Integrand size = 21, antiderivative size = 137 \[ \int \frac {\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {4 (a-a \cos (c+d x))^6}{3 a^8 d}-\frac {4 (a-a \cos (c+d x))^7}{a^9 d}+\frac {19 (a-a \cos (c+d x))^8}{4 a^{10} d}-\frac {25 (a-a \cos (c+d x))^9}{9 a^{11} d}+\frac {4 (a-a \cos (c+d x))^{10}}{5 a^{12} d}-\frac {(a-a \cos (c+d x))^{11}}{11 a^{13} d} \]
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Time = 0.24 (sec) , antiderivative size = 137, 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} \[ \int \frac {\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {(a-a \cos (c+d x))^{11}}{11 a^{13} d}+\frac {4 (a-a \cos (c+d x))^{10}}{5 a^{12} d}-\frac {25 (a-a \cos (c+d x))^9}{9 a^{11} d}+\frac {19 (a-a \cos (c+d x))^8}{4 a^{10} d}-\frac {4 (a-a \cos (c+d x))^7}{a^9 d}+\frac {4 (a-a \cos (c+d x))^6}{3 a^8 d} \]
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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 ^{11}(c+d x)}{(-a-a \cos (c+d x))^2} \, dx \\ & = \frac {\text {Subst}\left (\int \frac {(-a-x)^5 x^2 (-a+x)^3}{a^2} \, dx,x,-a \cos (c+d x)\right )}{a^{11} d} \\ & = \frac {\text {Subst}\left (\int (-a-x)^5 x^2 (-a+x)^3 \, dx,x,-a \cos (c+d x)\right )}{a^{13} d} \\ & = \frac {\text {Subst}\left (\int \left (-8 a^5 (-a-x)^5-28 a^4 (-a-x)^6-38 a^3 (-a-x)^7-25 a^2 (-a-x)^8-8 a (-a-x)^9-(-a-x)^{10}\right ) \, dx,x,-a \cos (c+d x)\right )}{a^{13} d} \\ & = \frac {4 (a-a \cos (c+d x))^6}{3 a^8 d}-\frac {4 (a-a \cos (c+d x))^7}{a^9 d}+\frac {19 (a-a \cos (c+d x))^8}{4 a^{10} d}-\frac {25 (a-a \cos (c+d x))^9}{9 a^{11} d}+\frac {4 (a-a \cos (c+d x))^{10}}{5 a^{12} d}-\frac {(a-a \cos (c+d x))^{11}}{11 a^{13} d} \\ \end{align*}
Time = 3.90 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.53 \[ \int \frac {\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {4 (2360+4038 \cos (c+d x)+2586 \cos (2 (c+d x))+1189 \cos (3 (c+d x))+342 \cos (4 (c+d x))+45 \cos (5 (c+d x))) \sin ^{12}\left (\frac {1}{2} (c+d x)\right )}{495 a^2 d} \]
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Time = 1.16 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.65
method | result | size |
derivativedivides | \(\frac {\frac {\cos \left (d x +c \right )^{11}}{11}-\frac {\cos \left (d x +c \right )^{10}}{5}-\frac {2 \cos \left (d x +c \right )^{9}}{9}+\frac {3 \cos \left (d x +c \right )^{8}}{4}-\cos \left (d x +c \right )^{6}+\frac {2 \cos \left (d x +c \right )^{5}}{5}+\frac {\cos \left (d x +c \right )^{4}}{2}-\frac {\cos \left (d x +c \right )^{3}}{3}}{d \,a^{2}}\) | \(89\) |
default | \(\frac {\frac {\cos \left (d x +c \right )^{11}}{11}-\frac {\cos \left (d x +c \right )^{10}}{5}-\frac {2 \cos \left (d x +c \right )^{9}}{9}+\frac {3 \cos \left (d x +c \right )^{8}}{4}-\cos \left (d x +c \right )^{6}+\frac {2 \cos \left (d x +c \right )^{5}}{5}+\frac {\cos \left (d x +c \right )^{4}}{2}-\frac {\cos \left (d x +c \right )^{3}}{3}}{d \,a^{2}}\) | \(89\) |
parallelrisch | \(\frac {-42470-3960 \cos \left (4 d x +4 c \right )-990 \cos \left (3 d x +3 c \right )+45 \cos \left (11 d x +11 c \right )-34650 \cos \left (d x +c \right )+990 \cos \left (8 d x +8 c \right )-1485 \cos \left (7 d x +7 c \right )-990 \cos \left (6 d x +6 c \right )+4257 \cos \left (5 d x +5 c \right )+13860 \cos \left (2 d x +2 c \right )+55 \cos \left (9 d x +9 c \right )-198 \cos \left (10 d x +10 c \right )}{506880 a^{2} d}\) | \(129\) |
