Integrand size = 12, antiderivative size = 106 \[ \int \frac {1}{(-5-3 \cos (c+d x))^4} \, dx=\frac {385 x}{32768}-\frac {385 \arctan \left (\frac {\sin (c+d x)}{3+\cos (c+d x)}\right )}{16384 d}-\frac {\sin (c+d x)}{16 d (5+3 \cos (c+d x))^3}-\frac {25 \sin (c+d x)}{512 d (5+3 \cos (c+d x))^2}-\frac {311 \sin (c+d x)}{8192 d (5+3 \cos (c+d x))} \]
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Time = 0.10 (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, 2737} \[ \int \frac {1}{(-5-3 \cos (c+d x))^4} \, dx=-\frac {385 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)+3}\right )}{16384 d}-\frac {311 \sin (c+d x)}{8192 d (3 \cos (c+d x)+5)}-\frac {25 \sin (c+d x)}{512 d (3 \cos (c+d x)+5)^2}-\frac {\sin (c+d x)}{16 d (3 \cos (c+d x)+5)^3}+\frac {385 x}{32768} \]
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Rule 12
Rule 2737
Rule 2743
Rule 2833
Rubi steps \begin{align*} \text {integral}& = -\frac {\sin (c+d x)}{16 d (5+3 \cos (c+d x))^3}-\frac {1}{48} \int \frac {15-6 \cos (c+d x)}{(-5-3 \cos (c+d x))^3} \, dx \\ & = -\frac {\sin (c+d x)}{16 d (5+3 \cos (c+d x))^3}-\frac {25 \sin (c+d x)}{512 d (5+3 \cos (c+d x))^2}+\frac {\int \frac {186-75 \cos (c+d x)}{(-5-3 \cos (c+d x))^2} \, dx}{1536} \\ & = -\frac {\sin (c+d x)}{16 d (5+3 \cos (c+d x))^3}-\frac {25 \sin (c+d x)}{512 d (5+3 \cos (c+d x))^2}-\frac {311 \sin (c+d x)}{8192 d (5+3 \cos (c+d x))}-\frac {\int \frac {1155}{-5-3 \cos (c+d x)} \, dx}{24576} \\ & = -\frac {\sin (c+d x)}{16 d (5+3 \cos (c+d x))^3}-\frac {25 \sin (c+d x)}{512 d (5+3 \cos (c+d x))^2}-\frac {311 \sin (c+d x)}{8192 d (5+3 \cos (c+d x))}-\frac {385 \int \frac {1}{-5-3 \cos (c+d x)} \, dx}{8192} \\ & = \frac {385 x}{32768}-\frac {385 \arctan \left (\frac {\sin (c+d x)}{3+\cos (c+d x)}\right )}{16384 d}-\frac {\sin (c+d x)}{16 d (5+3 \cos (c+d x))^3}-\frac {25 \sin (c+d x)}{512 d (5+3 \cos (c+d x))^2}-\frac {311 \sin (c+d x)}{8192 d (5+3 \cos (c+d x))} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(-5-3 \cos (c+d x))^4} \, dx=-\frac {770 \arctan \left (2 \cot \left (\frac {1}{2} (c+d x)\right )\right )+\frac {9 (4883 \sin (c+d x)+2340 \sin (2 (c+d x))+311 \sin (3 (c+d x)))}{(5+3 \cos (c+d x))^3}}{32768 d} \]
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Time = 0.82 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(\frac {\frac {-\frac {639 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {117 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}-\frac {369 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}}{8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+4\right )^{3}}+\frac {385 \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{16384}}{d}\) | \(75\) |
default | \(\frac {\frac {-\frac {639 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {117 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}-\frac {369 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}}{8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+4\right )^{3}}+\frac {385 \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{16384}}{d}\) | \(75\) |
risch | \(-\frac {i \left (10395 \,{\mathrm e}^{5 i \left (d x +c \right )}+86625 \,{\mathrm e}^{4 i \left (d x +c \right )}+239470 \,{\mathrm e}^{3 i \left (d x +c \right )}+218466 \,{\mathrm e}^{2 i \left (d x +c \right )}+73575 \,{\mathrm e}^{i \left (d x +c \right )}+8397\right )}{12288 d \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}+10 \,{\mathrm e}^{i \left (d x +c \right )}+3\right )^{3}}+\frac {385 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}+3\right )}{32768 d}-\frac {385 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {1}{3}\right )}{32768 d}\) | \(127\) |
parallelrisch | \(\frac {385 i \left (-770-27 \cos \left (3 d x +3 c \right )-981 \cos \left (d x +c \right )-270 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 i\right )+385 i \left (770+27 \cos \left (3 d x +3 c \right )+981 \cos \left (d x +c \right )+270 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 i\right )-175788 \sin \left (d x +c \right )-84240 \sin \left (2 d x +2 c \right )-11196 \sin \left (3 d x +3 c \right )}{32768 d \left (770+27 \cos \left (3 d x +3 c \right )+981 \cos \left (d x +c \right )+270 \cos \left (2 d x +2 c \right )\right )}\) | \(167\) |
