Optimal. Leaf size=25 \[ -\frac {\sinh ^5(x)}{5}-\frac {\sinh ^3(x)}{3}+\frac {\cosh ^5(x)}{5} \]
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Rubi [A] time = 0.18, 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 = {3518, 3108, 3107, 2565, 30, 2564, 14} \[ -\frac {\sinh ^5(x)}{5}-\frac {\sinh ^3(x)}{3}+\frac {\cosh ^5(x)}{5} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 2564
Rule 2565
Rule 3107
Rule 3108
Rule 3518
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 \operatorname {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,i \sinh (x)\right )\right )+\operatorname {Subst}\left (\int x^4 \, dx,x,\cosh (x)\right )\\ &=\frac {\cosh ^5(x)}{5}-i \operatorname {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 = 0.06, size = 34, normalized size = 1.36 \[ \frac {1}{120} (\cosh (x)-\sinh (x)) (10 \sinh (2 x)+\sinh (4 x)+20 \cosh (2 x)+4 \cosh (4 x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.37, size = 56, normalized size = 2.24 \[ \frac {\cosh \relax (x)^{4} + \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + {\left (6 \, \cosh \relax (x)^{2} + 5\right )} \sinh \relax (x)^{2} + 5 \, \cosh \relax (x)^{2} + {\left (\cosh \relax (x)^{3} + 5 \, \cosh \relax (x)\right )} \sinh \relax (x)}{30 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 25, normalized size = 1.00 \[ \frac {1}{240} \, {\left (10 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-5 \, x\right )} + \frac {1}{48} \, e^{\left (3 \, x\right )} + \frac {1}{8} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 82, normalized size = 3.28 \[ -\frac {1}{6 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {2}{5 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {4}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 27, normalized size = 1.08 \[ \frac {1}{48} \, {\left (6 \, e^{\left (-2 \, x\right )} + 1\right )} e^{\left (3 \, x\right )} + \frac {1}{24} \, e^{\left (-3 \, x\right )} + \frac {1}{80} \, e^{\left (-5 \, x\right )} \]
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 \[ \frac {{\mathrm {e}}^{-3\,x}}{24}+\frac {{\mathrm {e}}^{3\,x}}{48}+\frac {{\mathrm {e}}^{-5\,x}}{80}+\frac {{\mathrm {e}}^x}{8} \]
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 \[ \int \frac {\cosh ^{3}{\relax (x )}}{\coth {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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