Integrand size = 21, antiderivative size = 203 \[ \int (a+a \sec (c+d x))^3 \sin ^9(c+d x) \, dx=\frac {11 a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cos ^2(c+d x)}{d}-\frac {14 a^3 \cos ^3(c+d x)}{3 d}-\frac {7 a^3 \cos ^4(c+d x)}{2 d}+\frac {6 a^3 \cos ^5(c+d x)}{5 d}+\frac {11 a^3 \cos ^6(c+d x)}{6 d}+\frac {a^3 \cos ^7(c+d x)}{7 d}-\frac {3 a^3 \cos ^8(c+d x)}{8 d}-\frac {a^3 \cos ^9(c+d x)}{9 d}+\frac {a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \]
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Time = 0.24 (sec) , antiderivative size = 203, 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} \[ \int (a+a \sec (c+d x))^3 \sin ^9(c+d x) \, dx=-\frac {a^3 \cos ^9(c+d x)}{9 d}-\frac {3 a^3 \cos ^8(c+d x)}{8 d}+\frac {a^3 \cos ^7(c+d x)}{7 d}+\frac {11 a^3 \cos ^6(c+d x)}{6 d}+\frac {6 a^3 \cos ^5(c+d x)}{5 d}-\frac {7 a^3 \cos ^4(c+d x)}{2 d}-\frac {14 a^3 \cos ^3(c+d x)}{3 d}+\frac {3 a^3 \cos ^2(c+d x)}{d}+\frac {11 a^3 \cos (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \log (\cos (c+d x))}{d} \]
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Rule 12
Rule 90
Rule 2915
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-a-a \cos (c+d x))^3 \sin ^6(c+d x) \tan ^3(c+d x) \, dx \\ & = \frac {\text {Subst}\left (\int \frac {a^3 (-a-x)^4 (-a+x)^7}{x^3} \, dx,x,-a \cos (c+d x)\right )}{a^9 d} \\ & = \frac {\text {Subst}\left (\int \frac {(-a-x)^4 (-a+x)^7}{x^3} \, dx,x,-a \cos (c+d x)\right )}{a^6 d} \\ & = \frac {\text {Subst}\left (\int \left (-11 a^8-\frac {a^{11}}{x^3}+\frac {3 a^{10}}{x^2}+\frac {a^9}{x}+6 a^7 x+14 a^6 x^2-14 a^5 x^3-6 a^4 x^4+11 a^3 x^5-a^2 x^6-3 a x^7+x^8\right ) \, dx,x,-a \cos (c+d x)\right )}{a^6 d} \\ & = \frac {11 a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cos ^2(c+d x)}{d}-\frac {14 a^3 \cos ^3(c+d x)}{3 d}-\frac {7 a^3 \cos ^4(c+d x)}{2 d}+\frac {6 a^3 \cos ^5(c+d x)}{5 d}+\frac {11 a^3 \cos ^6(c+d x)}{6 d}+\frac {a^3 \cos ^7(c+d x)}{7 d}-\frac {3 a^3 \cos ^8(c+d x)}{8 d}-\frac {a^3 \cos ^9(c+d x)}{9 d}+\frac {a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \\ \end{align*}
Time = 1.93 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.73 \[ \int (a+a \sec (c+d x))^3 \sin ^9(c+d x) \, dx=\frac {a^3 (471450+11624760 \cos (c+d x)+2188872 \cos (3 (c+d x))+41160 \cos (4 (c+d x))-204156 \cos (5 (c+d x))-35805 \cos (6 (c+d x))+22972 \cos (7 (c+d x))+9030 \cos (8 (c+d x))-820 \cos (9 (c+d x))-945 \cos (10 (c+d x))-140 \cos (11 (c+d x))+645120 \log (\cos (c+d x))+210 \cos (2 (c+d x)) (-413+3072 \log (\cos (c+d x)))) \sec ^2(c+d x)}{1290240 d} \]
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Time = 3.02 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.07
method | result | size |
parallelrisch | \(\frac {a^{3} \left (-645120 \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+645120 \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+645120 \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-35805 \cos \left (6 d x +6 c \right )+22972 \cos \left (7 d x +7 c \right )+9030 \cos \left (8 d x +8 c \right )-820 \cos \left (9 d x +9 c \right )-945 \cos \left (10 d x +10 c \right )-140 \cos \left (11 d x +11 c \right )+11624760 \cos \left (d x +c \right )+6529934 \cos \left (2 d x +2 c \right )+2188872 \cos \left (3 d x +3 c \right )+41160 \cos \left (4 d x +4 c \right )-204156 \cos \left (5 d x +5 c \right )+7088114\right )}{645120 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(217\) |
derivativedivides | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{10}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{8}}{2}+\frac {2 \sin \left (d x +c \right )^{6}}{3}+\sin \left (d x +c \right )^{4}+2 \sin \left (d x +c \right )^{2}+4 \ln \left (\cos \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{10}}{\cos \left (d x +c \right )}+\left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{8}}{8}-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{3} \left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )}{9}}{d}\) | \(252\) |
default | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{10}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{8}}{2}+\frac {2 \sin \left (d x +c \right )^{6}}{3}+\sin \left (d x +c \right )^{4}+2 \sin \left (d x +c \right )^{2}+4 \ln \left (\cos \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{10}}{\cos \left (d x +c \right )}+\left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{8}}{8}-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{3} \left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )}{9}}{d}\) | \(252\) |
