Integrand size = 12, antiderivative size = 58 \[ \int \frac {1}{(-5+3 \sin (c+d x))^2} \, dx=\frac {5 x}{64}-\frac {5 \arctan \left (\frac {\cos (c+d x)}{3-\sin (c+d x)}\right )}{32 d}-\frac {3 \cos (c+d x)}{16 d (5-3 \sin (c+d x))} \]
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Time = 0.02 (sec) , antiderivative size = 58, 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, 2737} \[ \int \frac {1}{(-5+3 \sin (c+d x))^2} \, dx=-\frac {5 \arctan \left (\frac {\cos (c+d x)}{3-\sin (c+d x)}\right )}{32 d}-\frac {3 \cos (c+d x)}{16 d (5-3 \sin (c+d x))}+\frac {5 x}{64} \]
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Rule 12
Rule 2737
Rule 2743
Rubi steps \begin{align*} \text {integral}& = -\frac {3 \cos (c+d x)}{16 d (5-3 \sin (c+d x))}-\frac {1}{16} \int \frac {5}{-5+3 \sin (c+d x)} \, dx \\ & = -\frac {3 \cos (c+d x)}{16 d (5-3 \sin (c+d x))}-\frac {5}{16} \int \frac {1}{-5+3 \sin (c+d x)} \, dx \\ & = \frac {5 x}{64}-\frac {5 \arctan \left (\frac {\cos (c+d x)}{3-\sin (c+d x)}\right )}{32 d}-\frac {3 \cos (c+d x)}{16 d (5-3 \sin (c+d x))} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.57 \[ \int \frac {1}{(-5+3 \sin (c+d x))^2} \, dx=\frac {-25 \arctan \left (\frac {2 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}\right )+\frac {6 (-5+5 \cos (c+d x)+3 \sin (c+d x))}{-5+3 \sin (c+d x)}}{160 d} \]
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Time = 0.22 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {\frac {\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{200}-\frac {3}{40}}{\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{5}+1}+\frac {5 \arctan \left (\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}-\frac {3}{4}\right )}{32}}{d}\) | \(63\) |
default | \(\frac {\frac {\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{200}-\frac {3}{40}}{\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{5}+1}+\frac {5 \arctan \left (\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}-\frac {3}{4}\right )}{32}}{d}\) | \(63\) |
risch | \(-\frac {5 \,{\mathrm e}^{i \left (d x +c \right )}-3 i}{8 d \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}-3-10 i {\mathrm e}^{i \left (d x +c \right )}\right )}+\frac {5 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}-3 i\right )}{64 d}-\frac {5 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i}{3}\right )}{64 d}\) | \(86\) |
parallelrisch | \(\frac {\left (-75 i \sin \left (d x +c \right )+125 i\right ) \ln \left (5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3-4 i\right )+\left (75 i \sin \left (d x +c \right )-125 i\right ) \ln \left (5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3+4 i\right )+60 \cos \left (d x +c \right )-36 \sin \left (d x +c \right )+60}{960 d \sin \left (d x +c \right )-1600 d}\) | \(93\) |
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Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(-5+3 \sin (c+d x))^2} \, dx=\frac {5 \, {\left (3 \, \sin \left (d x + c\right ) - 5\right )} \arctan \left (\frac {5 \, \sin \left (d x + c\right ) - 3}{4 \, \cos \left (d x + c\right )}\right ) + 12 \, \cos \left (d x + c\right )}{64 \, {\left (3 \, d \sin \left (d x + c\right ) - 5 \, d\right )}} \]
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Result contains complex when optimal does not.
Time = 0.91 (sec) , antiderivative size = 384, normalized size of antiderivative = 6.62 \[ \int \frac {1}{(-5+3 \sin (c+d x))^2} \, dx=\begin {cases} \frac {x}{\left (-5 + 3 \sin {\left (2 \operatorname {atan}{\left (\frac {3}{5} - \frac {4 i}{5} \right )} \right )}\right )^{2}} & \text {for}\: c = - d x + 2 \operatorname {atan}{\left (\frac {3}{5} - \frac {4 i}{5} \right )} \\\frac {x}{\left (-5 + 3 \sin {\left (2 \operatorname {atan}{\left (\frac {3}{5} + \frac {4 i}{5} \right )} \right )}\right )^{2}} & \text {for}\: c = - d x + 2 \operatorname {atan}{\left (\frac {3}{5} + \frac {4 i}{5} \right )} \\\frac {x}{\left (3 \sin {\left (c \right )} - 5\right )^{2}} & \text {for}\: d = 0 \\\frac {125 \left (\operatorname {atan}{\left (\frac {5 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4} - \frac {3}{4} \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 )}}{800 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 960 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 800 d} - \frac {150 \left (\operatorname {atan}{\left (\frac {5 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4} - \frac {3}{4} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{800 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 960 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 800 d} + \frac {125 \left (\operatorname {atan}{\left (\frac {5 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4} - \frac {3}{4} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{800 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 960 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 800 d} + \frac {36 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{800 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 960 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 800 d} - \frac {60}{800 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 960 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 800 d} & \text {otherwise} \end {cases} \]
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Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.60 \[ \int \frac {1}{(-5+3 \sin (c+d x))^2} \, dx=-\frac {\frac {12 \, {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 5\right )}}{\frac {6 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {5 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 5} - 25 \, \arctan \left (\frac {5 \, \sin \left (d x + c\right )}{4 \, {\left (\cos \left (d x + c\right ) + 1\right )}} - \frac {3}{4}\right )}{160 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.66 \[ \int \frac {1}{(-5+3 \sin (c+d x))^2} \, dx=\frac {25 \, d x + 25 \, c + \frac {24 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5\right )}}{5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5} + 50 \, \arctan \left (\frac {3 \, \cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 3}{\cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right ) - 9}\right )}{320 \, d} \]
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Time = 0.00 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.43 \[ \int \frac {1}{(-5+3 \sin (c+d x))^2} \, dx=\frac {5\,\mathrm {atan}\left (\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}-\frac {3}{4}\right )}{32\,d}-\frac {5\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{32\,d}+\frac {\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{200}-\frac {3}{40}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}+1\right )} \]
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