Optimal. Leaf size=25 \[ \frac {\sinh (x)}{3 (1+\cosh (x))^2}+\frac {\sinh (x)}{3 (1+\cosh (x))} \]
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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2729, 2727}
\begin {gather*} \frac {\sinh (x)}{3 (\cosh (x)+1)}+\frac {\sinh (x)}{3 (\cosh (x)+1)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2727
Rule 2729
Rubi steps
\begin {align*} \int \frac {1}{(1+\cosh (x))^2} \, dx &=\frac {\sinh (x)}{3 (1+\cosh (x))^2}+\frac {1}{3} \int \frac {1}{1+\cosh (x)} \, dx\\ &=\frac {\sinh (x)}{3 (1+\cosh (x))^2}+\frac {\sinh (x)}{3 (1+\cosh (x))}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 16, normalized size = 0.64 \begin {gather*} \frac {(2+\cosh (x)) \sinh (x)}{3 (1+\cosh (x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 16, normalized size = 0.64
method | result | size |
risch | \(-\frac {2 \left (1+3 \,{\mathrm e}^{x}\right )}{3 \left (1+{\mathrm e}^{x}\right )^{3}}\) | \(15\) |
default | \(-\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{6}+\frac {\tanh \left (\frac {x}{2}\right )}{2}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs.
\(2 (21) = 42\).
time = 1.97, size = 49, normalized size = 1.96 \begin {gather*} \frac {2 \, e^{\left (-x\right )}}{3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1} + \frac {2}{3 \, {\left (3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1\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. 58 vs.
\(2 (21) = 42\).
time = 0.81, size = 58, normalized size = 2.32 \begin {gather*} -\frac {2 \, {\left (3 \, \cosh \left (x\right ) + 3 \, \sinh \left (x\right ) + 1\right )}}{3 \, {\left (\cosh \left (x\right )^{3} + 3 \, {\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} + 3 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + 3 \, \cosh \left (x\right ) + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 14, normalized size = 0.56 \begin {gather*} - \frac {\tanh ^{3}{\left (\frac {x}{2} \right )}}{6} + \frac {\tanh {\left (\frac {x}{2} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.06, size = 14, normalized size = 0.56 \begin {gather*} -\frac {2 \, {\left (3 \, e^{x} + 1\right )}}{3 \, {\left (e^{x} + 1\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.29, size = 14, normalized size = 0.56 \begin {gather*} -\frac {2\,\left (3\,{\mathrm {e}}^x+1\right )}{3\,{\left ({\mathrm {e}}^x+1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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