Integrand size = 12, antiderivative size = 81 \[ \int \frac {1}{(5+3 \cos (c+d x))^3} \, dx=\frac {59 x}{2048}-\frac {59 \arctan \left (\frac {\sin (c+d x)}{3+\cos (c+d x)}\right )}{1024 d}-\frac {3 \sin (c+d x)}{32 d (5+3 \cos (c+d x))^2}-\frac {45 \sin (c+d x)}{512 d (5+3 \cos (c+d x))} \]
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Time = 0.07 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2743, 2833, 12, 2736} \[ \int \frac {1}{(5+3 \cos (c+d x))^3} \, dx=-\frac {59 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)+3}\right )}{1024 d}-\frac {45 \sin (c+d x)}{512 d (3 \cos (c+d x)+5)}-\frac {3 \sin (c+d x)}{32 d (3 \cos (c+d x)+5)^2}+\frac {59 x}{2048} \]
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Rule 12
Rule 2736
Rule 2743
Rule 2833
Rubi steps \begin{align*} \text {integral}& = -\frac {3 \sin (c+d x)}{32 d (5+3 \cos (c+d x))^2}-\frac {1}{32} \int \frac {-10+3 \cos (c+d x)}{(5+3 \cos (c+d x))^2} \, dx \\ & = -\frac {3 \sin (c+d x)}{32 d (5+3 \cos (c+d x))^2}-\frac {45 \sin (c+d x)}{512 d (5+3 \cos (c+d x))}+\frac {1}{512} \int \frac {59}{5+3 \cos (c+d x)} \, dx \\ & = -\frac {3 \sin (c+d x)}{32 d (5+3 \cos (c+d x))^2}-\frac {45 \sin (c+d x)}{512 d (5+3 \cos (c+d x))}+\frac {59}{512} \int \frac {1}{5+3 \cos (c+d x)} \, dx \\ & = \frac {59 x}{2048}-\frac {59 \arctan \left (\frac {\sin (c+d x)}{3+\cos (c+d x)}\right )}{1024 d}-\frac {3 \sin (c+d x)}{32 d (5+3 \cos (c+d x))^2}-\frac {45 \sin (c+d x)}{512 d (5+3 \cos (c+d x))} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.69 \[ \int \frac {1}{(5+3 \cos (c+d x))^3} \, dx=-\frac {59 \arctan \left (2 \cot \left (\frac {1}{2} (c+d x)\right )\right )+\frac {3 (182 \sin (c+d x)+45 \sin (2 (c+d x)))}{(5+3 \cos (c+d x))^2}}{1024 d} \]
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Time = 0.72 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(\frac {\frac {-\frac {69 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}-\frac {51 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}}{4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+4\right )^{2}}+\frac {59 \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{1024}}{d}\) | \(62\) |
default | \(\frac {\frac {-\frac {69 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}-\frac {51 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}}{4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+4\right )^{2}}+\frac {59 \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{1024}}{d}\) | \(62\) |
risch | \(-\frac {3 i \left (59 \,{\mathrm e}^{3 i \left (d x +c \right )}+295 \,{\mathrm e}^{2 i \left (d x +c \right )}+241 \,{\mathrm e}^{i \left (d x +c \right )}+45\right )}{256 d \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}+10 \,{\mathrm e}^{i \left (d x +c \right )}+3\right )^{2}}-\frac {59 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {1}{3}\right )}{2048 d}+\frac {59 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}+3\right )}{2048 d}\) | \(105\) |
parallelrisch | \(\frac {59 i \left (-59-9 \cos \left (2 d x +2 c \right )-60 \cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 i\right )+59 i \left (59+9 \cos \left (2 d x +2 c \right )+60 \cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 i\right )-2184 \sin \left (d x +c \right )-540 \sin \left (2 d x +2 c \right )}{2048 d \left (59+9 \cos \left (2 d x +2 c \right )+60 \cos \left (d x +c \right )\right )}\) | \(123\) |
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Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(5+3 \cos (c+d x))^3} \, dx=-\frac {59 \, {\left (9 \, \cos \left (d x + c\right )^{2} + 30 \, \cos \left (d x + c\right ) + 25\right )} \arctan \left (\frac {5 \, \cos \left (d x + c\right ) + 3}{4 \, \sin \left (d x + c\right )}\right ) + 12 \, {\left (45 \, \cos \left (d x + c\right ) + 91\right )} \sin \left (d x + c\right )}{2048 \, {\left (9 \, d \cos \left (d x + c\right )^{2} + 30 \, d \cos \left (d x + c\right ) + 25 \, d\right )}} \]
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Result contains complex when optimal does not.
Time = 1.59 (sec) , antiderivative size = 359, normalized size of antiderivative = 4.43 \[ \int \frac {1}{(5+3 \cos (c+d x))^3} \, dx=\begin {cases} \frac {x}{\left (5 + 3 \cosh {\left (2 \operatorname {atanh}{\left (2 \right )} \right )}\right )^{3}} & \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 )^{3}} & \text {for}\: d = 0 \\\frac {59 \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 )}}{1024 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8192 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} + \frac {472 \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 )}}{1024 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8192 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} + \frac {944 \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 )}{1024 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8192 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} - \frac {138 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1024 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8192 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} - \frac {408 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1024 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8192 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} & \text {otherwise} \end {cases} \]
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Time = 0.34 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.37 \[ \int \frac {1}{(5+3 \cos (c+d x))^3} \, dx=-\frac {\frac {6 \, {\left (\frac {68 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {23 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{\frac {8 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {\sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 16} - 59 \, \arctan \left (\frac {\sin \left (d x + c\right )}{2 \, {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}{1024 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(5+3 \cos (c+d x))^3} \, dx=\frac {59 \, d x + 59 \, c - \frac {12 \, {\left (23 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 68 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 4\right )}^{2}} - 118 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 3}\right )}{2048 \, d} \]
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Time = 15.51 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.19 \[ \int \frac {1}{(5+3 \cos (c+d x))^3} \, dx=\frac {59\,\mathrm {atan}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{1024\,d}-\frac {59\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{1024\,d}-\frac {\frac {69\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{512}+\frac {51\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+16\right )} \]
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