Integrand size = 21, antiderivative size = 114 \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {4 (a-a \cos (c+d x))^5}{5 a^7 d}-\frac {2 (a-a \cos (c+d x))^6}{a^8 d}+\frac {13 (a-a \cos (c+d x))^7}{7 a^9 d}-\frac {3 (a-a \cos (c+d x))^8}{4 a^{10} d}+\frac {(a-a \cos (c+d x))^9}{9 a^{11} d} \]
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Time = 0.22 (sec) , antiderivative size = 114, 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, 90} \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {(a-a \cos (c+d x))^9}{9 a^{11} d}-\frac {3 (a-a \cos (c+d x))^8}{4 a^{10} d}+\frac {13 (a-a \cos (c+d x))^7}{7 a^9 d}-\frac {2 (a-a \cos (c+d x))^6}{a^8 d}+\frac {4 (a-a \cos (c+d x))^5}{5 a^7 d} \]
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Rule 12
Rule 90
Rule 2915
Rule 3957
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos ^2(c+d x) \sin ^9(c+d x)}{(-a-a \cos (c+d x))^2} \, dx \\ & = \frac {\text {Subst}\left (\int \frac {(-a-x)^4 x^2 (-a+x)^2}{a^2} \, dx,x,-a \cos (c+d x)\right )}{a^9 d} \\ & = \frac {\text {Subst}\left (\int (-a-x)^4 x^2 (-a+x)^2 \, dx,x,-a \cos (c+d x)\right )}{a^{11} d} \\ & = \frac {\text {Subst}\left (\int \left (4 a^4 (-a-x)^4+12 a^3 (-a-x)^5+13 a^2 (-a-x)^6+6 a (-a-x)^7+(-a-x)^8\right ) \, dx,x,-a \cos (c+d x)\right )}{a^{11} d} \\ & = \frac {4 (a-a \cos (c+d x))^5}{5 a^7 d}-\frac {2 (a-a \cos (c+d x))^6}{a^8 d}+\frac {13 (a-a \cos (c+d x))^7}{7 a^9 d}-\frac {3 (a-a \cos (c+d x))^8}{4 a^{10} d}+\frac {(a-a \cos (c+d x))^9}{9 a^{11} d} \\ \end{align*}
Time = 2.93 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.54 \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {2 (992+1615 \cos (c+d x)+970 \cos (2 (c+d x))+385 \cos (3 (c+d x))+70 \cos (4 (c+d x))) \sin ^{10}\left (\frac {1}{2} (c+d x)\right )}{315 a^2 d} \]
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Time = 0.96 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.69
method | result | size |
derivativedivides | \(\frac {-\frac {\cos \left (d x +c \right )^{9}}{9}+\frac {\cos \left (d x +c \right )^{8}}{4}+\frac {\cos \left (d x +c \right )^{7}}{7}-\frac {2 \cos \left (d x +c \right )^{6}}{3}+\frac {\cos \left (d x +c \right )^{5}}{5}+\frac {\cos \left (d x +c \right )^{4}}{2}-\frac {\cos \left (d x +c \right )^{3}}{3}}{d \,a^{2}}\) | \(79\) |
default | \(\frac {-\frac {\cos \left (d x +c \right )^{9}}{9}+\frac {\cos \left (d x +c \right )^{8}}{4}+\frac {\cos \left (d x +c \right )^{7}}{7}-\frac {2 \cos \left (d x +c \right )^{6}}{3}+\frac {\cos \left (d x +c \right )^{5}}{5}+\frac {\cos \left (d x +c \right )^{4}}{2}-\frac {\cos \left (d x +c \right )^{3}}{3}}{d \,a^{2}}\) | \(79\) |
parallelrisch | \(\frac {-\frac {64}{315}-\frac {128 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{3}-\frac {64 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{35}-\frac {256 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{15}-\frac {128 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{5}-\frac {256 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{35}}{d \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{9} a^{2}}\) | \(91\) |
norman | \(\frac {-\frac {64}{315 a d}-\frac {128 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{3 a d}-\frac {64 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{35 d a}-\frac {256 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{35 d a}-\frac {256 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{15 d a}-\frac {128 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{5 d a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{9} a}\) | \(124\) |
risch | \(-\frac {13 \cos \left (d x +c \right )}{128 a^{2} d}-\frac {\cos \left (9 d x +9 c \right )}{2304 d \,a^{2}}+\frac {\cos \left (8 d x +8 c \right )}{512 d \,a^{2}}-\frac {3 \cos \left (7 d x +7 c \right )}{1792 d \,a^{2}}-\frac {\cos \left (6 d x +6 c \right )}{192 d \,a^{2}}+\frac {\cos \left (5 d x +5 c \right )}{80 d \,a^{2}}-\frac {\cos \left (4 d x +4 c \right )}{128 d \,a^{2}}-\frac {\cos \left (3 d x +3 c \right )}{96 d \,a^{2}}+\frac {3 \cos \left (2 d x +2 c \right )}{64 d \,a^{2}}\) | \(152\) |
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Time = 0.27 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.69 \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {140 \, \cos \left (d x + c\right )^{9} - 315 \, \cos \left (d x + c\right )^{8} - 180 \, \cos \left (d x + c\right )^{7} + 840 \, \cos \left (d x + c\right )^{6} - 252 \, \cos \left (d x + c\right )^{5} - 630 \, \cos \left (d x + c\right )^{4} + 420 \, \cos \left (d x + c\right )^{3}}{1260 \, a^{2} d} \]
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Timed out. \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.69 \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {140 \, \cos \left (d x + c\right )^{9} - 315 \, \cos \left (d x + c\right )^{8} - 180 \, \cos \left (d x + c\right )^{7} + 840 \, \cos \left (d x + c\right )^{6} - 252 \, \cos \left (d x + c\right )^{5} - 630 \, \cos \left (d x + c\right )^{4} + 420 \, \cos \left (d x + c\right )^{3}}{1260 \, a^{2} d} \]
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Time = 0.37 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.24 \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {64 \, {\left (\frac {9 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {36 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {84 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {126 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {210 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 1\right )}}{315 \, a^{2} d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{9}} \]
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Time = 0.07 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.84 \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\frac {{\cos \left (c+d\,x\right )}^4}{2\,a^2}-\frac {{\cos \left (c+d\,x\right )}^3}{3\,a^2}+\frac {{\cos \left (c+d\,x\right )}^5}{5\,a^2}-\frac {2\,{\cos \left (c+d\,x\right )}^6}{3\,a^2}+\frac {{\cos \left (c+d\,x\right )}^7}{7\,a^2}+\frac {{\cos \left (c+d\,x\right )}^8}{4\,a^2}-\frac {{\cos \left (c+d\,x\right )}^9}{9\,a^2}}{d} \]
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