Integrand size = 29, antiderivative size = 174 \[ \int \frac {\sec ^6(c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 \text {arctanh}(\sin (c+d x))}{256 a d}-\frac {\sec ^8(c+d x)}{8 a d}+\frac {\sec ^{10}(c+d x)}{10 a d}-\frac {3 \sec (c+d x) \tan (c+d x)}{256 a d}-\frac {\sec ^3(c+d x) \tan (c+d x)}{128 a d}-\frac {\sec ^5(c+d x) \tan (c+d x)}{160 a d}+\frac {3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}-\frac {\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d} \]
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Time = 0.18 (sec) , antiderivative size = 174, 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, 2686, 14, 2691, 3853, 3855} \[ \int \frac {\sec ^6(c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 \text {arctanh}(\sin (c+d x))}{256 a d}+\frac {\sec ^{10}(c+d x)}{10 a d}-\frac {\sec ^8(c+d x)}{8 a d}-\frac {\tan ^3(c+d x) \sec ^7(c+d x)}{10 a d}+\frac {3 \tan (c+d x) \sec ^7(c+d x)}{80 a d}-\frac {\tan (c+d x) \sec ^5(c+d x)}{160 a d}-\frac {\tan (c+d x) \sec ^3(c+d x)}{128 a d}-\frac {3 \tan (c+d x) \sec (c+d x)}{256 a d} \]
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Rule 14
Rule 2686
Rule 2691
Rule 2914
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^8(c+d x) \tan ^3(c+d x) \, dx}{a}-\frac {\int \sec ^7(c+d x) \tan ^4(c+d x) \, dx}{a} \\ & = -\frac {\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}+\frac {3 \int \sec ^7(c+d x) \tan ^2(c+d x) \, dx}{10 a}+\frac {\text {Subst}\left (\int x^7 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{a d} \\ & = \frac {3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}-\frac {\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}-\frac {3 \int \sec ^7(c+d x) \, dx}{80 a}+\frac {\text {Subst}\left (\int \left (-x^7+x^9\right ) \, dx,x,\sec (c+d x)\right )}{a d} \\ & = -\frac {\sec ^8(c+d x)}{8 a d}+\frac {\sec ^{10}(c+d x)}{10 a d}-\frac {\sec ^5(c+d x) \tan (c+d x)}{160 a d}+\frac {3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}-\frac {\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}-\frac {\int \sec ^5(c+d x) \, dx}{32 a} \\ & = -\frac {\sec ^8(c+d x)}{8 a d}+\frac {\sec ^{10}(c+d x)}{10 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)}{160 a d}+\frac {3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}-\frac {\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}-\frac {3 \int \sec ^3(c+d x) \, dx}{128 a} \\ & = -\frac {\sec ^8(c+d x)}{8 a d}+\frac {\sec ^{10}(c+d x)}{10 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)}{160 a d}+\frac {3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}-\frac {\sec ^7(c+d x) \tan ^3(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 {\sec ^8(c+d x)}{8 a d}+\frac {\sec ^{10}(c+d x)}{10 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)}{160 a d}+\frac {3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}-\frac {\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d} \\ \end{align*}
Time = 1.80 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.60 \[ \int \frac {\sec ^6(c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {30 \text {arctanh}(\sin (c+d x))-\frac {10}{(-1+\sin (c+d x))^4}+\frac {15}{(-1+\sin (c+d x))^2}-\frac {30}{-1+\sin (c+d x)}-\frac {16}{(1+\sin (c+d x))^5}+\frac {10}{(1+\sin (c+d x))^4}+\frac {20}{(1+\sin (c+d x))^3}+\frac {15}{(1+\sin (c+d x))^2}}{2560 a d} \]
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Time = 1.72 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.66
method | result | size |
derivativedivides | \(\frac {\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}-\frac {3}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {3}{256 \left (\sin \left (d x +c \right )-1\right )}+\frac {3 \ln \left (\sin \left (d x +c \right )-1\right )}{512}+\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {1}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{128 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {3}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {3 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\) | \(115\) |
default | \(\frac {\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}-\frac {3}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {3}{256 \left (\sin \left (d x +c \right )-1\right )}+\frac {3 \ln \left (\sin \left (d x +c \right )-1\right )}{512}+\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {1}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{128 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {3}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {3 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\) | \(115\) |
