Optimal. Leaf size=93 \[ \frac {2 \sinh (c+d x)}{35 d (\cosh (c+d x)+1)}+\frac {2 \sinh (c+d x)}{35 d (\cosh (c+d x)+1)^2}+\frac {3 \sinh (c+d x)}{35 d (\cosh (c+d x)+1)^3}+\frac {\sinh (c+d x)}{7 d (\cosh (c+d x)+1)^4} \]
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Rubi [A] time = 0.05, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2650, 2648} \[ \frac {2 \sinh (c+d x)}{35 d (\cosh (c+d x)+1)}+\frac {2 \sinh (c+d x)}{35 d (\cosh (c+d x)+1)^2}+\frac {3 \sinh (c+d x)}{35 d (\cosh (c+d x)+1)^3}+\frac {\sinh (c+d x)}{7 d (\cosh (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2650
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 = 0.09, size = 54, normalized size = 0.58 \[ \frac {56 \sinh (c+d x)+28 \sinh (2 (c+d x))+8 \sinh (3 (c+d x))+\sinh (4 (c+d x))}{140 d (\cosh (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 347, normalized size = 3.73 \[ -\frac {4 \, {\left (35 \, \cosh \left (d x + c\right )^{2} + 10 \, {\left (7 \, \cosh \left (d x + c\right ) + 2\right )} \sinh \left (d x + c\right ) + 35 \, \sinh \left (d x + c\right )^{2} + 22 \, \cosh \left (d x + c\right ) + 7\right )}}{35 \, {\left (d \cosh \left (d x + c\right )^{6} + d \sinh \left (d x + c\right )^{6} + 7 \, d \cosh \left (d x + c\right )^{5} + {\left (6 \, d \cosh \left (d x + c\right ) + 7 \, d\right )} \sinh \left (d x + c\right )^{5} + 21 \, d \cosh \left (d x + c\right )^{4} + {\left (15 \, d \cosh \left (d x + c\right )^{2} + 35 \, d \cosh \left (d x + c\right ) + 21 \, d\right )} \sinh \left (d x + c\right )^{4} + 35 \, d \cosh \left (d x + c\right )^{3} + {\left (20 \, d \cosh \left (d x + c\right )^{3} + 70 \, d \cosh \left (d x + c\right )^{2} + 84 \, d \cosh \left (d x + c\right ) + 35 \, d\right )} \sinh \left (d x + c\right )^{3} + 35 \, d \cosh \left (d x + c\right )^{2} + {\left (15 \, d \cosh \left (d x + c\right )^{4} + 70 \, d \cosh \left (d x + c\right )^{3} + 126 \, d \cosh \left (d x + c\right )^{2} + 105 \, d \cosh \left (d x + c\right ) + 35 \, d\right )} \sinh \left (d x + c\right )^{2} + 22 \, d \cosh \left (d x + c\right ) + {\left (6 \, d \cosh \left (d x + c\right )^{5} + 35 \, d \cosh \left (d x + c\right )^{4} + 84 \, d \cosh \left (d x + c\right )^{3} + 105 \, d \cosh \left (d x + c\right )^{2} + 70 \, d \cosh \left (d x + c\right ) + 20 \, d\right )} \sinh \left (d x + c\right ) + 7 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 47, normalized size = 0.51 \[ -\frac {4 \, {\left (35 \, e^{\left (3 \, d x + 3 \, c\right )} + 21 \, e^{\left (2 \, d x + 2 \, c\right )} + 7 \, e^{\left (d x + c\right )} + 1\right )}}{35 \, d {\left (e^{\left (d x + c\right )} + 1\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 56, normalized size = 0.60 \[ \frac {-\frac {\left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56}+\frac {3 \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40}-\frac {\left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 364, normalized size = 3.91 \[ \frac {4 \, e^{\left (-d x - c\right )}}{5 \, d {\left (7 \, e^{\left (-d x - c\right )} + 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} + 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} + 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} + 1\right )}} + \frac {12 \, e^{\left (-2 \, d x - 2 \, c\right )}}{5 \, d {\left (7 \, e^{\left (-d x - c\right )} + 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} + 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} + 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} + 1\right )}} + \frac {4 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (7 \, e^{\left (-d x - c\right )} + 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} + 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} + 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} + 1\right )}} + \frac {4}{35 \, d {\left (7 \, e^{\left (-d x - c\right )} + 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} + 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} + 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 283, normalized size = 3.04 \[ -\frac {4}{35\,d\,\left (4\,{\mathrm {e}}^{c+d\,x}+6\,{\mathrm {e}}^{2\,c+2\,d\,x}+4\,{\mathrm {e}}^{3\,c+3\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {16\,{\mathrm {e}}^{c+d\,x}}{35\,d\,\left (5\,{\mathrm {e}}^{c+d\,x}+10\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{3\,c+3\,d\,x}+5\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{5\,c+5\,d\,x}+1\right )}-\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}}{7\,d\,\left (6\,{\mathrm {e}}^{c+d\,x}+15\,{\mathrm {e}}^{2\,c+2\,d\,x}+20\,{\mathrm {e}}^{3\,c+3\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,d\,x}+6\,{\mathrm {e}}^{5\,c+5\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {16\,{\mathrm {e}}^{3\,c+3\,d\,x}}{7\,d\,\left (7\,{\mathrm {e}}^{c+d\,x}+21\,{\mathrm {e}}^{2\,c+2\,d\,x}+35\,{\mathrm {e}}^{3\,c+3\,d\,x}+35\,{\mathrm {e}}^{4\,c+4\,d\,x}+21\,{\mathrm {e}}^{5\,c+5\,d\,x}+7\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{7\,c+7\,d\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.19, size = 68, normalized size = 0.73 \[ \begin {cases} - \frac {\tanh ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 d} + \frac {3 \tanh ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 d} - \frac {\tanh ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 d} + \frac {\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 d} & \text {for}\: d \neq 0 \\\frac {x}{\left (\cosh {\relax (c )} + 1\right )^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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