Integrand size = 21, antiderivative size = 154 \[ \int \frac {\tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {63 \text {arctanh}(\sin (c+d x))}{256 a d}-\frac {63 \sec (c+d x) \tan (c+d x)}{256 a d}+\frac {21 \sec (c+d x) \tan ^3(c+d x)}{128 a d}-\frac {21 \sec (c+d x) \tan ^5(c+d x)}{160 a d}+\frac {9 \sec (c+d x) \tan ^7(c+d x)}{80 a d}-\frac {\sec (c+d x) \tan ^9(c+d x)}{10 a d}+\frac {\tan ^{10}(c+d x)}{10 a d} \]
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Time = 0.15 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2785, 2687, 30, 2691, 3855} \[ \int \frac {\tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {63 \text {arctanh}(\sin (c+d x))}{256 a d}+\frac {\tan ^{10}(c+d x)}{10 a d}-\frac {\tan ^9(c+d x) \sec (c+d x)}{10 a d}+\frac {9 \tan ^7(c+d x) \sec (c+d x)}{80 a d}-\frac {21 \tan ^5(c+d x) \sec (c+d x)}{160 a d}+\frac {21 \tan ^3(c+d x) \sec (c+d x)}{128 a d}-\frac {63 \tan (c+d x) \sec (c+d x)}{256 a d} \]
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Rule 30
Rule 2687
Rule 2691
Rule 2785
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^2(c+d x) \tan ^9(c+d x) \, dx}{a}-\frac {\int \sec (c+d x) \tan ^{10}(c+d x) \, dx}{a} \\ & = -\frac {\sec (c+d x) \tan ^9(c+d x)}{10 a d}+\frac {9 \int \sec (c+d x) \tan ^8(c+d x) \, dx}{10 a}+\frac {\text {Subst}\left (\int x^9 \, dx,x,\tan (c+d x)\right )}{a d} \\ & = \frac {9 \sec (c+d x) \tan ^7(c+d x)}{80 a d}-\frac {\sec (c+d x) \tan ^9(c+d x)}{10 a d}+\frac {\tan ^{10}(c+d x)}{10 a d}-\frac {63 \int \sec (c+d x) \tan ^6(c+d x) \, dx}{80 a} \\ & = -\frac {21 \sec (c+d x) \tan ^5(c+d x)}{160 a d}+\frac {9 \sec (c+d x) \tan ^7(c+d x)}{80 a d}-\frac {\sec (c+d x) \tan ^9(c+d x)}{10 a d}+\frac {\tan ^{10}(c+d x)}{10 a d}+\frac {21 \int \sec (c+d x) \tan ^4(c+d x) \, dx}{32 a} \\ & = \frac {21 \sec (c+d x) \tan ^3(c+d x)}{128 a d}-\frac {21 \sec (c+d x) \tan ^5(c+d x)}{160 a d}+\frac {9 \sec (c+d x) \tan ^7(c+d x)}{80 a d}-\frac {\sec (c+d x) \tan ^9(c+d x)}{10 a d}+\frac {\tan ^{10}(c+d x)}{10 a d}-\frac {63 \int \sec (c+d x) \tan ^2(c+d x) \, dx}{128 a} \\ & = -\frac {63 \sec (c+d x) \tan (c+d x)}{256 a d}+\frac {21 \sec (c+d x) \tan ^3(c+d x)}{128 a d}-\frac {21 \sec (c+d x) \tan ^5(c+d x)}{160 a d}+\frac {9 \sec (c+d x) \tan ^7(c+d x)}{80 a d}-\frac {\sec (c+d x) \tan ^9(c+d x)}{10 a d}+\frac {\tan ^{10}(c+d x)}{10 a d}+\frac {63 \int \sec (c+d x) \, dx}{256 a} \\ & = \frac {63 \text {arctanh}(\sin (c+d x))}{256 a d}-\frac {63 \sec (c+d x) \tan (c+d x)}{256 a d}+\frac {21 \sec (c+d x) \tan ^3(c+d x)}{128 a d}-\frac {21 \sec (c+d x) \tan ^5(c+d x)}{160 a d}+\frac {9 \sec (c+d x) \tan ^7(c+d x)}{80 a d}-\frac {\sec (c+d x) \tan ^9(c+d x)}{10 a d}+\frac {\tan ^{10}(c+d x)}{10 a d} \\ \end{align*}
Time = 1.55 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.79 \[ \int \frac {\tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {630 \text {arctanh}(\sin (c+d x))+\frac {2 \left (128-187 \sin (c+d x)-827 \sin ^2(c+d x)+643 \sin ^3(c+d x)+1923 \sin ^4(c+d x)-765 \sin ^5(c+d x)-2045 \sin ^6(c+d x)+325 \sin ^7(c+d x)+965 \sin ^8(c+d x)\right )}{(-1+\sin (c+d x))^4 (1+\sin (c+d x))^5}}{2560 a d} \]
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Time = 1.81 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(\frac {\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}+\frac {1}{32 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {57}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {65}{256 \left (\sin \left (d x +c \right )-1\right )}-\frac {63 \ln \left (\sin \left (d x +c \right )-1\right )}{512}+\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {13}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {23}{128 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {187}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {1}{2 \sin \left (d x +c \right )+2}+\frac {63 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\) | \(139\) |