parts | \(-\frac {a^{3} \left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )}{9 d}+\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{10}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{8}}{2}+\frac {2 \sin \left (d x +c \right )^{6}}{3}+\sin \left (d x +c \right )^{4}+2 \sin \left (d x +c \right )^{2}+4 \ln \left (\cos \left (d x +c \right )\right )\right )}{d}+\frac {3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{8}}{8}-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}+\frac {3 a^{3} \left (\frac {\sin \left (d x +c \right )^{10}}{\cos \left (d x +c \right )}+\left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )\right )}{d}\) | \(260\) |
risch | \(-\frac {2 i a^{3} c}{d}-i a^{3} x -\frac {a^{3} \cos \left (9 d x +9 c \right )}{2304 d}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}-\frac {25 a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{64 d}+\frac {57 a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{256 d}+\frac {1059 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{256 d}+\frac {1059 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{256 d}+\frac {57 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{256 d}-\frac {25 a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{64 d}+\frac {2 a^{3} \left (3 \,{\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {3 a^{3} \cos \left (8 d x +8 c \right )}{1024 d}-\frac {3 a^{3} \cos \left (7 d x +7 c \right )}{1792 d}+\frac {13 a^{3} \cos \left (6 d x +6 c \right )}{384 d}+\frac {3 a^{3} \cos \left (5 d x +5 c \right )}{40 d}-\frac {45 a^{3} \cos \left (4 d x +4 c \right )}{256 d}\) | \(295\) |
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Time = 0.31 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.90 \[ \int (a+a \sec (c+d x))^3 \sin ^9(c+d x) \, dx=-\frac {35840 \, a^{3} \cos \left (d x + c\right )^{11} + 120960 \, a^{3} \cos \left (d x + c\right )^{10} - 46080 \, a^{3} \cos \left (d x + c\right )^{9} - 591360 \, a^{3} \cos \left (d x + c\right )^{8} - 387072 \, a^{3} \cos \left (d x + c\right )^{7} + 1128960 \, a^{3} \cos \left (d x + c\right )^{6} + 1505280 \, a^{3} \cos \left (d x + c\right )^{5} - 967680 \, a^{3} \cos \left (d x + c\right )^{4} - 3548160 \, a^{3} \cos \left (d x + c\right )^{3} - 322560 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) + 212205 \, a^{3} \cos \left (d x + c\right )^{2} - 967680 \, a^{3} \cos \left (d x + c\right ) - 161280 \, a^{3}}{322560 \, d \cos \left (d x + c\right )^{2}} \]
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Timed out. \[ \int (a+a \sec (c+d x))^3 \sin ^9(c+d x) \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.78 \[ \int (a+a \sec (c+d x))^3 \sin ^9(c+d x) \, dx=-\frac {280 \, a^{3} \cos \left (d x + c\right )^{9} + 945 \, a^{3} \cos \left (d x + c\right )^{8} - 360 \, a^{3} \cos \left (d x + c\right )^{7} - 4620 \, a^{3} \cos \left (d x + c\right )^{6} - 3024 \, a^{3} \cos \left (d x + c\right )^{5} + 8820 \, a^{3} \cos \left (d x + c\right )^{4} + 11760 \, a^{3} \cos \left (d x + c\right )^{3} - 7560 \, a^{3} \cos \left (d x + c\right )^{2} - 27720 \, a^{3} \cos \left (d x + c\right ) - 2520 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) - \frac {1260 \, {\left (6 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )}}{\cos \left (d x + c\right )^{2}}}{2520 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 396 vs. \(2 (187) = 374\).
Time = 0.49 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.95 \[ \int (a+a \sec (c+d x))^3 \sin ^9(c+d x) \, dx=-\frac {2520 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 2520 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \frac {1260 \, {\left (9 \, a^{3} + \frac {2 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {3 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}} + \frac {45257 \, a^{3} - \frac {392193 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {1467972 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {3001908 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3232782 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {2359854 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {1190196 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {397764 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {79281 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {7129 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{9}}}{2520 \, d} \]
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Time = 12.63 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.77 \[ \int (a+a \sec (c+d x))^3 \sin ^9(c+d x) \, dx=\frac {\frac {3\,a^3\,\cos \left (c+d\,x\right )+\frac {a^3}{2}}{{\cos \left (c+d\,x\right )}^2}+11\,a^3\,\cos \left (c+d\,x\right )+3\,a^3\,{\cos \left (c+d\,x\right )}^2-\frac {14\,a^3\,{\cos \left (c+d\,x\right )}^3}{3}-\frac {7\,a^3\,{\cos \left (c+d\,x\right )}^4}{2}+\frac {6\,a^3\,{\cos \left (c+d\,x\right )}^5}{5}+\frac {11\,a^3\,{\cos \left (c+d\,x\right )}^6}{6}+\frac {a^3\,{\cos \left (c+d\,x\right )}^7}{7}-\frac {3\,a^3\,{\cos \left (c+d\,x\right )}^8}{8}-\frac {a^3\,{\cos \left (c+d\,x\right )}^9}{9}+a^3\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
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