Integrand size = 21, antiderivative size = 109 \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\cos ^4(c+d x)}{4 a^3 d}+\frac {3 \cos ^5(c+d x)}{5 a^3 d}-\frac {\cos ^6(c+d x)}{3 a^3 d}-\frac {2 \cos ^7(c+d x)}{7 a^3 d}+\frac {3 \cos ^8(c+d x)}{8 a^3 d}-\frac {\cos ^9(c+d x)}{9 a^3 d} \]
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Time = 0.22 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2915, 12, 76} \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\cos ^9(c+d x)}{9 a^3 d}+\frac {3 \cos ^8(c+d x)}{8 a^3 d}-\frac {2 \cos ^7(c+d x)}{7 a^3 d}-\frac {\cos ^6(c+d x)}{3 a^3 d}+\frac {3 \cos ^5(c+d x)}{5 a^3 d}-\frac {\cos ^4(c+d x)}{4 a^3 d} \]
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Rule 12
Rule 76
Rule 2915
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos ^3(c+d x) \sin ^9(c+d x)}{(-a-a \cos (c+d x))^3} \, dx \\ & = \frac {\text {Subst}\left (\int \frac {(-a-x)^4 x^3 (-a+x)}{a^3} \, dx,x,-a \cos (c+d x)\right )}{a^9 d} \\ & = \frac {\text {Subst}\left (\int (-a-x)^4 x^3 (-a+x) \, dx,x,-a \cos (c+d x)\right )}{a^{12} d} \\ & = \frac {\text {Subst}\left (\int \left (-a^5 x^3-3 a^4 x^4-2 a^3 x^5+2 a^2 x^6+3 a x^7+x^8\right ) \, dx,x,-a \cos (c+d x)\right )}{a^{12} d} \\ & = -\frac {\cos ^4(c+d x)}{4 a^3 d}+\frac {3 \cos ^5(c+d x)}{5 a^3 d}-\frac {\cos ^6(c+d x)}{3 a^3 d}-\frac {2 \cos ^7(c+d x)}{7 a^3 d}+\frac {3 \cos ^8(c+d x)}{8 a^3 d}-\frac {\cos ^9(c+d x)}{9 a^3 d} \\ \end{align*}
Time = 1.86 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.92 \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {34771-52920 \cos (c+d x)+37800 \cos (2 (c+d x))-18480 \cos (3 (c+d x))+3780 \cos (4 (c+d x))+3024 \cos (5 (c+d x))-4200 \cos (6 (c+d x))+2700 \cos (7 (c+d x))-945 \cos (8 (c+d x))+140 \cos (9 (c+d x))}{322560 a^3 d} \]
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Time = 0.99 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.63
method | result | size |
derivativedivides | \(\frac {-\frac {\cos \left (d x +c \right )^{9}}{9}+\frac {3 \cos \left (d x +c \right )^{8}}{8}-\frac {2 \cos \left (d x +c \right )^{7}}{7}-\frac {\cos \left (d x +c \right )^{6}}{3}+\frac {3 \cos \left (d x +c \right )^{5}}{5}-\frac {\cos \left (d x +c \right )^{4}}{4}}{d \,a^{3}}\) | \(69\) |
default | \(\frac {-\frac {\cos \left (d x +c \right )^{9}}{9}+\frac {3 \cos \left (d x +c \right )^{8}}{8}-\frac {2 \cos \left (d x +c \right )^{7}}{7}-\frac {\cos \left (d x +c \right )^{6}}{3}+\frac {3 \cos \left (d x +c \right )^{5}}{5}-\frac {\cos \left (d x +c \right )^{4}}{4}}{d \,a^{3}}\) | \(69\) |
parallelrisch | \(\frac {-3780 \cos \left (4 d x +4 c \right )+18480 \cos \left (3 d x +3 c \right )+52920 \cos \left (d x +c \right )+945 \cos \left (8 d x +8 c \right )-2700 \cos \left (7 d x +7 c \right )+4200 \cos \left (6 d x +6 c \right )-3024 \cos \left (5 d x +5 c \right )+101971-37800 \cos \left (2 d x +2 c \right )-140 \cos \left (9 d x +9 c \right )}{322560 a^{3} d}\) | \(107\) |
risch | \(\frac {21 \cos \left (d x +c \right )}{128 a^{3} d}-\frac {\cos \left (9 d x +9 c \right )}{2304 d \,a^{3}}+\frac {3 \cos \left (8 d x +8 c \right )}{1024 d \,a^{3}}-\frac {15 \cos \left (7 d x +7 c \right )}{1792 d \,a^{3}}+\frac {5 \cos \left (6 d x +6 c \right )}{384 d \,a^{3}}-\frac {3 \cos \left (5 d x +5 c \right )}{320 d \,a^{3}}-\frac {3 \cos \left (4 d x +4 c \right )}{256 d \,a^{3}}+\frac {11 \cos \left (3 d x +3 c \right )}{192 d \,a^{3}}-\frac {15 \cos \left (2 d x +2 c \right )}{128 d \,a^{3}}\) | \(152\) |
norman | \(\frac {\frac {128}{315 a d}+\frac {32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{a d}+\frac {32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{3 a d}+\frac {64 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d a}+\frac {128 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{35 d a}+\frac {512 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{35 d a}+\frac {512 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{15 d a}+\frac {256 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{5 d a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{9} a^{2}}\) | \(162\) |
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Time = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.63 \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {280 \, \cos \left (d x + c\right )^{9} - 945 \, \cos \left (d x + c\right )^{8} + 720 \, \cos \left (d x + c\right )^{7} + 840 \, \cos \left (d x + c\right )^{6} - 1512 \, \cos \left (d x + c\right )^{5} + 630 \, \cos \left (d x + c\right )^{4}}{2520 \, a^{3} d} \]
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Timed out. \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.63 \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {280 \, \cos \left (d x + c\right )^{9} - 945 \, \cos \left (d x + c\right )^{8} + 720 \, \cos \left (d x + c\right )^{7} + 840 \, \cos \left (d x + c\right )^{6} - 1512 \, \cos \left (d x + c\right )^{5} + 630 \, \cos \left (d x + c\right )^{4}}{2520 \, a^{3} d} \]
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Time = 0.42 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.70 \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {32 \, {\left (\frac {36 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {144 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {336 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {504 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {630 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {105 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {315 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 4\right )}}{315 \, a^{3} d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{9}} \]
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Time = 13.18 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.77 \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {{\cos \left (c+d\,x\right )}^4}{4\,a^3}-\frac {3\,{\cos \left (c+d\,x\right )}^5}{5\,a^3}+\frac {{\cos \left (c+d\,x\right )}^6}{3\,a^3}+\frac {2\,{\cos \left (c+d\,x\right )}^7}{7\,a^3}-\frac {3\,{\cos \left (c+d\,x\right )}^8}{8\,a^3}+\frac {{\cos \left (c+d\,x\right )}^9}{9\,a^3}}{d} \]
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