Optimal. Leaf size=101 \[ -\frac {\sinh (c+d x)}{7 d (1-\cosh (c+d x))^4}-\frac {3 \sinh (c+d x)}{35 d (1-\cosh (c+d x))^3}-\frac {2 \sinh (c+d x)}{35 d (1-\cosh (c+d x))^2}-\frac {2 \sinh (c+d x)}{35 d (1-\cosh (c+d x))} \]
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Rubi [A]
time = 0.04, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 {2 \sinh (c+d x)}{35 d (1-\cosh (c+d x))}-\frac {2 \sinh (c+d x)}{35 d (1-\cosh (c+d x))^2}-\frac {3 \sinh (c+d x)}{35 d (1-\cosh (c+d x))^3}-\frac {\sinh (c+d x)}{7 d (1-\cosh (c+d x))^4} \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))^4} \, dx &=-\frac {\sinh (c+d x)}{7 d (1-\cosh (c+d x))^4}+\frac {3}{7} \int \frac {1}{(1-\cosh (c+d x))^3} \, dx\\ &=-\frac {\sinh (c+d x)}{7 d (1-\cosh (c+d x))^4}-\frac {3 \sinh (c+d x)}{35 d (1-\cosh (c+d x))^3}+\frac {6}{35} \int \frac {1}{(1-\cosh (c+d x))^2} \, dx\\ &=-\frac {\sinh (c+d x)}{7 d (1-\cosh (c+d x))^4}-\frac {3 \sinh (c+d x)}{35 d (1-\cosh (c+d x))^3}-\frac {2 \sinh (c+d x)}{35 d (1-\cosh (c+d x))^2}+\frac {2}{35} \int \frac {1}{1-\cosh (c+d x)} \, dx\\ &=-\frac {\sinh (c+d x)}{7 d (1-\cosh (c+d x))^4}-\frac {3 \sinh (c+d x)}{35 d (1-\cosh (c+d x))^3}-\frac {2 \sinh (c+d x)}{35 d (1-\cosh (c+d x))^2}-\frac {2 \sinh (c+d x)}{35 d (1-\cosh (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 51, normalized size = 0.50 \begin {gather*} \frac {(-32+29 \cosh (c+d x)-8 \cosh (2 (c+d x))+\cosh (3 (c+d x))) \sinh (c+d x)}{70 d (-1+\cosh (c+d x))^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.91, size = 58, normalized size = 0.57
method | result | size |
risch | \(-\frac {4 \left (35 \,{\mathrm e}^{3 d x +3 c}-21 \,{\mathrm e}^{2 d x +2 c}+7 \,{\mathrm e}^{d x +c}-1\right )}{35 d \left ({\mathrm e}^{d x +c}-1\right )^{7}}\) | \(48\) |
derivativedivides | \(\frac {\frac {3}{40 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {1}{8 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{8 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {1}{56 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}}{d}\) | \(58\) |
default | \(\frac {\frac {3}{40 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {1}{8 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{8 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {1}{56 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}}{d}\) | \(58\) |
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. 364 vs.
\(2 (85) = 170\).
time = 0.28, size = 364, normalized size = 3.60 \begin {gather*} \frac {4 \, e^{\left (-d x - c\right )}}{5 \, d {\left (7 \, e^{\left (-d x - c\right )} - 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} - 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} - 1\right )}} - \frac {12 \, e^{\left (-2 \, d x - 2 \, c\right )}}{5 \, d {\left (7 \, e^{\left (-d x - c\right )} - 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} - 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} - 1\right )}} + \frac {4 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (7 \, e^{\left (-d x - c\right )} - 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} - 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} - 1\right )}} - \frac {4}{35 \, d {\left (7 \, e^{\left (-d x - c\right )} - 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} - 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, 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. 347 vs.