risch | \(\frac {i \left (268 \,{\mathrm e}^{5 i \left (d x +c \right )}+230 i {\mathrm e}^{14 i \left (d x +c \right )}+268 \,{\mathrm e}^{13 i \left (d x +c \right )}-11364 \,{\mathrm e}^{11 i \left (d x +c \right )}+13770 \,{\mathrm e}^{9 i \left (d x +c \right )}+30 i {\mathrm e}^{16 i \left (d x +c \right )}+1482 i {\mathrm e}^{8 i \left (d x +c \right )}+15 \,{\mathrm e}^{i \left (d x +c \right )}-766 i {\mathrm e}^{6 i \left (d x +c \right )}+766 i {\mathrm e}^{12 i \left (d x +c \right )}-1482 i {\mathrm e}^{10 i \left (d x +c \right )}-230 i {\mathrm e}^{4 i \left (d x +c \right )}+15 \,{\mathrm e}^{17 i \left (d x +c \right )}+100 \,{\mathrm e}^{15 i \left (d x +c \right )}-30 i {\mathrm e}^{2 i \left (d x +c \right )}+100 \,{\mathrm e}^{3 i \left (d x +c \right )}-11364 \,{\mathrm e}^{7 i \left (d x +c \right )}\right )}{640 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{10} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{8} d a}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{256 d a}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{256 a d}\) | \(277\) |
parallelrisch | \(\frac {\left (15 \cos \left (10 d x +10 c \right )+3150 \cos \left (2 d x +2 c \right )+1800 \cos \left (4 d x +4 c \right )+675 \cos \left (6 d x +6 c \right )+150 \cos \left (8 d x +8 c \right )+1890\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (-15 \cos \left (10 d x +10 c \right )-3150 \cos \left (2 d x +2 c \right )-1800 \cos \left (4 d x +4 c \right )-675 \cos \left (6 d x +6 c \right )-150 \cos \left (8 d x +8 c \right )-1890\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+20200 \sin \left (3 d x +3 c \right )-1256 \sin \left (5 d x +5 c \right )-290 \sin \left (7 d x +7 c \right )-30 \sin \left (9 d x +9 c \right )+32 \cos \left (10 d x +10 c \right )-34240 \cos \left (2 d x +2 c \right )+3840 \cos \left (4 d x +4 c \right )+1440 \cos \left (6 d x +6 c \right )+320 \cos \left (8 d x +8 c \right )-44340 \sin \left (d x +c \right )+28608}{1280 a d \left (\cos \left (10 d x +10 c \right )+10 \cos \left (8 d x +8 c \right )+45 \cos \left (6 d x +6 c \right )+120 \cos \left (4 d x +4 c \right )+210 \cos \left (2 d x +2 c \right )+126\right )}\) | \(315\) |
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Time = 0.30 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.07 \[ \int \frac {\sec ^6(c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {30 \, \cos \left (d x + c\right )^{8} - 10 \, \cos \left (d x + c\right )^{6} - 4 \, \cos \left (d x + c\right )^{4} - 368 \, \cos \left (d x + c\right )^{2} - 15 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (15 \, \cos \left (d x + c\right )^{6} + 10 \, \cos \left (d x + c\right )^{4} + 8 \, \cos \left (d x + c\right )^{2} - 16\right )} \sin \left (d x + c\right ) + 288}{2560 \, {\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\right )}} \]
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Timed out. \[ \int \frac {\sec ^6(c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.23 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.23 \[ \int \frac {\sec ^6(c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{8} + 15 \, \sin \left (d x + c\right )^{7} - 55 \, \sin \left (d x + c\right )^{6} - 55 \, \sin \left (d x + c\right )^{5} + 73 \, \sin \left (d x + c\right )^{4} + 73 \, \sin \left (d x + c\right )^{3} + 143 \, \sin \left (d x + c\right )^{2} - 17 \, \sin \left (d x + c\right ) - 32\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} - \frac {15 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {15 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{2560 \, d} \]
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Time = 0.53 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.90 \[ \int \frac {\sec ^6(c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {5 \, {\left (25 \, \sin \left (d x + c\right )^{4} - 124 \, \sin \left (d x + c\right )^{3} + 234 \, \sin \left (d x + c\right )^{2} - 196 \, \sin \left (d x + c\right ) + 53\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac {137 \, \sin \left (d x + c\right )^{5} + 685 \, \sin \left (d x + c\right )^{4} + 1310 \, \sin \left (d x + c\right )^{3} + 1110 \, \sin \left (d x + c\right )^{2} + 305 \, \sin \left (d x + c\right ) + 21}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{10240 \, d} \]
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Time = 18.55 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.85 \[ \int \frac {\sec ^6(c+d x) \tan ^3(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 {233\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{64}+\frac {323\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{160}+\frac {2687\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{320}-\frac {231\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{160}+\frac {5349\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{320}+\frac {353\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}+\frac {5349\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{320}-\frac {231\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{160}+\frac {2687\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{320}+\frac {323\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}+\frac {233\,{\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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