Optimal. Leaf size=25 \[ \frac {\cosh ^5(x)}{5}-\frac {\sinh ^3(x)}{3}-\frac {\sinh ^5(x)}{5} \]
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Rubi [A]
time = 0.12, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3599, 3187,
3186, 2645, 30, 2644, 14} \begin {gather*} -\frac {\sinh ^5(x)}{5}-\frac {\sinh ^3(x)}{3}+\frac {\cosh ^5(x)}{5} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 2644
Rule 2645
Rule 3186
Rule 3187
Rule 3599
Rubi steps
\begin {align*} \int \frac {\cosh ^3(x)}{1+\coth (x)} \, dx &=-\left (i \int \frac {\cosh ^3(x) \sinh (x)}{-i \cosh (x)-i \sinh (x)} \, dx\right )\\ &=-\int \cosh ^3(x) \sinh (x) (-\cosh (x)+\sinh (x)) \, dx\\ &=i \int \left (-i \cosh ^4(x) \sinh (x)+i \cosh ^3(x) \sinh ^2(x)\right ) \, dx\\ &=\int \cosh ^4(x) \sinh (x) \, dx-\int \cosh ^3(x) \sinh ^2(x) \, dx\\ &=-\left (i \text {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,i \sinh (x)\right )\right )+\text {Subst}\left (\int x^4 \, dx,x,\cosh (x)\right )\\ &=\frac {\cosh ^5(x)}{5}-i \text {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,i \sinh (x)\right )\\ &=\frac {\cosh ^5(x)}{5}-\frac {\sinh ^3(x)}{3}-\frac {\sinh ^5(x)}{5}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 34, normalized size = 1.36 \begin {gather*} \frac {1}{120} (\cosh (x)-\sinh (x)) (20 \cosh (2 x)+4 \cosh (4 x)+10 \sinh (2 x)+\sinh (4 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. \(81\) vs.
\(2(19)=38\).
time = 0.38, size = 82, normalized size = 3.28
method | result | size |
risch | \(\frac {{\mathrm e}^{3 x}}{48}+\frac {{\mathrm e}^{x}}{8}+\frac {{\mathrm e}^{-3 x}}{24}+\frac {{\mathrm e}^{-5 x}}{80}\) | \(24\) |
default | \(-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {2}{5 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {4}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {1}{6 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 27, normalized size = 1.08 \begin {gather*} \frac {1}{48} \, {\left (6 \, e^{\left (-2 \, x\right )} + 1\right )} e^{\left (3 \, x\right )} + \frac {1}{24} \, 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. 56 vs.
\(2 (19) = 38\).
time = 0.36, size = 56, normalized size = 2.24 \begin {gather*} \frac {\cosh \left (x\right )^{4} + \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} + 5\right )} \sinh \left (x\right )^{2} + 5 \, \cosh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{3} + 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right )}{30 \, {\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 {\cosh ^{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 = 0.41, size = 25, normalized size = 1.00 \begin {gather*} \frac {1}{240} \, {\left (10 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-5 \, x\right )} + \frac {1}{48} \, e^{\left (3 \, x\right )} + \frac {1}{8} \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.32, size = 23, normalized size = 0.92 \begin {gather*} \frac {{\mathrm {e}}^{-3\,x}}{24}+\frac {{\mathrm {e}}^{3\,x}}{48}+\frac {{\mathrm {e}}^{-5\,x}}{80}+\frac {{\mathrm {e}}^x}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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