Integrand size = 29, antiderivative size = 176 \[ \int \frac {\sec ^3(c+d x) \tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 \text {arctanh}(\sin (c+d x))}{256 a d}-\frac {3 \sec (c+d x) \tan (c+d x)}{256 a d}-\frac {\sec ^3(c+d x) \tan (c+d x)}{128 a d}+\frac {\sec ^5(c+d x) \tan (c+d x)}{32 a d}-\frac {\sec ^5(c+d x) \tan ^3(c+d x)}{16 a d}+\frac {\sec ^5(c+d x) \tan ^5(c+d x)}{10 a d}-\frac {\tan ^8(c+d x)}{8 a d}-\frac {\tan ^{10}(c+d x)}{10 a d} \]
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Time = 0.21 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2914, 2691, 3853, 3855, 2687, 14} \[ \int \frac {\sec ^3(c+d x) \tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 \text {arctanh}(\sin (c+d x))}{256 a d}-\frac {\tan ^{10}(c+d x)}{10 a d}-\frac {\tan ^8(c+d x)}{8 a d}+\frac {\tan ^5(c+d x) \sec ^5(c+d x)}{10 a d}-\frac {\tan ^3(c+d x) \sec ^5(c+d x)}{16 a d}+\frac {\tan (c+d x) \sec ^5(c+d x)}{32 a d}-\frac {\tan (c+d x) \sec ^3(c+d x)}{128 a d}-\frac {3 \tan (c+d x) \sec (c+d x)}{256 a d} \]
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Rule 14
Rule 2687
Rule 2691
Rule 2914
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^5(c+d x) \tan ^6(c+d x) \, dx}{a}-\frac {\int \sec ^4(c+d x) \tan ^7(c+d x) \, dx}{a} \\ & = \frac {\sec ^5(c+d x) \tan ^5(c+d x)}{10 a d}-\frac {\int \sec ^5(c+d x) \tan ^4(c+d x) \, dx}{2 a}-\frac {\text {Subst}\left (\int x^7 \left (1+x^2\right ) \, dx,x,\tan (c+d x)\right )}{a d} \\ & = -\frac {\sec ^5(c+d x) \tan ^3(c+d x)}{16 a d}+\frac {\sec ^5(c+d x) \tan ^5(c+d x)}{10 a d}+\frac {3 \int \sec ^5(c+d x) \tan ^2(c+d x) \, dx}{16 a}-\frac {\text {Subst}\left (\int \left (x^7+x^9\right ) \, dx,x,\tan (c+d x)\right )}{a d} \\ & = \frac {\sec ^5(c+d x) \tan (c+d x)}{32 a d}-\frac {\sec ^5(c+d x) \tan ^3(c+d x)}{16 a d}+\frac {\sec ^5(c+d x) \tan ^5(c+d x)}{10 a d}-\frac {\tan ^8(c+d x)}{8 a d}-\frac {\tan ^{10}(c+d x)}{10 a d}-\frac {\int \sec ^5(c+d x) \, dx}{32 a} \\ & = -\frac {\sec ^3(c+d x) \tan (c+d x)}{128 a d}+\frac {\sec ^5(c+d x) \tan (c+d x)}{32 a d}-\frac {\sec ^5(c+d x) \tan ^3(c+d x)}{16 a d}+\frac {\sec ^5(c+d x) \tan ^5(c+d x)}{10 a d}-\frac {\tan ^8(c+d x)}{8 a d}-\frac {\tan ^{10}(c+d x)}{10 a d}-\frac {3 \int \sec ^3(c+d x) \, dx}{128 a} \\ & = -\frac {3 \sec (c+d x) \tan (c+d x)}{256 a d}-\frac {\sec ^3(c+d x) \tan (c+d x)}{128 a d}+\frac {\sec ^5(c+d x) \tan (c+d x)}{32 a d}-\frac {\sec ^5(c+d x) \tan ^3(c+d x)}{16 a d}+\frac {\sec ^5(c+d x) \tan ^5(c+d x)}{10 a d}-\frac {\tan ^8(c+d x)}{8 a d}-\frac {\tan ^{10}(c+d x)}{10 a d}-\frac {3 \int \sec (c+d x) \, dx}{256 a} \\ & = -\frac {3 \text {arctanh}(\sin (c+d x))}{256 a d}-\frac {3 \sec (c+d x) \tan (c+d x)}{256 a d}-\frac {\sec ^3(c+d x) \tan (c+d x)}{128 a d}+\frac {\sec ^5(c+d x) \tan (c+d x)}{32 a d}-\frac {\sec ^5(c+d x) \tan ^3(c+d x)}{16 a d}+\frac {\sec ^5(c+d x) \tan ^5(c+d x)}{10 a d}-\frac {\tan ^8(c+d x)}{8 a d}-\frac {\tan ^{10}(c+d x)}{10 a d} \\ \end{align*}
Time = 1.72 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.69 \[ \int \frac {\sec ^3(c+d x) \tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {30 \text {arctanh}(\sin (c+d x))-\frac {2 \left (32+47 \sin (c+d x)-113 \sin ^2(c+d x)-183 \sin ^3(c+d x)+137 \sin ^4(c+d x)+265 \sin ^5(c+d x)-55 \sin ^6(c+d x)+15 \sin ^7(c+d x)+15 \sin ^8(c+d x)\right )}{(-1+\sin (c+d x))^4 (1+\sin (c+d x))^5}}{2560 a d} \]
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Time = 1.69 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}+\frac {1}{64 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {9}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {1}{128 \left (\sin \left (d x +c \right )-1\right )}+\frac {3 \ln \left (\sin \left (d x +c \right )-1\right )}{512}-\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {7}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {5}{128 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {5}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5}{256 \left (1+\sin \left (d x +c \right )\right )}-\frac {3 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\) | \(139\) |
default | \(\frac {\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}+\frac {1}{64 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {9}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {1}{128 \left (\sin \left (d x +c \right )-1\right )}+\frac {3 \ln \left (\sin \left (d x +c \right )-1\right )}{512}-\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {7}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {5}{128 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {5}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5}{256 \left (1+\sin \left (d x +c \right )\right )}-\frac {3 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\) | \(139\) |
