Integrand size = 12, antiderivative size = 106 \[ \int \frac {1}{(5+3 \cosh (c+d x))^4} \, dx=\frac {385 x}{32768}-\frac {385 \text {arctanh}\left (\frac {\sinh (c+d x)}{3+\cosh (c+d x)}\right )}{16384 d}-\frac {\sinh (c+d x)}{16 d (5+3 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (5+3 \cosh (c+d x))^2}-\frac {311 \sinh (c+d x)}{8192 d (5+3 \cosh (c+d x))} \]
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Time = 0.07 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2743, 2833, 12, 2736} \[ \int \frac {1}{(5+3 \cosh (c+d x))^4} \, dx=-\frac {385 \text {arctanh}\left (\frac {\sinh (c+d x)}{\cosh (c+d x)+3}\right )}{16384 d}-\frac {311 \sinh (c+d x)}{8192 d (3 \cosh (c+d x)+5)}-\frac {25 \sinh (c+d x)}{512 d (3 \cosh (c+d x)+5)^2}-\frac {\sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)^3}+\frac {385 x}{32768} \]
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Rule 12
Rule 2736
Rule 2743
Rule 2833
Rubi steps \begin{align*} \text {integral}& = -\frac {\sinh (c+d x)}{16 d (5+3 \cosh (c+d x))^3}-\frac {1}{48} \int \frac {-15+6 \cosh (c+d x)}{(5+3 \cosh (c+d x))^3} \, dx \\ & = -\frac {\sinh (c+d x)}{16 d (5+3 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (5+3 \cosh (c+d x))^2}+\frac {\int \frac {186-75 \cosh (c+d x)}{(5+3 \cosh (c+d x))^2} \, dx}{1536} \\ & = -\frac {\sinh (c+d x)}{16 d (5+3 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (5+3 \cosh (c+d x))^2}-\frac {311 \sinh (c+d x)}{8192 d (5+3 \cosh (c+d x))}-\frac {\int -\frac {1155}{5+3 \cosh (c+d x)} \, dx}{24576} \\ & = -\frac {\sinh (c+d x)}{16 d (5+3 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (5+3 \cosh (c+d x))^2}-\frac {311 \sinh (c+d x)}{8192 d (5+3 \cosh (c+d x))}+\frac {385 \int \frac {1}{5+3 \cosh (c+d x)} \, dx}{8192} \\ & = \frac {385 x}{32768}-\frac {385 \text {arctanh}\left (\frac {\sinh (c+d x)}{3+\cosh (c+d x)}\right )}{16384 d}-\frac {\sinh (c+d x)}{16 d (5+3 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (5+3 \cosh (c+d x))^2}-\frac {311 \sinh (c+d x)}{8192 d (5+3 \cosh (c+d x))} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(296\) vs. \(2(106)=212\).
Time = 0.27 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.79 \[ \int \frac {1}{(5+3 \cosh (c+d x))^4} \, dx=-\frac {296450 \log \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )\right )+10395 \cosh (3 (c+d x)) \log \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )\right )+377685 \cosh (c+d x) \left (\log \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )+103950 \cosh (2 (c+d x)) \left (\log \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )-296450 \log \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )-10395 \cosh (3 (c+d x)) \log \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )+175788 \sinh (c+d x)+84240 \sinh (2 (c+d x))+11196 \sinh (3 (c+d x))}{131072 d (5+3 \cosh (c+d x))^3} \]
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Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.06
method | result | size |
risch | \(\frac {10395 \,{\mathrm e}^{5 d x +5 c}+86625 \,{\mathrm e}^{4 d x +4 c}+239470 \,{\mathrm e}^{3 d x +3 c}+218466 \,{\mathrm e}^{2 d x +2 c}+73575 \,{\mathrm e}^{d x +c}+8397}{12288 d \left (3 \,{\mathrm e}^{2 d x +2 c}+10 \,{\mathrm e}^{d x +c}+3\right )^{3}}-\frac {385 \ln \left ({\mathrm e}^{d x +c}+3\right )}{32768 d}+\frac {385 \ln \left (\frac {1}{3}+{\mathrm e}^{d x +c}\right )}{32768 d}\) | \(112\) |
derivativedivides | \(\frac {\frac {9}{2048 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{3}}+\frac {81}{4096 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}+\frac {639}{16384 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}-\frac {385 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{32768}+\frac {9}{2048 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{3}}-\frac {81}{4096 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}+\frac {639}{16384 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}+\frac {385 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{32768}}{d}\) | \(124\) |
default | \(\frac {\frac {9}{2048 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{3}}+\frac {81}{4096 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}+\frac {639}{16384 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}-\frac {385 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{32768}+\frac {9}{2048 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{3}}-\frac {81}{4096 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}+\frac {639}{16384 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}+\frac {385 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{32768}}{d}\) | \(124\) |
parallelrisch | \(\frac {\left (-377685 \cosh \left (d x +c \right )-103950 \cosh \left (2 d x +2 c \right )-10395 \cosh \left (3 d x +3 c \right )-296450\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )+\left (377685 \cosh \left (d x +c \right )+103950 \cosh \left (2 d x +2 c \right )+10395 \cosh \left (3 d x +3 c \right )+296450\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )-175788 \sinh \left (d x +c \right )-84240 \sinh \left (2 d x +2 c \right )-11196 \sinh \left (3 d x +3 c \right )}{32768 d \left (770+27 \cosh \left (3 d x +3 c \right )+981 \cosh \left (d x +c \right )+270 \cosh \left (2 d x +2 c \right )\right )}\) | \(161\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1078 vs. \(2 (96) = 192\).
