Integrand size = 19, antiderivative size = 152 \[ \int (a+a \sec (c+d x)) \sin ^9(c+d x) \, dx=-\frac {a \cos (c+d x)}{d}+\frac {2 a \cos ^2(c+d x)}{d}+\frac {4 a \cos ^3(c+d x)}{3 d}-\frac {3 a \cos ^4(c+d x)}{2 d}-\frac {6 a \cos ^5(c+d x)}{5 d}+\frac {2 a \cos ^6(c+d x)}{3 d}+\frac {4 a \cos ^7(c+d x)}{7 d}-\frac {a \cos ^8(c+d x)}{8 d}-\frac {a \cos ^9(c+d x)}{9 d}-\frac {a \log (\cos (c+d x))}{d} \]
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Time = 0.13 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3957, 2915, 12, 90} \[ \int (a+a \sec (c+d x)) \sin ^9(c+d x) \, dx=-\frac {a \cos ^9(c+d x)}{9 d}-\frac {a \cos ^8(c+d x)}{8 d}+\frac {4 a \cos ^7(c+d x)}{7 d}+\frac {2 a \cos ^6(c+d x)}{3 d}-\frac {6 a \cos ^5(c+d x)}{5 d}-\frac {3 a \cos ^4(c+d x)}{2 d}+\frac {4 a \cos ^3(c+d x)}{3 d}+\frac {2 a \cos ^2(c+d x)}{d}-\frac {a \cos (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]
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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)) \sin ^8(c+d x) \tan (c+d x) \, dx \\ & = \frac {\text {Subst}\left (\int \frac {a (-a-x)^4 (-a+x)^5}{x} \, dx,x,-a \cos (c+d x)\right )}{a^9 d} \\ & = \frac {\text {Subst}\left (\int \frac {(-a-x)^4 (-a+x)^5}{x} \, dx,x,-a \cos (c+d x)\right )}{a^8 d} \\ & = \frac {\text {Subst}\left (\int \left (a^8-\frac {a^9}{x}+4 a^7 x-4 a^6 x^2-6 a^5 x^3+6 a^4 x^4+4 a^3 x^5-4 a^2 x^6-a x^7+x^8\right ) \, dx,x,-a \cos (c+d x)\right )}{a^8 d} \\ & = -\frac {a \cos (c+d x)}{d}+\frac {2 a \cos ^2(c+d x)}{d}+\frac {4 a \cos ^3(c+d x)}{3 d}-\frac {3 a \cos ^4(c+d x)}{2 d}-\frac {6 a \cos ^5(c+d x)}{5 d}+\frac {2 a \cos ^6(c+d x)}{3 d}+\frac {4 a \cos ^7(c+d x)}{7 d}-\frac {a \cos ^8(c+d x)}{8 d}-\frac {a \cos ^9(c+d x)}{9 d}-\frac {a \log (\cos (c+d x))}{d} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.70 \[ \int (a+a \sec (c+d x)) \sin ^9(c+d x) \, dx=-\frac {a \left (39690 \cos (c+d x)-161280 \cos ^2(c+d x)+120960 \cos ^4(c+d x)-53760 \cos ^6(c+d x)+10080 \cos ^8(c+d x)-8820 \cos (3 (c+d x))+2268 \cos (5 (c+d x))-405 \cos (7 (c+d x))+35 \cos (9 (c+d x))+80640 \log (\cos (c+d x))\right )}{80640 d} \]
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Time = 1.58 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\sin \left (d x +c \right )^{8}}{8}-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a \left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )}{9}}{d}\) | \(107\) |
default | \(\frac {a \left (-\frac {\sin \left (d x +c \right )^{8}}{8}-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a \left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )}{9}}{d}\) | \(107\) |
parts | \(-\frac {a \left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )}{9 d}+\frac {a \left (-\frac {\sin \left (d x +c \right )^{8}}{8}-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(109\) |
parallelrisch | \(-\frac {a \left (-322560 \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+322560 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+322560 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+269777+28980 \cos \left (4 d x +4 c \right )-35280 \cos \left (3 d x +3 c \right )+158760 \cos \left (d x +c \right )+315 \cos \left (8 d x +8 c \right )-1620 \cos \left (7 d x +7 c \right )-4200 \cos \left (6 d x +6 c \right )+9072 \cos \left (5 d x +5 c \right )-163800 \cos \left (2 d x +2 c \right )+140 \cos \left (9 d x +9 c \right )\right )}{322560 d}\) | \(147\) |
