Integrand size = 12, antiderivative size = 108 \[ \int \frac {1}{(-5+3 \cos (c+d x))^4} \, dx=\frac {385 x}{32768}+\frac {385 \arctan \left (\frac {\sin (c+d x)}{3-\cos (c+d x)}\right )}{16384 d}+\frac {\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}+\frac {25 \sin (c+d x)}{512 d (5-3 \cos (c+d x))^2}+\frac {311 \sin (c+d x)}{8192 d (5-3 \cos (c+d x))} \]
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Time = 0.16 (sec) , antiderivative size = 108, 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} \[ \int \frac {1}{(-5+3 \cos (c+d x))^4} \, dx=\frac {385 \arctan \left (\frac {\sin (c+d x)}{3-\cos (c+d x)}\right )}{16384 d}+\frac {311 \sin (c+d x)}{8192 d (5-3 \cos (c+d x))}+\frac {25 \sin (c+d x)}{512 d (5-3 \cos (c+d x))^2}+\frac {\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}+\frac {385 x}{32768} \]
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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 = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(-5+3 \cos (c+d x))^4} \, dx=\frac {770 \arctan \left (2 \tan \left (\frac {1}{2} (c+d x)\right )\right )-\frac {9 (4883 \sin (c+d x)-2340 \sin (2 (c+d x))+311 \sin (3 (c+d x)))}{(-5+3 \cos (c+d x))^3}}{32768 d} \]
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Time = 0.67 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(\frac {\frac {\frac {369 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512}+\frac {117 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}+\frac {639 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8192}}{{\left (4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right )}^{3}}+\frac {385 \arctan \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16384}}{d}\) | \(77\) |
default | \(\frac {\frac {\frac {369 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512}+\frac {117 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}+\frac {639 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8192}}{{\left (4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right )}^{3}}+\frac {385 \arctan \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16384}}{d}\) | \(77\) |
risch | \(\frac {i \left (10395 \,{\mathrm e}^{5 i \left (d x +c \right )}-86625 \,{\mathrm e}^{4 i \left (d x +c \right )}+239470 \,{\mathrm e}^{3 i \left (d x +c \right )}-218466 \,{\mathrm e}^{2 i \left (d x +c \right )}+73575 \,{\mathrm e}^{i \left (d x +c \right )}-8397\right )}{12288 d \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}-10 \,{\mathrm e}^{i \left (d x +c \right )}+3\right )^{3}}-\frac {385 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {1}{3}\right )}{32768 d}+\frac {385 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}-3\right )}{32768 d}\) | \(127\) |
parallelrisch | \(\frac {385 i \left (770-27 \cos \left (3 d x +3 c \right )-981 \cos \left (d x +c \right )+270 \cos \left (2 d x +2 c \right )\right ) \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )+385 i \left (27 \cos \left (3 d x +3 c \right )+981 \cos \left (d x +c \right )-270 \cos \left (2 d x +2 c \right )-770\right ) \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )-175788 \sin \left (d x +c \right )+84240 \sin \left (2 d x +2 c \right )-11196 \sin \left (3 d x +3 c \right )}{32768 d \left (27 \cos \left (3 d x +3 c \right )+981 \cos \left (d x +c \right )-270 \cos \left (2 d x +2 c \right )-770\right )}\) | \(171\) |
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Time = 0.26 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.12 \[ \int \frac {1}{(-5+3 \cos (c+d x))^4} \, dx=-\frac {385 \, {\left (27 \, \cos \left (d x + c\right )^{3} - 135 \, \cos \left (d x + c\right )^{2} + 225 \, \cos \left (d x + c\right ) - 125\right )} \arctan \left (\frac {5 \, \cos \left (d x + c\right ) - 3}{4 \, \sin \left (d x + c\right )}\right ) + 36 \, {\left (311 \, \cos \left (d x + c\right )^{2} - 1170 \, \cos \left (d x + c\right ) + 1143\right )} \sin \left (d x + c\right )}{32768 \, {\left (27 \, d \cos \left (d x + c\right )^{3} - 135 \, d \cos \left (d x + c\right )^{2} + 225 \, d \cos \left (d x + c\right ) - 125 \, d\right )}} \]
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Result contains complex when optimal does not.
Time = 2.98 (sec) , antiderivative size = 597, normalized size of antiderivative = 5.53 \[ \int \frac {1}{(-5+3 \cos (c+d x))^4} \, dx=\begin {cases} \frac {x}{\left (-5 + 3 \cosh {\left (2 \operatorname {atanh}{\left (\frac {1}{2} \right )} \right )}\right )^{4}} & \text {for}\: c = - d x - 2 i \operatorname {atanh}{\left (\frac {1}{2} \right )} \vee c = - d x + 2 i \operatorname {atanh}{\left (\frac {1}{2} \right )} \\\frac {x}{\left (3 \cos {\left (c \right )} - 5\right )^{4}} & \text {for}\: d = 0 \\\frac {24640 \left (\operatorname {atan}{\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} \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 )}}{1048576 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} + \frac {18480 \left (\operatorname {atan}{\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} \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 )}}{1048576 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} + \frac {4620 \left (\operatorname {atan}{\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} \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 )}}{1048576 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} + \frac {385 \left (\operatorname {atan}{\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{1048576 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} + \frac {11808 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1048576 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} + \frac {7488 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1048576 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} + \frac {1278 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1048576 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} & \text {otherwise} \end {cases} \]
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Time = 0.40 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.41 \[ \int \frac {1}{(-5+3 \cos (c+d x))^4} \, dx=\frac {\frac {18 \, {\left (\frac {71 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {416 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {656 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{\frac {12 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {48 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {64 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 1} + 385 \, \arctan \left (\frac {2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{16384 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(-5+3 \cos (c+d x))^4} \, dx=\frac {385 \, d x + 385 \, c + \frac {36 \, {\left (656 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 416 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 71 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}} - 770 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) - 3}\right )}{32768 \, d} \]
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Time = 0.00 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(-5+3 \cos (c+d x))^4} \, dx=\frac {385\,\mathrm {atan}\left (2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{16384\,d}-\frac {385\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{16384\,d}+\frac {\frac {369\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{512}+\frac {117\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{256}+\frac {639\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8192}}{d\,{\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3} \]
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