Optimal. Leaf size=17 \[ \frac {\coth ^2(x)}{2}-\frac {\coth ^3(x)}{3} \]
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Rubi [A] time = 0.04, 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} \[ \frac {\coth ^2(x)}{2}-\frac {\coth ^3(x)}{3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 848
Rule 3516
Rubi steps
\begin {align*} \int \frac {\text {csch}^4(x)}{1+\tanh (x)} \, dx &=-\operatorname {Subst}\left (\int \frac {-1+x^2}{x^4 (1+x)} \, dx,x,\tanh (x)\right )\\ &=-\operatorname {Subst}\left (\int \frac {-1+x}{x^4} \, dx,x,\tanh (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (-\frac {1}{x^4}+\frac {1}{x^3}\right ) \, dx,x,\tanh (x)\right )\\ &=\frac {\coth ^2(x)}{2}-\frac {\coth ^3(x)}{3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 20, normalized size = 1.18 \[ -\frac {1}{6} \text {csch}(x) (2 \cosh (x)+(2 \coth (x)-3) \text {csch}(x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 84, normalized size = 4.94 \[ -\frac {4 \, {\left (2 \, \cosh \relax (x) + \sinh \relax (x)\right )}}{3 \, {\left (\cosh \relax (x)^{5} + 5 \, \cosh \relax (x) \sinh \relax (x)^{4} + \sinh \relax (x)^{5} + {\left (10 \, \cosh \relax (x)^{2} - 3\right )} \sinh \relax (x)^{3} - 3 \, \cosh \relax (x)^{3} + {\left (10 \, \cosh \relax (x)^{3} - 9 \, \cosh \relax (x)\right )} \sinh \relax (x)^{2} + {\left (5 \, \cosh \relax (x)^{4} - 9 \, \cosh \relax (x)^{2} + 4\right )} \sinh \relax (x) + 2 \, \cosh \relax (x)\right )}} \]
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 = 48, normalized size = 2.82 \[ -\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{24}+\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{8}-\frac {\tanh \left (\frac {x}{2}\right )}{8}-\frac {1}{24 \tanh \left (\frac {x}{2}\right )^{3}}-\frac {1}{8 \tanh \left (\frac {x}{2}\right )}+\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.30, size = 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 = 1.06, 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 \[ \int \frac {\operatorname {csch}^{4}{\relax (x )}}{\tanh {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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