{\relax (x )}}{\tanh {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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