Integrand size = 29, antiderivative size = 109 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {51 x}{8 a^3}+\frac {7 \cos (c+d x)}{a^3 d}-\frac {\cos ^3(c+d x)}{a^3 d}-\frac {19 \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{4 a^3 d}+\frac {4 \cos (c+d x)}{a^3 d (1+\sin (c+d x))} \]
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Time = 0.19 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2954, 2951, 2718, 2715, 8, 2713, 2727} \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\cos ^3(c+d x)}{a^3 d}+\frac {7 \cos (c+d x)}{a^3 d}-\frac {\sin ^3(c+d x) \cos (c+d x)}{4 a^3 d}-\frac {19 \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac {4 \cos (c+d x)}{a^3 d (\sin (c+d x)+1)}+\frac {51 x}{8 a^3} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2718
Rule 2727
Rule 2951
Rule 2954
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sin (c+d x) (a-a \sin (c+d x))^3 \tan ^2(c+d x) \, dx}{a^6} \\ & = \frac {\int \left (4 a-4 a \sin (c+d x)+4 a \sin ^2(c+d x)-3 a \sin ^3(c+d x)+a \sin ^4(c+d x)-\frac {4 a}{1+\sin (c+d x)}\right ) \, dx}{a^4} \\ & = \frac {4 x}{a^3}+\frac {\int \sin ^4(c+d x) \, dx}{a^3}-\frac {3 \int \sin ^3(c+d x) \, dx}{a^3}-\frac {4 \int \sin (c+d x) \, dx}{a^3}+\frac {4 \int \sin ^2(c+d x) \, dx}{a^3}-\frac {4 \int \frac {1}{1+\sin (c+d x)} \, dx}{a^3} \\ & = \frac {4 x}{a^3}+\frac {4 \cos (c+d x)}{a^3 d}-\frac {2 \cos (c+d x) \sin (c+d x)}{a^3 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{4 a^3 d}+\frac {4 \cos (c+d x)}{a^3 d (1+\sin (c+d x))}+\frac {3 \int \sin ^2(c+d x) \, dx}{4 a^3}+\frac {2 \int 1 \, dx}{a^3}+\frac {3 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d} \\ & = \frac {6 x}{a^3}+\frac {7 \cos (c+d x)}{a^3 d}-\frac {\cos ^3(c+d x)}{a^3 d}-\frac {19 \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{4 a^3 d}+\frac {4 \cos (c+d x)}{a^3 d (1+\sin (c+d x))}+\frac {3 \int 1 \, dx}{8 a^3} \\ & = \frac {51 x}{8 a^3}+\frac {7 \cos (c+d x)}{a^3 d}-\frac {\cos ^3(c+d x)}{a^3 d}-\frac {19 \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{4 a^3 d}+\frac {4 \cos (c+d x)}{a^3 d (1+\sin (c+d x))} \\ \end{align*}
Time = 1.15 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.79 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {2040 d x \cos \left (\frac {d x}{2}\right )+997 \cos \left (c+\frac {d x}{2}\right )+800 \cos \left (c+\frac {3 d x}{2}\right )+160 \cos \left (3 c+\frac {5 d x}{2}\right )-35 \cos \left (3 c+\frac {7 d x}{2}\right )-5 \cos \left (5 c+\frac {9 d x}{2}\right )-3563 \sin \left (\frac {d x}{2}\right )+2040 d x \sin \left (c+\frac {d x}{2}\right )+800 \sin \left (2 c+\frac {3 d x}{2}\right )-160 \sin \left (2 c+\frac {5 d x}{2}\right )-35 \sin \left (4 c+\frac {7 d x}{2}\right )+5 \sin \left (4 c+\frac {9 d x}{2}\right )}{320 a^3 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]
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Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.06
method | result | size |
risch | \(\frac {51 x}{8 a^{3}}+\frac {25 \,{\mathrm e}^{i \left (d x +c \right )}}{8 d \,a^{3}}+\frac {25 \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d \,a^{3}}+\frac {8}{d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}+\frac {\sin \left (4 d x +4 c \right )}{32 d \,a^{3}}-\frac {\cos \left (3 d x +3 c \right )}{4 d \,a^{3}}-\frac {5 \sin \left (2 d x +2 c \right )}{4 d \,a^{3}}\) | \(115\) |
derivativedivides | \(\frac {\frac {8 \left (\frac {19 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {27 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {9 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {27 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {19 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}+\frac {3}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {51 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {16}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2}}{d \,a^{3}}\) | \(143\) |
default | \(\frac {\frac {8 \left (\frac {19 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {27 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {9 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {27 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {19 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}+\frac {3}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {51 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {16}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2}}{d \,a^{3}}\) | \(143\) |
parallelrisch | \(\frac {\left (408 d x +280\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (408 d x -632\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+160 \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-32 \sin \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )-7 \sin \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )+\sin \left (\frac {9 d x}{2}+\frac {9 c}{2}\right )+160 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+32 \cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )-7 \cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )-\cos \left (\frac {9 d x}{2}+\frac {9 c}{2}\right )}{64 d \,a^{3} \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(149\) |
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Time = 0.26 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.32 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {2 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{4} - 15 \, \cos \left (d x + c\right )^{3} - 51 \, d x - {\left (51 \, d x + 67\right )} \cos \left (d x + c\right ) - 56 \, \cos \left (d x + c\right )^{2} - {\left (2 \, \cos \left (d x + c\right )^{4} - 6 \, \cos \left (d x + c\right )^{3} + 51 \, d x - 21 \, \cos \left (d x + c\right )^{2} + 35 \, \cos \left (d x + c\right ) - 32\right )} \sin \left (d x + c\right ) - 32}{8 \, {\left (a^{3} d \cos \left (d x + c\right ) + a^{3} d \sin \left (d x + c\right ) + a^{3} d\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 3578 vs. \(2 (100) = 200\).
Time = 58.11 (sec) , antiderivative size = 3578, normalized size of antiderivative = 32.83 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (103) = 206\).
Time = 0.30 (sec) , antiderivative size = 398, normalized size of antiderivative = 3.65 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {\frac {29 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {269 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {133 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {309 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {171 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {187 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {51 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {51 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + 80}{a^{3} + \frac {a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {4 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {4 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {4 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {4 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {a^{3} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}} + \frac {51 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{4 \, d} \]
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Time = 0.35 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.33 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {51 \, {\left (d x + c\right )}}{a^{3}} + \frac {64}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}} + \frac {2 \, {\left (19 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 32 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 27 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 144 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 27 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 160 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 19 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 48\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a^{3}}}{8 \, d} \]
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Time = 14.21 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.34 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {51\,x}{8\,a^3}+\frac {\frac {51\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4}+\frac {51\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {187\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{4}+\frac {171\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+\frac {309\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{4}+\frac {133\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}+\frac {269\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4}+\frac {29\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+20}{a^3\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^4} \]
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