Optimal. Leaf size=19 \[ \frac {2 \cosh (x)}{3}-\frac {\sinh (x)}{3 (1+\coth (x))} \]
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Rubi [A]
time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3583, 2718}
\begin {gather*} \frac {2 \cosh (x)}{3}-\frac {\sinh (x)}{3 (\coth (x)+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3583
Rubi steps
\begin {align*} \int \frac {\sinh (x)}{1+\coth (x)} \, dx &=-\frac {\sinh (x)}{3 (1+\coth (x))}+\frac {2}{3} \int \sinh (x) \, dx\\ &=\frac {2 \cosh (x)}{3}-\frac {\sinh (x)}{3 (1+\coth (x))}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 21, normalized size = 1.11 \begin {gather*} \frac {1}{12} \left (9 \cosh (x)-\cosh (3 x)+4 \sinh ^3(x)\right ) \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. \(39\) vs.
\(2(15)=30\).
time = 0.51, size = 40, normalized size = 2.11
method | result | size |
risch | \(\frac {{\mathrm e}^{x}}{4}+\frac {{\mathrm e}^{-x}}{2}-\frac {{\mathrm e}^{-3 x}}{12}\) | \(18\) |
default | \(-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {2}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{2 \tanh \left (\frac {x}{2}\right )+2}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 17, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, e^{\left (-x\right )} - \frac {1}{12} \, 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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Fricas [A]
time = 0.35, size = 25, normalized size = 1.32 \begin {gather*} \frac {\cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 3}{6 \, {\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 {\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 = 0.42, size = 19, normalized size = 1.00 \begin {gather*} \frac {1}{12} \, {\left (6 \, e^{\left (2 \, x\right )} - 1\right )} 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 = 1.24, size = 17, normalized size = 0.89 \begin {gather*} \frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^{-3\,x}}{12}+\frac {{\mathrm {e}}^x}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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