\(2 (85) = 170\).
time = 0.38, size = 347, normalized size = 3.44 \begin {gather*} -\frac {4 \, {\left (35 \, \cosh \left (d x + c\right )^{2} + 10 \, {\left (7 \, \cosh \left (d x + c\right ) - 2\right )} \sinh \left (d x + c\right ) + 35 \, \sinh \left (d x + c\right )^{2} - 22 \, \cosh \left (d x + c\right ) + 7\right )}}{35 \, {\left (d \cosh \left (d x + c\right )^{6} + d \sinh \left (d x + c\right )^{6} - 7 \, d \cosh \left (d x + c\right )^{5} + {\left (6 \, d \cosh \left (d x + c\right ) - 7 \, d\right )} \sinh \left (d x + c\right )^{5} + 21 \, d \cosh \left (d x + c\right )^{4} + {\left (15 \, d \cosh \left (d x + c\right )^{2} - 35 \, d \cosh \left (d x + c\right ) + 21 \, d\right )} \sinh \left (d x + c\right )^{4} - 35 \, d \cosh \left (d x + c\right )^{3} + {\left (20 \, d \cosh \left (d x + c\right )^{3} - 70 \, d \cosh \left (d x + c\right )^{2} + 84 \, d \cosh \left (d x + c\right ) - 35 \, d\right )} \sinh \left (d x + c\right )^{3} + 35 \, d \cosh \left (d x + c\right )^{2} + {\left (15 \, d \cosh \left (d x + c\right )^{4} - 70 \, d \cosh \left (d x + c\right )^{3} + 126 \, d \cosh \left (d x + c\right )^{2} - 105 \, d \cosh \left (d x + c\right ) + 35 \, d\right )} \sinh \left (d x + c\right )^{2} - 22 \, d \cosh \left (d x + c\right ) + {\left (6 \, d \cosh \left (d x + c\right )^{5} - 35 \, d \cosh \left (d x + c\right )^{4} + 84 \, d \cosh \left (d x + c\right )^{3} - 105 \, d \cosh \left (d x + c\right )^{2} + 70 \, d \cosh \left (d x + c\right ) - 20 \, d\right )} \sinh \left (d x + c\right ) + 7 \, d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.03, size = 73, normalized size = 0.72 \begin {gather*} \begin {cases} \frac {1}{8 d \tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}} - \frac {1}{8 d \tanh ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}} + \frac {3}{40 d \tanh ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}} - \frac {1}{56 d \tanh ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}} & \text {for}\: d \neq 0 \\\frac {x}{\left (1 - \cosh {\left (c \right )}\right )^{4}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 47, normalized size = 0.47 \begin {gather*} -\frac {4 \, {\left (35 \, e^{\left (3 \, d x + 3 \, c\right )} - 21 \, e^{\left (2 \, d x + 2 \, c\right )} + 7 \, e^{\left (d x + c\right )} - 1\right )}}{35 \, d {\left (e^{\left (d x + c\right )} - 1\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 283, normalized size = 2.80 \begin {gather*} -\frac {4}{35\,d\,\left (6\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{c+d\,x}-4\,{\mathrm {e}}^{3\,c+3\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {16\,{\mathrm {e}}^{c+d\,x}}{35\,d\,\left (5\,{\mathrm {e}}^{c+d\,x}-10\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{3\,c+3\,d\,x}-5\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{5\,c+5\,d\,x}-1\right )}-\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}}{7\,d\,\left (15\,{\mathrm {e}}^{2\,c+2\,d\,x}-6\,{\mathrm {e}}^{c+d\,x}-20\,{\mathrm {e}}^{3\,c+3\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,d\,x}-6\,{\mathrm {e}}^{5\,c+5\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {16\,{\mathrm {e}}^{3\,c+3\,d\,x}}{7\,d\,\left (7\,{\mathrm {e}}^{c+d\,x}-21\,{\mathrm {e}}^{2\,c+2\,d\,x}+35\,{\mathrm {e}}^{3\,c+3\,d\,x}-35\,{\mathrm {e}}^{4\,c+4\,d\,x}+21\,{\mathrm {e}}^{5\,c+5\,d\,x}-7\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{7\,c+7\,d\,x}-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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