Optimal. Leaf size=17 \[ \frac {\tanh ^2(x)}{2}-\frac {\tanh ^3(x)}{3} \]
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Rubi [A] time = 0.05, antiderivative size = 17, 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 = {3516, 848, 43} \[ \frac {\tanh ^2(x)}{2}-\frac {\tanh ^3(x)}{3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 848
Rule 3516
Rubi steps
\begin {align*} \int \frac {\text {sech}^4(x)}{1+\coth (x)} \, dx &=-\operatorname {Subst}\left (\int \frac {-1+x^2}{x^4 (1+x)} \, dx,x,\coth (x)\right )\\ &=-\operatorname {Subst}\left (\int \frac {-1+x}{x^4} \, dx,x,\coth (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (-\frac {1}{x^4}+\frac {1}{x^3}\right ) \, dx,x,\coth (x)\right )\\ &=\frac {\tanh ^2(x)}{2}-\frac {\tanh ^3(x)}{3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 17, normalized size = 1.00 \[ \frac {1}{6} \left (-2 \tanh ^3(x)-3 \text {sech}^2(x)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.37, size = 84, normalized size = 4.94 \[ -\frac {4 \, {\left (\cosh \relax (x) + 2 \, \sinh \relax (x)\right )}}{3 \, {\left (\cosh \relax (x)^{5} + 5 \, \cosh \relax (x) \sinh \relax (x)^{4} + \sinh \relax (x)^{5} + {\left (10 \, \cosh \relax (x)^{2} + 3\right )} \sinh \relax (x)^{3} + 3 \, \cosh \relax (x)^{3} + {\left (10 \, \cosh \relax (x)^{3} + 9 \, \cosh \relax (x)\right )} \sinh \relax (x)^{2} + {\left (5 \, \cosh \relax (x)^{4} + 9 \, \cosh \relax (x)^{2} + 2\right )} \sinh \relax (x) + 4 \, \cosh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 18, normalized size = 1.06 \[ -\frac {2 \, {\left (3 \, e^{\left (2 \, x\right )} - 1\right )}}{3 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 38, normalized size = 2.24 \[ -\frac {4 \left (-\frac {\left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{2}+\frac {2 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3}-\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{2}\right )}{\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.30, size = 75, normalized size = 4.41 \[ -\frac {2 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1} - \frac {4 \, e^{\left (-4 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1} - \frac {2}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 18, normalized size = 1.06 \[ -\frac {2\,\left (3\,{\mathrm {e}}^{2\,x}-1\right )}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^3} \]
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 \[ \int \frac {\operatorname {sech}^{4}{\relax (x )}}{\coth {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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