Optimal. Leaf size=29 \[ \frac {4 \cosh ^3(x)}{15}-\frac {4 \cosh (x)}{5}-\frac {\sinh ^3(x)}{5 (\coth (x)+1)} \]
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Rubi [A] time = 0.05, antiderivative size = 29, 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 = {3502, 2633} \[ \frac {4 \cosh ^3(x)}{15}-\frac {4 \cosh (x)}{5}-\frac {\sinh ^3(x)}{5 (\coth (x)+1)} \]
Antiderivative was successfully verified.
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Rule 2633
Rule 3502
Rubi steps
\begin {align*} \int \frac {\sinh ^3(x)}{1+\coth (x)} \, dx &=-\frac {\sinh ^3(x)}{5 (1+\coth (x))}+\frac {4}{5} \int \sinh ^3(x) \, dx\\ &=-\frac {\sinh ^3(x)}{5 (1+\coth (x))}-\frac {4}{5} \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (x)\right )\\ &=-\frac {4 \cosh (x)}{5}+\frac {4 \cosh ^3(x)}{15}-\frac {\sinh ^3(x)}{5 (1+\coth (x))}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 36, normalized size = 1.24 \[ \frac {\text {csch}(x) (-40 \sinh (2 x)+4 \sinh (4 x)-20 \cosh (2 x)+\cosh (4 x)-45)}{120 (\coth (x)+1)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 60, normalized size = 2.07 \[ \frac {\cosh \relax (x)^{4} + 16 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 10\right )} \sinh \relax (x)^{2} - 20 \, \cosh \relax (x)^{2} + 16 \, {\left (\cosh \relax (x)^{3} - 5 \, \cosh \relax (x)\right )} \sinh \relax (x) - 45}{120 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 31, normalized size = 1.07 \[ -\frac {1}{240} \, {\left (90 \, e^{\left (4 \, x\right )} - 20 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-5 \, x\right )} + \frac {1}{48} \, e^{\left (3 \, x\right )} - \frac {1}{4} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 80, normalized size = 2.76 \[ -\frac {1}{6 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {2}{5 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 33, normalized size = 1.14 \[ -\frac {1}{48} \, {\left (12 \, e^{\left (-2 \, x\right )} - 1\right )} e^{\left (3 \, x\right )} - \frac {3}{8} \, e^{\left (-x\right )} + \frac {1}{12} \, e^{\left (-3 \, x\right )} - \frac {1}{80} \, e^{\left (-5 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 29, normalized size = 1.00 \[ \frac {{\mathrm {e}}^{-3\,x}}{12}-\frac {3\,{\mathrm {e}}^{-x}}{8}+\frac {{\mathrm {e}}^{3\,x}}{48}-\frac {{\mathrm {e}}^{-5\,x}}{80}-\frac {{\mathrm {e}}^x}{4} \]
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 \[ \int \frac {\sinh ^{3}{\relax (x )}}{\coth {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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