Optimal. Leaf size=17 \[ \frac {\cosh ^3(x)}{3}-\frac {\sinh ^3(x)}{3} \]
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Rubi [A] time = 0.11, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3518, 3108, 3107, 2565, 30, 2564} \[ \frac {\cosh ^3(x)}{3}-\frac {\sinh ^3(x)}{3} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2564
Rule 2565
Rule 3107
Rule 3108
Rule 3518
Rubi steps
\begin {align*} \int \frac {\cosh (x)}{1+\coth (x)} \, dx &=-\left (i \int \frac {\cosh (x) \sinh (x)}{-i \cosh (x)-i \sinh (x)} \, dx\right )\\ &=-\int \cosh (x) \sinh (x) (-\cosh (x)+\sinh (x)) \, dx\\ &=i \int \left (-i \cosh ^2(x) \sinh (x)+i \cosh (x) \sinh ^2(x)\right ) \, dx\\ &=\int \cosh ^2(x) \sinh (x) \, dx-\int \cosh (x) \sinh ^2(x) \, dx\\ &=-\left (i \operatorname {Subst}\left (\int x^2 \, dx,x,i \sinh (x)\right )\right )+\operatorname {Subst}\left (\int x^2 \, dx,x,\cosh (x)\right )\\ &=\frac {\cosh ^3(x)}{3}-\frac {\sinh ^3(x)}{3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 19, normalized size = 1.12 \[ \frac {1}{12} \left (-4 \sinh ^3(x)+3 \cosh (x)+\cosh (3 x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 23, normalized size = 1.35 \[ \frac {\cosh \relax (x)^{2} + \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}{3 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 11, normalized size = 0.65 \[ \frac {1}{12} \, e^{\left (-3 \, x\right )} + \frac {1}{4} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 42, normalized size = 2.47 \[ -\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {2}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{2 \tanh \left (\frac {x}{2}\right )+2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 11, normalized size = 0.65 \[ \frac {1}{12} \, e^{\left (-3 \, x\right )} + \frac {1}{4} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 11, normalized size = 0.65 \[ \frac {{\mathrm {e}}^{-3\,x}}{12}+\frac {{\mathrm {e}}^x}{4} \]
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 \[ \int \frac {\cosh {\relax (x )}}{\coth {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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