Optimal. Leaf size=11 \[ \coth (x)-\frac {\coth ^2(x)}{2} \]
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Rubi [A] time = 0.03, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3487} \[ \coth (x)-\frac {\coth ^2(x)}{2} \]
Antiderivative was successfully verified.
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Rule 3487
Rubi steps
\begin {align*} \int \frac {\text {csch}^4(x)}{1+\coth (x)} \, dx &=\operatorname {Subst}(\int (1-x) \, dx,x,\coth (x))\\ &=\coth (x)-\frac {\coth ^2(x)}{2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 11, normalized size = 1.00 \[ \coth (x)-\frac {\text {csch}^2(x)}{2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 55, normalized size = 5.00 \[ -\frac {2}{\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 10, normalized size = 0.91 \[ -\frac {2}{{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 32, normalized size = 2.91 \[ -\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{8}+\frac {\tanh \left (\frac {x}{2}\right )}{2}+\frac {1}{2 \tanh \left (\frac {x}{2}\right )}-\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.30, size = 41, normalized size = 3.73 \[ \frac {4 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \frac {2}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 16, normalized size = 1.45 \[ -\frac {2}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1} \]
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 \[ \int \frac {\operatorname {csch}^{4}{\relax (x )}}{\coth {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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