Integrand size = 27, antiderivative size = 124 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {x}{8 a^2}-\frac {2 \cos ^7(c+d x)}{35 a^2 d}-\frac {\cos (c+d x) \sin (c+d x)}{8 a^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{12 a^2 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{15 a^2 d}-\frac {\cos ^9(c+d x)}{5 d (a+a \sin (c+d x))^2} \]
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Time = 0.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2938, 2761, 2715, 8} \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {2 \cos ^7(c+d x)}{35 a^2 d}-\frac {\sin (c+d x) \cos ^5(c+d x)}{15 a^2 d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{12 a^2 d}-\frac {\sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac {x}{8 a^2}-\frac {\cos ^9(c+d x)}{5 d (a \sin (c+d x)+a)^2} \]
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Rule 8
Rule 2715
Rule 2761
Rule 2938
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^9(c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac {2 \int \frac {\cos ^8(c+d x)}{a+a \sin (c+d x)} \, dx}{5 a} \\ & = -\frac {2 \cos ^7(c+d x)}{35 a^2 d}-\frac {\cos ^9(c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac {2 \int \cos ^6(c+d x) \, dx}{5 a^2} \\ & = -\frac {2 \cos ^7(c+d x)}{35 a^2 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{15 a^2 d}-\frac {\cos ^9(c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac {\int \cos ^4(c+d x) \, dx}{3 a^2} \\ & = -\frac {2 \cos ^7(c+d x)}{35 a^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{12 a^2 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{15 a^2 d}-\frac {\cos ^9(c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac {\int \cos ^2(c+d x) \, dx}{4 a^2} \\ & = -\frac {2 \cos ^7(c+d x)}{35 a^2 d}-\frac {\cos (c+d x) \sin (c+d x)}{8 a^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{12 a^2 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{15 a^2 d}-\frac {\cos ^9(c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac {\int 1 \, dx}{8 a^2} \\ & = -\frac {x}{8 a^2}-\frac {2 \cos ^7(c+d x)}{35 a^2 d}-\frac {\cos (c+d x) \sin (c+d x)}{8 a^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{12 a^2 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{15 a^2 d}-\frac {\cos ^9(c+d x)}{5 d (a+a \sin (c+d x))^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(418\) vs. \(2(124)=248\).
Time = 3.77 (sec) , antiderivative size = 418, normalized size of antiderivative = 3.37 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {70 (7+24 d x) \cos \left (\frac {c}{2}\right )+1155 \cos \left (\frac {c}{2}+d x\right )+1155 \cos \left (\frac {3 c}{2}+d x\right )+210 \cos \left (\frac {3 c}{2}+2 d x\right )-210 \cos \left (\frac {5 c}{2}+2 d x\right )+525 \cos \left (\frac {5 c}{2}+3 d x\right )+525 \cos \left (\frac {7 c}{2}+3 d x\right )-210 \cos \left (\frac {7 c}{2}+4 d x\right )+210 \cos \left (\frac {9 c}{2}+4 d x\right )+63 \cos \left (\frac {9 c}{2}+5 d x\right )+63 \cos \left (\frac {11 c}{2}+5 d x\right )-70 \cos \left (\frac {11 c}{2}+6 d x\right )+70 \cos \left (\frac {13 c}{2}+6 d x\right )-15 \cos \left (\frac {13 c}{2}+7 d x\right )-15 \cos \left (\frac {15 c}{2}+7 d x\right )-490 \sin \left (\frac {c}{2}\right )+1680 d x \sin \left (\frac {c}{2}\right )-1155 \sin \left (\frac {c}{2}+d x\right )+1155 \sin \left (\frac {3 c}{2}+d x\right )+210 \sin \left (\frac {3 c}{2}+2 d x\right )+210 \sin \left (\frac {5 c}{2}+2 d x\right )-525 \sin \left (\frac {5 c}{2}+3 d x\right )+525 \sin \left (\frac {7 c}{2}+3 d x\right )-210 \sin \left (\frac {7 c}{2}+4 d x\right )-210 \sin \left (\frac {9 c}{2}+4 d x\right )-63 \sin \left (\frac {9 c}{2}+5 d x\right )+63 \sin \left (\frac {11 c}{2}+5 d x\right )-70 \sin \left (\frac {11 c}{2}+6 d x\right )-70 \sin \left (\frac {13 c}{2}+6 d x\right )+15 \sin \left (\frac {13 c}{2}+7 d x\right )-15 \sin \left (\frac {15 c}{2}+7 d x\right )}{13440 a^2 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \]
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Time = 0.30 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.72
method | result | size |
parallelrisch | \(\frac {-840 d x +15 \cos \left (7 d x +7 c \right )-63 \cos \left (5 d x +5 c \right )-525 \cos \left (3 d x +3 c \right )-1155 \cos \left (d x +c \right )+70 \sin \left (6 d x +6 c \right )+210 \sin \left (4 d x +4 c \right )-210 \sin \left (2 d x +2 c \right )-1728}{6720 d \,a^{2}}\) | \(89\) |
risch | \(-\frac {x}{8 a^{2}}-\frac {11 \cos \left (d x +c \right )}{64 a^{2} d}+\frac {\cos \left (7 d x +7 c \right )}{448 d \,a^{2}}+\frac {\sin \left (6 d x +6 c \right )}{96 d \,a^{2}}-\frac {3 \cos \left (5 d x +5 c \right )}{320 d \,a^{2}}+\frac {\sin \left (4 d x +4 c \right )}{32 d \,a^{2}}-\frac {5 \cos \left (3 d x +3 c \right )}{64 d \,a^{2}}-\frac {\sin \left (2 d x +2 c \right )}{32 d \,a^{2}}\) | \(124\) |
derivativedivides | \(\frac {\frac {4 \left (-\frac {\left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {\left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {11 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12}-2 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {31 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48}-\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-4 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {31 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48}-\frac {7 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10}-\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12}-\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16}-\frac {9}{70}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d \,a^{2}}\) | \(194\) |
default | \(\frac {\frac {4 \left (-\frac {\left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {\left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {11 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12}-2 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {31 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48}-\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-4 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {31 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48}-\frac {7 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10}-\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12}-\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16}-\frac {9}{70}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d \,a^{2}}\) | \(194\) |
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Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.56 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {120 \, \cos \left (d x + c\right )^{7} - 336 \, \cos \left (d x + c\right )^{5} - 105 \, d x + 35 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{840 \, a^{2} d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 3196 vs. \(2 (112) = 224\).
Time = 83.56 (sec) , antiderivative size = 3196, normalized size of antiderivative = 25.77 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 436 vs. \(2 (112) = 224\).
Time = 0.36 (sec) , antiderivative size = 436, normalized size of antiderivative = 3.52 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {672 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1540 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {1176 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1085 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {6720 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {840 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {1085 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {3360 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {1540 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {840 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {105 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - 216}{a^{2} + \frac {7 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {21 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {35 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {35 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {21 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {7 \, a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {a^{2} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}}} - \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{420 \, d} \]
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Time = 0.32 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.55 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {105 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 840 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 1540 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 3360 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 1085 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 840 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 6720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1085 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1176 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1540 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 672 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 216\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7} a^{2}}}{840 \, d} \]
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Time = 13.99 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.50 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {x}{8\,a^2}-\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{4}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{3}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {31\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{12}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {31\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{12}+\frac {14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}+\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {18}{35}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \]
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