Optimal. Leaf size=24 \[ \frac {1}{2} \text {ArcTan}(\sinh (x))+\frac {\text {sech}^3(x)}{3}+\frac {1}{2} \text {sech}(x) \tanh (x) \]
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Rubi [A]
time = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3582, 3853,
3855} \begin {gather*} \frac {1}{2} \text {ArcTan}(\sinh (x))+\frac {\text {sech}^3(x)}{3}+\frac {1}{2} \tanh (x) \text {sech}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 3582
Rule 3853
Rule 3855
Rubi steps
\begin {align*} \int \frac {\text {sech}^5(x)}{1+\tanh (x)} \, dx &=\frac {\text {sech}^3(x)}{3}+\int \text {sech}^3(x) \, dx\\ &=\frac {\text {sech}^3(x)}{3}+\frac {1}{2} \text {sech}(x) \tanh (x)+\frac {1}{2} \int \text {sech}(x) \, dx\\ &=\frac {1}{2} \tan ^{-1}(\sinh (x))+\frac {\text {sech}^3(x)}{3}+\frac {1}{2} \text {sech}(x) \tanh (x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 24, normalized size = 1.00 \begin {gather*} \text {ArcTan}\left (\tanh \left (\frac {x}{2}\right )\right )+\frac {\text {sech}^3(x)}{3}+\frac {1}{2} \text {sech}(x) \tanh (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(40\) vs.
\(2(18)=36\).
time = 0.57, size = 41, normalized size = 1.71
method | result | size |
default | \(\frac {-\left (\tanh ^{5}\left (\frac {x}{2}\right )\right )+2 \left (\tanh ^{4}\left (\frac {x}{2}\right )\right )+\tanh \left (\frac {x}{2}\right )+\frac {2}{3}}{\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}+\arctan \left (\tanh \left (\frac {x}{2}\right )\right )\) | \(41\) |
risch | \(\frac {{\mathrm e}^{x} \left (3 \,{\mathrm e}^{4 x}+8 \,{\mathrm e}^{2 x}-3\right )}{3 \left (1+{\mathrm e}^{2 x}\right )^{3}}+\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{2}-\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{2}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs.
\(2 (18) = 36\).
time = 0.46, size = 49, normalized size = 2.04 \begin {gather*} \frac {3 \, e^{\left (-x\right )} + 8 \, e^{\left (-3 \, x\right )} - 3 \, e^{\left (-5 \, x\right )}}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1\right )}} - \arctan \left (e^{\left (-x\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 288 vs.
\(2 (18) = 36\).
time = 0.42, size = 288, normalized size = 12.00 \begin {gather*} \frac {3 \, \cosh \left (x\right )^{5} + 15 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + 3 \, \sinh \left (x\right )^{5} + 2 \, {\left (15 \, \cosh \left (x\right )^{2} + 4\right )} \sinh \left (x\right )^{3} + 8 \, \cosh \left (x\right )^{3} + 6 \, {\left (5 \, \cosh \left (x\right )^{3} + 4 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 3 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{4} + 3 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \cosh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )^{2} + 6 \, {\left (\cosh \left (x\right )^{5} + 2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + 3 \, {\left (5 \, \cosh \left (x\right )^{4} + 8 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) - 3 \, \cosh \left (x\right )}{3 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{4} + 3 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \cosh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )^{2} + 6 \, {\left (\cosh \left (x\right )^{5} + 2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {sech}^{5}{\left (x \right )}}{\tanh {\left (x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 31, normalized size = 1.29 \begin {gather*} \frac {3 \, e^{\left (5 \, x\right )} + 8 \, e^{\left (3 \, x\right )} - 3 \, e^{x}}{3 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} + \arctan \left (e^{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 61, normalized size = 2.54 \begin {gather*} \mathrm {atan}\left ({\mathrm {e}}^x\right )+\frac {{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}+1}-\frac {8\,{\mathrm {e}}^x}{3\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )}+\frac {2\,{\mathrm {e}}^x}{3\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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