default | \(\frac {\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}+\frac {1}{32 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {57}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {65}{256 \left (\sin \left (d x +c \right )-1\right )}-\frac {63 \ln \left (\sin \left (d x +c \right )-1\right )}{512}+\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {13}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {23}{128 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {187}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {1}{2 \sin \left (d x +c \right )+2}+\frac {63 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\) | \(139\) |
risch | \(\frac {i \left (8708 \,{\mathrm e}^{5 i \left (d x +c \right )}+1570 i {\mathrm e}^{14 i \left (d x +c \right )}+650 i {\mathrm e}^{16 i \left (d x +c \right )}-658 i {\mathrm e}^{8 i \left (d x +c \right )}-3626 i {\mathrm e}^{6 i \left (d x +c \right )}+3626 i {\mathrm e}^{12 i \left (d x +c \right )}+658 i {\mathrm e}^{10 i \left (d x +c \right )}+15470 \,{\mathrm e}^{9 i \left (d x +c \right )}+965 \,{\mathrm e}^{i \left (d x +c \right )}+8708 \,{\mathrm e}^{13 i \left (d x +c \right )}-1484 \,{\mathrm e}^{11 i \left (d x +c \right )}+965 \,{\mathrm e}^{17 i \left (d x +c \right )}+460 \,{\mathrm e}^{15 i \left (d x +c \right )}-1570 i {\mathrm e}^{4 i \left (d x +c \right )}-650 i {\mathrm e}^{2 i \left (d x +c \right )}+460 \,{\mathrm e}^{3 i \left (d x +c \right )}-1484 \,{\mathrm e}^{7 i \left (d x +c \right )}\right )}{640 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{10} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{8} d a}-\frac {63 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{256 d a}+\frac {63 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{256 a d}\) | \(277\) |
parallelrisch | \(\frac {\left (-8820 \sin \left (3 d x +3 c \right )-6300 \sin \left (5 d x +5 c \right )-2205 \sin \left (7 d x +7 c \right )-315 \sin \left (9 d x +9 c \right )-35280 \cos \left (2 d x +2 c \right )-17640 \cos \left (4 d x +4 c \right )-5040 \cos \left (6 d x +6 c \right )-630 \cos \left (8 d x +8 c \right )-4410 \sin \left (d x +c \right )-22050\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (8820 \sin \left (3 d x +3 c \right )+6300 \sin \left (5 d x +5 c \right )+2205 \sin \left (7 d x +7 c \right )+315 \sin \left (9 d x +9 c \right )+35280 \cos \left (2 d x +2 c \right )+17640 \cos \left (4 d x +4 c \right )+5040 \cos \left (6 d x +6 c \right )+630 \cos \left (8 d x +8 c \right )+4410 \sin \left (d x +c \right )+22050\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-2016 \sin \left (3 d x +3 c \right )+600 \sin \left (5 d x +5 c \right )+9 \sin \left (7 d x +7 c \right )+187 \sin \left (9 d x +9 c \right )+17976 \cos \left (2 d x +2 c \right )+27888 \cos \left (4 d x +4 c \right )+3912 \cos \left (6 d x +6 c \right )+2304 \cos \left (8 d x +8 c \right )+1302 \sin \left (d x +c \right )+28560}{1280 a d \left (70+\sin \left (9 d x +9 c \right )+7 \sin \left (7 d x +7 c \right )+20 \sin \left (5 d x +5 c \right )+28 \sin \left (3 d x +3 c \right )+14 \sin \left (d x +c \right )+2 \cos \left (8 d x +8 c \right )+16 \cos \left (6 d x +6 c \right )+56 \cos \left (4 d x +4 c \right )+112 \cos \left (2 d x +2 c \right )\right )}\) | \(427\) |
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Time = 0.29 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.21 \[ \int \frac {\tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {1930 \, \cos \left (d x + c\right )^{8} - 3630 \, \cos \left (d x + c\right )^{6} + 3156 \, \cos \left (d x + c\right )^{4} - 1488 \, \cos \left (d x + c\right )^{2} + 315 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 315 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (325 \, \cos \left (d x + c\right )^{6} - 210 \, \cos \left (d x + c\right )^{4} + 88 \, \cos \left (d x + c\right )^{2} - 16\right )} \sin \left (d x + c\right ) + 288}{2560 \, {\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\right )}} \]