Time = 0.25 (sec) , antiderivative size = 1078, normalized size of antiderivative = 10.17 \[ \int \frac {1}{(5+3 \cosh (c+d x))^4} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 784 vs. \(2 (94) = 188\).
Time = 2.69 (sec) , antiderivative size = 784, normalized size of antiderivative = 7.40 \[ \int \frac {1}{(5+3 \cosh (c+d x))^4} \, dx=\text {Too large to display} \]
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Time = 0.19 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.59 \[ \int \frac {1}{(5+3 \cosh (c+d x))^4} \, dx=-\frac {385 \, \log \left (3 \, e^{\left (-d x - c\right )} + 1\right )}{32768 \, d} + \frac {385 \, \log \left (e^{\left (-d x - c\right )} + 3\right )}{32768 \, d} - \frac {73575 \, e^{\left (-d x - c\right )} + 218466 \, e^{\left (-2 \, d x - 2 \, c\right )} + 239470 \, e^{\left (-3 \, d x - 3 \, c\right )} + 86625 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10395 \, e^{\left (-5 \, d x - 5 \, c\right )} + 8397}{12288 \, d {\left (270 \, e^{\left (-d x - c\right )} + 981 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1540 \, e^{\left (-3 \, d x - 3 \, c\right )} + 981 \, e^{\left (-4 \, d x - 4 \, c\right )} + 270 \, e^{\left (-5 \, d x - 5 \, c\right )} + 27 \, e^{\left (-6 \, d x - 6 \, c\right )} + 27\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.03 \[ \int \frac {1}{(5+3 \cosh (c+d x))^4} \, dx=\frac {\frac {8 \, {\left (10395 \, e^{\left (5 \, d x + 5 \, c\right )} + 86625 \, e^{\left (4 \, d x + 4 \, c\right )} + 239470 \, e^{\left (3 \, d x + 3 \, c\right )} + 218466 \, e^{\left (2 \, d x + 2 \, c\right )} + 73575 \, e^{\left (d x + c\right )} + 8397\right )}}{{\left (3 \, e^{\left (2 \, d x + 2 \, c\right )} + 10 \, e^{\left (d x + c\right )} + 3\right )}^{3}} + 1155 \, \log \left (3 \, e^{\left (d x + c\right )} + 1\right ) - 1155 \, \log \left (e^{\left (d x + c\right )} + 3\right )}{98304 \, d} \]
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Time = 0.12 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.13 \[ \int \frac {1}{(5+3 \cosh (c+d x))^4} \, dx=\frac {\frac {385\,{\mathrm {e}}^{c+d\,x}}{4096\,d}+\frac {1925}{12288\,d}}{10\,{\mathrm {e}}^{c+d\,x}+3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3}-\frac {385\,\mathrm {atan}\left (\left (\frac {5}{4\,d}+\frac {3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c}{4\,d}\right )\,\sqrt {-d^2}\right )}{16384\,\sqrt {-d^2}}-\frac {\frac {385\,{\mathrm {e}}^{c+d\,x}}{1152\,d}+\frac {3461}{3456\,d}}{60\,{\mathrm {e}}^{c+d\,x}+118\,{\mathrm {e}}^{2\,c+2\,d\,x}+60\,{\mathrm {e}}^{3\,c+3\,d\,x}+9\,{\mathrm {e}}^{4\,c+4\,d\,x}+9}+\frac {\frac {365\,{\mathrm {e}}^{c+d\,x}}{54\,d}+\frac {41}{18\,d}}{270\,{\mathrm {e}}^{c+d\,x}+981\,{\mathrm {e}}^{2\,c+2\,d\,x}+1540\,{\mathrm {e}}^{3\,c+3\,d\,x}+981\,{\mathrm {e}}^{4\,c+4\,d\,x}+270\,{\mathrm {e}}^{5\,c+5\,d\,x}+27\,{\mathrm {e}}^{6\,c+6\,d\,x}+27} \]
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