Optimal. Leaf size=51 \[ -\frac {\sinh (c+d x)}{3 d (1-\cosh (c+d x))^2}-\frac {\sinh (c+d x)}{3 d (1-\cosh (c+d x))} \]
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Rubi [A]
time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2729, 2727}
\begin {gather*} -\frac {\sinh (c+d x)}{3 d (1-\cosh (c+d x))}-\frac {\sinh (c+d x)}{3 d (1-\cosh (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2727
Rule 2729
Rubi steps
\begin {align*} \int \frac {1}{(1-\cosh (c+d x))^2} \, dx &=-\frac {\sinh (c+d x)}{3 d (1-\cosh (c+d x))^2}+\frac {1}{3} \int \frac {1}{1-\cosh (c+d x)} \, dx\\ &=-\frac {\sinh (c+d x)}{3 d (1-\cosh (c+d x))^2}-\frac {\sinh (c+d x)}{3 d (1-\cosh (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 31, normalized size = 0.61 \begin {gather*} \frac {(-2+\cosh (c+d x)) \sinh (c+d x)}{3 d (-1+\cosh (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.95, size = 32, normalized size = 0.63
method | result | size |
risch | \(-\frac {2 \left (-1+3 \,{\mathrm e}^{d x +c}\right )}{3 d \left ({\mathrm e}^{d x +c}-1\right )^{3}}\) | \(26\) |
derivativedivides | \(\frac {\frac {1}{2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{d}\) | \(32\) |
default | \(\frac {\frac {1}{2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{d}\) | \(32\) |
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. 90 vs.
\(2 (43) = 86\).
time = 0.26, size = 90, normalized size = 1.76 \begin {gather*} \frac {2 \, e^{\left (-d x - c\right )}}{d {\left (3 \, e^{\left (-d x - c\right )} - 3 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-3 \, d x - 3 \, c\right )} - 1\right )}} - \frac {2}{3 \, d {\left (3 \, e^{\left (-d x - c\right )} - 3 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-3 \, d x - 3 \, c\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. 117 vs.
\(2 (43) = 86\).
time = 0.41, size = 117, normalized size = 2.29 \begin {gather*} -\frac {2 \, {\left (3 \, \cosh \left (d x + c\right ) + 3 \, \sinh \left (d x + c\right ) - 1\right )}}{3 \, {\left (d \cosh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{3} - 3 \, d \cosh \left (d x + c\right )^{2} + 3 \, {\left (d \cosh \left (d x + c\right ) - d\right )} \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \left (d x + c\right ) + 3 \, {\left (d \cosh \left (d x + c\right )^{2} - 2 \, d \cosh \left (d x + c\right ) + d\right )} \sinh \left (d x + c\right ) - d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.66, size = 39, normalized size = 0.76 \begin {gather*} \begin {cases} \frac {1}{2 d \tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}} - \frac {1}{6 d \tanh ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}} & \text {for}\: d \neq 0 \\\frac {x}{\left (1 - \cosh {\left (c \right )}\right )^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 25, normalized size = 0.49 \begin {gather*} -\frac {2 \, {\left (3 \, e^{\left (d x + c\right )} - 1\right )}}{3 \, d {\left (e^{\left (d x + c\right )} - 1\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 25, normalized size = 0.49 \begin {gather*} -\frac {2\,\left (3\,{\mathrm {e}}^{c+d\,x}-1\right )}{3\,d\,{\left ({\mathrm {e}}^{c+d\,x}-1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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