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Time = 0.26 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.14 \[ \int \frac {1}{(-5-3 \cos (c+d x))^4} \, dx=-\frac {385 \, {\left (27 \, \cos \left (d x + c\right )^{3} + 135 \, \cos \left (d x + c\right )^{2} + 225 \, \cos \left (d x + c\right ) + 125\right )} \arctan \left (\frac {5 \, \cos \left (d x + c\right ) + 3}{4 \, \sin \left (d x + c\right )}\right ) + 36 \, {\left (311 \, \cos \left (d x + c\right )^{2} + 1170 \, \cos \left (d x + c\right ) + 1143\right )} \sin \left (d x + c\right )}{32768 \, {\left (27 \, d \cos \left (d x + c\right )^{3} + 135 \, d \cos \left (d x + c\right )^{2} + 225 \, d \cos \left (d x + c\right ) + 125 \, d\right )}} \]
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Result contains complex when optimal does not.
Time = 3.15 (sec) , antiderivative size = 595, normalized size of antiderivative = 5.61 \[ \int \frac {1}{(-5-3 \cos (c+d x))^4} \, dx=\begin {cases} \frac {x}{\left (-5 - 3 \cosh {\left (2 \operatorname {atanh}{\left (2 \right )} \right )}\right )^{4}} & \text {for}\: c = - d x - 2 i \operatorname {atanh}{\left (2 \right )} \vee c = - d x + 2 i \operatorname {atanh}{\left (2 \right )} \\\frac {x}{\left (- 3 \cos {\left (c \right )} - 5\right )^{4}} & \text {for}\: d = 0 \\\frac {385 \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16384 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1048576 d} + \frac {4620 \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16384 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1048576 d} + \frac {18480 \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16384 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1048576 d} + \frac {24640 \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{16384 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1048576 d} - \frac {1278 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16384 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1048576 d} - \frac {7488 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16384 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1048576 d} - \frac {11808 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16384 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1048576 d} & \text {otherwise} \end {cases} \]
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Time = 0.33 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.42 \[ \int \frac {1}{(-5-3 \cos (c+d x))^4} \, dx=-\frac {\frac {18 \, {\left (\frac {656 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {416 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {71 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{\frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {12 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {\sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 64} - 385 \, \arctan \left (\frac {\sin \left (d x + c\right )}{2 \, {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}{16384 \, d} \]
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Time = 0.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(-5-3 \cos (c+d x))^4} \, dx=\frac {385 \, d x + 385 \, c - \frac {36 \, {\left (71 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 416 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 656 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 4\right )}^{3}} - 770 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 3}\right )}{32768 \, d} \]
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Time = 0.00 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(-5-3 \cos (c+d x))^4} \, dx=\frac {385\,\mathrm {atan}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{16384\,d}-\frac {385\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{16384\,d}-\frac {\frac {639\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8192}+\frac {117\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{256}+\frac {369\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{512}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+4\right )}^3} \]
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