Integrand size = 12, antiderivative size = 56 \[ \int \frac {1}{(5+3 \cosh (c+d x))^2} \, dx=\frac {5 x}{64}-\frac {5 \text {arctanh}\left (\frac {\sinh (c+d x)}{3+\cosh (c+d x)}\right )}{32 d}-\frac {3 \sinh (c+d x)}{16 d (5+3 \cosh (c+d x))} \]
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Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2743, 12, 2736} \[ \int \frac {1}{(5+3 \cosh (c+d x))^2} \, dx=-\frac {5 \text {arctanh}\left (\frac {\sinh (c+d x)}{\cosh (c+d x)+3}\right )}{32 d}-\frac {3 \sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)}+\frac {5 x}{64} \]
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Rule 12
Rule 2736
Rule 2743
Rubi steps \begin{align*} \text {integral}& = -\frac {3 \sinh (c+d x)}{16 d (5+3 \cosh (c+d x))}-\frac {1}{16} \int -\frac {5}{5+3 \cosh (c+d x)} \, dx \\ & = -\frac {3 \sinh (c+d x)}{16 d (5+3 \cosh (c+d x))}+\frac {5}{16} \int \frac {1}{5+3 \cosh (c+d x)} \, dx \\ & = \frac {5 x}{64}-\frac {5 \text {arctanh}\left (\frac {\sinh (c+d x)}{3+\cosh (c+d x)}\right )}{32 d}-\frac {3 \sinh (c+d x)}{16 d (5+3 \cosh (c+d x))} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(144\) vs. \(2(56)=112\).
Time = 0.11 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.57 \[ \int \frac {1}{(5+3 \cosh (c+d x))^2} \, dx=\frac {-15 \cosh (c+d x) \left (\log \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )+25 \left (-\log \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )\right )+\log \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )-12 \sinh (c+d x)}{64 d (5+3 \cosh (c+d x))} \]
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Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.14
method | result | size |
derivativedivides | \(\frac {\frac {3}{32 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}-\frac {5 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{64}+\frac {3}{32 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}+\frac {5 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{64}}{d}\) | \(64\) |
default | \(\frac {\frac {3}{32 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}-\frac {5 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{64}+\frac {3}{32 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}+\frac {5 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{64}}{d}\) | \(64\) |
risch | \(\frac {5 \,{\mathrm e}^{d x +c}+3}{8 d \left (3 \,{\mathrm e}^{2 d x +2 c}+10 \,{\mathrm e}^{d x +c}+3\right )}+\frac {5 \ln \left (\frac {1}{3}+{\mathrm e}^{d x +c}\right )}{64 d}-\frac {5 \ln \left ({\mathrm e}^{d x +c}+3\right )}{64 d}\) | \(68\) |
parallelrisch | \(\frac {-15 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right ) \cosh \left (d x +c \right )+15 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right ) \cosh \left (d x +c \right )-12 \sinh \left (d x +c \right )-25 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )+25 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{64 d \left (5+3 \cosh \left (d x +c \right )\right )}\) | \(95\) |
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Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (50) = 100\).
Time = 0.27 (sec) , antiderivative size = 212, normalized size of antiderivative = 3.79 \[ \int \frac {1}{(5+3 \cosh (c+d x))^2} \, dx=\frac {5 \, {\left (3 \, \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, \cosh \left (d x + c\right ) + 5\right )} \sinh \left (d x + c\right ) + 3 \, \sinh \left (d x + c\right )^{2} + 10 \, \cosh \left (d x + c\right ) + 3\right )} \log \left (3 \, \cosh \left (d x + c\right ) + 3 \, \sinh \left (d x + c\right ) + 1\right ) - 5 \, {\left (3 \, \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, \cosh \left (d x + c\right ) + 5\right )} \sinh \left (d x + c\right ) + 3 \, \sinh \left (d x + c\right )^{2} + 10 \, \cosh \left (d x + c\right ) + 3\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 3\right ) + 40 \, \cosh \left (d x + c\right ) + 40 \, \sinh \left (d x + c\right ) + 24}{64 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} + 3 \, d \sinh \left (d x + c\right )^{2} + 10 \, d \cosh \left (d x + c\right ) + 2 \, {\left (3 \, d \cosh \left (d x + c\right ) + 5 \, d\right )} \sinh \left (d x + c\right ) + 3 \, d\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (48) = 96\).
Time = 0.73 (sec) , antiderivative size = 199, normalized size of antiderivative = 3.55 \[ \int \frac {1}{(5+3 \cosh (c+d x))^2} \, dx=\begin {cases} - \frac {5 \log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )} \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{64 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 256 d} + \frac {20 \log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )}}{64 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 256 d} + \frac {5 \log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )} \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{64 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 256 d} - \frac {20 \log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )}}{64 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 256 d} + \frac {12 \tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{64 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 256 d} & \text {for}\: d \neq 0 \\\frac {x}{\left (3 \cosh {\left (c \right )} + 5\right )^{2}} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.45 \[ \int \frac {1}{(5+3 \cosh (c+d x))^2} \, dx=-\frac {5 \, \log \left (3 \, e^{\left (-d x - c\right )} + 1\right )}{64 \, d} + \frac {5 \, \log \left (e^{\left (-d x - c\right )} + 3\right )}{64 \, d} - \frac {5 \, e^{\left (-d x - c\right )} + 3}{8 \, d {\left (10 \, e^{\left (-d x - c\right )} + 3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.16 \[ \int \frac {1}{(5+3 \cosh (c+d x))^2} \, dx=\frac {\frac {8 \, {\left (5 \, e^{\left (d x + c\right )} + 3\right )}}{3 \, e^{\left (2 \, d x + 2 \, c\right )} + 10 \, e^{\left (d x + c\right )} + 3} + 5 \, \log \left (3 \, e^{\left (d x + c\right )} + 1\right ) - 5 \, \log \left (e^{\left (d x + c\right )} + 3\right )}{64 \, d} \]
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Time = 1.81 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.38 \[ \int \frac {1}{(5+3 \cosh (c+d x))^2} \, dx=\frac {\frac {5\,{\mathrm {e}}^{c+d\,x}}{8\,d}+\frac {3}{8\,d}}{10\,{\mathrm {e}}^{c+d\,x}+3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3}-\frac {5\,\mathrm {atan}\left (\left (\frac {5}{4\,d}+\frac {3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c}{4\,d}\right )\,\sqrt {-d^2}\right )}{32\,\sqrt {-d^2}} \]
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