risch | \(i a x +\frac {2 i a c}{d}+\frac {65 a \,{\mathrm e}^{2 i \left (d x +c \right )}}{256 d}+\frac {65 a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{256 d}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}-\frac {63 a \cos \left (d x +c \right )}{128 d}-\frac {a \cos \left (9 d x +9 c \right )}{2304 d}-\frac {a \cos \left (8 d x +8 c \right )}{1024 d}+\frac {9 a \cos \left (7 d x +7 c \right )}{1792 d}+\frac {5 a \cos \left (6 d x +6 c \right )}{384 d}-\frac {9 a \cos \left (5 d x +5 c \right )}{320 d}-\frac {23 a \cos \left (4 d x +4 c \right )}{256 d}+\frac {7 a \cos \left (3 d x +3 c \right )}{64 d}\) | \(180\) |
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Time = 0.31 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.76 \[ \int (a+a \sec (c+d x)) \sin ^9(c+d x) \, dx=-\frac {280 \, a \cos \left (d x + c\right )^{9} + 315 \, a \cos \left (d x + c\right )^{8} - 1440 \, a \cos \left (d x + c\right )^{7} - 1680 \, a \cos \left (d x + c\right )^{6} + 3024 \, a \cos \left (d x + c\right )^{5} + 3780 \, a \cos \left (d x + c\right )^{4} - 3360 \, a \cos \left (d x + c\right )^{3} - 5040 \, a \cos \left (d x + c\right )^{2} + 2520 \, a \cos \left (d x + c\right ) + 2520 \, a \log \left (-\cos \left (d x + c\right )\right )}{2520 \, d} \]
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Timed out. \[ \int (a+a \sec (c+d x)) \sin ^9(c+d x) \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.74 \[ \int (a+a \sec (c+d x)) \sin ^9(c+d x) \, dx=-\frac {280 \, a \cos \left (d x + c\right )^{9} + 315 \, a \cos \left (d x + c\right )^{8} - 1440 \, a \cos \left (d x + c\right )^{7} - 1680 \, a \cos \left (d x + c\right )^{6} + 3024 \, a \cos \left (d x + c\right )^{5} + 3780 \, a \cos \left (d x + c\right )^{4} - 3360 \, a \cos \left (d x + c\right )^{3} - 5040 \, a \cos \left (d x + c\right )^{2} + 2520 \, a \cos \left (d x + c\right ) + 2520 \, a \log \left (\cos \left (d x + c\right )\right )}{2520 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 293 vs. \(2 (138) = 276\).
Time = 0.33 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.93 \[ \int (a+a \sec (c+d x)) \sin ^9(c+d x) \, dx=\frac {2520 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 2520 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {9177 \, a - \frac {87633 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {375732 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {953988 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1594782 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {1336734 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {781956 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {302004 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {69201 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {7129 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{9}}}{2520 \, d} \]
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Time = 0.14 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.73 \[ \int (a+a \sec (c+d x)) \sin ^9(c+d x) \, dx=-\frac {a\,\cos \left (c+d\,x\right )-2\,a\,{\cos \left (c+d\,x\right )}^2-\frac {4\,a\,{\cos \left (c+d\,x\right )}^3}{3}+\frac {3\,a\,{\cos \left (c+d\,x\right )}^4}{2}+\frac {6\,a\,{\cos \left (c+d\,x\right )}^5}{5}-\frac {2\,a\,{\cos \left (c+d\,x\right )}^6}{3}-\frac {4\,a\,{\cos \left (c+d\,x\right )}^7}{7}+\frac {a\,{\cos \left (c+d\,x\right )}^8}{8}+\frac {a\,{\cos \left (c+d\,x\right )}^9}{9}+a\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
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