Optimal. Leaf size=29 \[ -\frac {4 \cosh (x)}{5}+\frac {4 \cosh ^3(x)}{15}-\frac {\sinh ^3(x)}{5 (1+\coth (x))} \]
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Rubi [A]
time = 0.03, 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 = {3583, 2713}
\begin {gather*} \frac {4 \cosh ^3(x)}{15}-\frac {4 \cosh (x)}{5}-\frac {\sinh ^3(x)}{5 (\coth (x)+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rule 3583
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} \text {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 = 0.06, size = 36, normalized size = 1.24 \begin {gather*} \frac {\text {csch}(x) (-45-20 \cosh (2 x)+\cosh (4 x)-40 \sinh (2 x)+4 \sinh (4 x))}{120 (1+\coth (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. \(79\) vs.
\(2(23)=46\).
time = 0.50, size = 80, normalized size = 2.76
method | result | size |
risch | \(\frac {{\mathrm e}^{3 x}}{48}-\frac {{\mathrm e}^{x}}{4}-\frac {3 \,{\mathrm e}^{-x}}{8}+\frac {{\mathrm e}^{-3 x}}{12}-\frac {{\mathrm e}^{-5 x}}{80}\) | \(30\) |
default | \(-\frac {1}{6 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {2}{5 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 33, normalized size = 1.14 \begin {gather*} -\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 )} \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. 60 vs.
\(2 (23) = 46\).
time = 0.36, size = 60, normalized size = 2.07 \begin {gather*} \frac {\cosh \left (x\right )^{4} + 16 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 10\right )} \sinh \left (x\right )^{2} - 20 \, \cosh \left (x\right )^{2} + 16 \, {\left (\cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 45}{120 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\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 {\sinh ^{3}{\left (x \right )}}{\coth {\left (x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 31, normalized size = 1.07 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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