risch | \(-\frac {35 \cos \left (d x +c \right )}{512 a^{2} d}+\frac {\cos \left (11 d x +11 c \right )}{11264 d \,a^{2}}-\frac {\cos \left (10 d x +10 c \right )}{2560 d \,a^{2}}+\frac {\cos \left (9 d x +9 c \right )}{9216 d \,a^{2}}+\frac {\cos \left (8 d x +8 c \right )}{512 d \,a^{2}}-\frac {3 \cos \left (7 d x +7 c \right )}{1024 d \,a^{2}}-\frac {\cos \left (6 d x +6 c \right )}{512 d \,a^{2}}+\frac {43 \cos \left (5 d x +5 c \right )}{5120 d \,a^{2}}-\frac {\cos \left (4 d x +4 c \right )}{128 d \,a^{2}}-\frac {\cos \left (3 d x +3 c \right )}{512 d \,a^{2}}+\frac {7 \cos \left (2 d x +2 c \right )}{256 d \,a^{2}}\) | \(186\) |
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Time = 0.30 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.65 \[ \int \frac {\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {180 \, \cos \left (d x + c\right )^{11} - 396 \, \cos \left (d x + c\right )^{10} - 440 \, \cos \left (d x + c\right )^{9} + 1485 \, \cos \left (d x + c\right )^{8} - 1980 \, \cos \left (d x + c\right )^{6} + 792 \, \cos \left (d x + c\right )^{5} + 990 \, \cos \left (d x + c\right )^{4} - 660 \, \cos \left (d x + c\right )^{3}}{1980 \, a^{2} d} \]
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Timed out. \[ \int \frac {\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.65 \[ \int \frac {\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {180 \, \cos \left (d x + c\right )^{11} - 396 \, \cos \left (d x + c\right )^{10} - 440 \, \cos \left (d x + c\right )^{9} + 1485 \, \cos \left (d x + c\right )^{8} - 1980 \, \cos \left (d x + c\right )^{6} + 792 \, \cos \left (d x + c\right )^{5} + 990 \, \cos \left (d x + c\right )^{4} - 660 \, \cos \left (d x + c\right )^{3}}{1980 \, a^{2} d} \]
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Time = 0.38 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.35 \[ \int \frac {\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {64 \, {\left (\frac {11 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {55 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {165 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {330 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {462 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {198 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {990 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 1\right )}}{495 \, a^{2} d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{11}} \]
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Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.80 \[ \int \frac {\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {{\cos \left (c+d\,x\right )}^3}{3\,a^2}-\frac {{\cos \left (c+d\,x\right )}^4}{2\,a^2}-\frac {2\,{\cos \left (c+d\,x\right )}^5}{5\,a^2}+\frac {{\cos \left (c+d\,x\right )}^6}{a^2}-\frac {3\,{\cos \left (c+d\,x\right )}^8}{4\,a^2}+\frac {2\,{\cos \left (c+d\,x\right )}^9}{9\,a^2}+\frac {{\cos \left (c+d\,x\right )}^{10}}{5\,a^2}-\frac {{\cos \left (c+d\,x\right )}^{11}}{11\,a^2}}{d} \]
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