risch | \(\frac {i \left (-2330 i {\mathrm e}^{14 i \left (d x +c \right )}+15 \,{\mathrm e}^{17 i \left (d x +c \right )}+30 i {\mathrm e}^{16 i \left (d x +c \right )}+100 \,{\mathrm e}^{15 i \left (d x +c \right )}+10698 i {\mathrm e}^{8 i \left (d x +c \right )}+1292 \,{\mathrm e}^{13 i \left (d x +c \right )}-5374 i {\mathrm e}^{6 i \left (d x +c \right )}+924 \,{\mathrm e}^{11 i \left (d x +c \right )}+5374 i {\mathrm e}^{12 i \left (d x +c \right )}+3530 \,{\mathrm e}^{9 i \left (d x +c \right )}-10698 i {\mathrm e}^{10 i \left (d x +c \right )}+924 \,{\mathrm e}^{7 i \left (d x +c \right )}+2330 i {\mathrm e}^{4 i \left (d x +c \right )}+1292 \,{\mathrm e}^{5 i \left (d x +c \right )}-30 i {\mathrm e}^{2 i \left (d x +c \right )}+100 \,{\mathrm e}^{3 i \left (d x +c \right )}+15 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{640 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{10} d a}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{256 d a}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{256 a d}\) | \(277\) |
parallelrisch | \(\frac {\left (15 \cos \left (10 d x +10 c \right )+3150 \cos \left (2 d x +2 c \right )+1800 \cos \left (4 d x +4 c \right )+675 \cos \left (6 d x +6 c \right )+150 \cos \left (8 d x +8 c \right )+1890\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (-15 \cos \left (10 d x +10 c \right )-3150 \cos \left (2 d x +2 c \right )-1800 \cos \left (4 d x +4 c \right )-675 \cos \left (6 d x +6 c \right )-150 \cos \left (8 d x +8 c \right )-1890\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-20760 \sin \left (3 d x +3 c \right )+6936 \sin \left (5 d x +5 c \right )-290 \sin \left (7 d x +7 c \right )-30 \sin \left (9 d x +9 c \right )-32 \cos \left (10 d x +10 c \right )+29120 \cos \left (2 d x +2 c \right )-14080 \cos \left (4 d x +4 c \right )+3680 \cos \left (6 d x +6 c \right )-320 \cos \left (8 d x +8 c \right )+37580 \sin \left (d x +c \right )-18368}{1280 a d \left (\cos \left (10 d x +10 c \right )+10 \cos \left (8 d x +8 c \right )+45 \cos \left (6 d x +6 c \right )+120 \cos \left (4 d x +4 c \right )+210 \cos \left (2 d x +2 c \right )+126\right )}\) | \(315\) |
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Time = 0.29 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.06 \[ \int \frac {\sec ^3(c+d x) \tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {30 \, \cos \left (d x + c\right )^{8} - 10 \, \cos \left (d x + c\right )^{6} + 124 \, \cos \left (d x + c\right )^{4} - 112 \, \cos \left (d x + c\right )^{2} - 15 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (15 \, \cos \left (d x + c\right )^{6} - 310 \, \cos \left (d x + c\right )^{4} + 392 \, \cos \left (d x + c\right )^{2} - 144\right )} \sin \left (d x + c\right ) + 32}{2560 \, {\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\right )}} \]
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Timed out. \[ \int \frac {\sec ^3(c+d x) \tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.22 \[ \int \frac {\sec ^3(c+d x) \tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{8} + 15 \, \sin \left (d x + c\right )^{7} - 55 \, \sin \left (d x + c\right )^{6} + 265 \, \sin \left (d x + c\right )^{5} + 137 \, \sin \left (d x + c\right )^{4} - 183 \, \sin \left (d x + c\right )^{3} - 113 \, \sin \left (d x + c\right )^{2} + 47 \, \sin \left (d x + c\right ) + 32\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} - \frac {15 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {15 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{2560 \, d} \]
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Time = 0.45 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.89 \[ \int \frac {\sec ^3(c+d x) \tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {5 \, {\left (25 \, \sin \left (d x + c\right )^{4} - 84 \, \sin \left (d x + c\right )^{3} + 66 \, \sin \left (d x + c\right )^{2} - 12 \, \sin \left (d x + c\right ) - 3\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac {137 \, \sin \left (d x + c\right )^{5} + 885 \, \sin \left (d x + c\right )^{4} + 2270 \, \sin \left (d x + c\right )^{3} + 2470 \, \sin \left (d x + c\right )^{2} + 1265 \, \sin \left (d x + c\right ) + 253}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{10240 \, d} \]
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Time = 18.12 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.82 \[ \int \frac {\sec ^3(c+d x) \tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}+\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{64}-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{32}-\frac {23\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{64}+\frac {67\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{160}+\frac {383\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{320}+\frac {2841\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{160}+\frac {741\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{320}+\frac {1377\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}+\frac {741\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{320}+\frac {2841\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{160}+\frac {383\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{320}+\frac {67\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}-\frac {23\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64}-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32}+\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64}+\frac {3\,\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 )}-\frac {3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,a\,d} \]
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