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Timed out. \[ \int \frac {\tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.23 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.39 \[ \int \frac {\tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {2 \, {\left (965 \, \sin \left (d x + c\right )^{8} + 325 \, \sin \left (d x + c\right )^{7} - 2045 \, \sin \left (d x + c\right )^{6} - 765 \, \sin \left (d x + c\right )^{5} + 1923 \, \sin \left (d x + c\right )^{4} + 643 \, \sin \left (d x + c\right )^{3} - 827 \, \sin \left (d x + c\right )^{2} - 187 \, \sin \left (d x + c\right ) + 128\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} + \frac {315 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {315 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{2560 \, d} \]
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Time = 0.43 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.01 \[ \int \frac {\tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {1260 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {1260 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {5 \, {\left (525 \, \sin \left (d x + c\right )^{4} - 1580 \, \sin \left (d x + c\right )^{3} + 1818 \, \sin \left (d x + c\right )^{2} - 932 \, \sin \left (d x + c\right ) + 177\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac {2877 \, \sin \left (d x + c\right )^{5} + 9265 \, \sin \left (d x + c\right )^{4} + 12030 \, \sin \left (d x + c\right )^{3} + 7430 \, \sin \left (d x + c\right )^{2} + 1965 \, \sin \left (d x + c\right ) + 113}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{10240 \, d} \]
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Time = 17.98 (sec) , antiderivative size = 497, normalized size of antiderivative = 3.23 \[ \int \frac {\tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {63\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,a\,d}-\frac {\frac {63\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}+\frac {63\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{64}-\frac {105\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{32}-\frac {483\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{64}+\frac {1407\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{160}+\frac {8043\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{320}-\frac {1779\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{160}-\frac {15159\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{320}+\frac {245\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}-\frac {15159\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{320}-\frac {1779\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{160}+\frac {8043\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{320}+\frac {1407\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}-\frac {483\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64}-\frac {105\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32}+\frac {63\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64}+\frac {63\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}-7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+20\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+56\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-28\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-112\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+14\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+140\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+14\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-112\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-28\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+56\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+20\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )} \]
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