Integrand size = 27, antiderivative size = 154 \[ \int \frac {\sec ^8(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {7 \text {arctanh}(\sin (c+d x))}{256 a d}+\frac {\sec ^{10}(c+d x)}{10 a d}+\frac {7 \sec (c+d x) \tan (c+d x)}{256 a d}+\frac {7 \sec ^3(c+d x) \tan (c+d x)}{384 a d}+\frac {7 \sec ^5(c+d x) \tan (c+d x)}{480 a d}+\frac {\sec ^7(c+d x) \tan (c+d x)}{80 a d}-\frac {\sec ^9(c+d x) \tan (c+d x)}{10 a d} \]
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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 = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2914, 2686, 30, 2691, 3853, 3855} \[ \int \frac {\sec ^8(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {7 \text {arctanh}(\sin (c+d x))}{256 a d}+\frac {\sec ^{10}(c+d x)}{10 a d}-\frac {\tan (c+d x) \sec ^9(c+d x)}{10 a d}+\frac {\tan (c+d x) \sec ^7(c+d x)}{80 a d}+\frac {7 \tan (c+d x) \sec ^5(c+d x)}{480 a d}+\frac {7 \tan (c+d x) \sec ^3(c+d x)}{384 a d}+\frac {7 \tan (c+d x) \sec (c+d x)}{256 a d} \]
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Rule 30
Rule 2686
Rule 2691
Rule 2914
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^{10}(c+d x) \tan (c+d x) \, dx}{a}-\frac {\int \sec ^9(c+d x) \tan ^2(c+d x) \, dx}{a} \\ & = -\frac {\sec ^9(c+d x) \tan (c+d x)}{10 a d}+\frac {\int \sec ^9(c+d x) \, dx}{10 a}+\frac {\text {Subst}\left (\int x^9 \, dx,x,\sec (c+d x)\right )}{a d} \\ & = \frac {\sec ^{10}(c+d x)}{10 a d}+\frac {\sec ^7(c+d x) \tan (c+d x)}{80 a d}-\frac {\sec ^9(c+d x) \tan (c+d x)}{10 a d}+\frac {7 \int \sec ^7(c+d x) \, dx}{80 a} \\ & = \frac {\sec ^{10}(c+d x)}{10 a d}+\frac {7 \sec ^5(c+d x) \tan (c+d x)}{480 a d}+\frac {\sec ^7(c+d x) \tan (c+d x)}{80 a d}-\frac {\sec ^9(c+d x) \tan (c+d x)}{10 a d}+\frac {7 \int \sec ^5(c+d x) \, dx}{96 a} \\ & = \frac {\sec ^{10}(c+d x)}{10 a d}+\frac {7 \sec ^3(c+d x) \tan (c+d x)}{384 a d}+\frac {7 \sec ^5(c+d x) \tan (c+d x)}{480 a d}+\frac {\sec ^7(c+d x) \tan (c+d x)}{80 a d}-\frac {\sec ^9(c+d x) \tan (c+d x)}{10 a d}+\frac {7 \int \sec ^3(c+d x) \, dx}{128 a} \\ & = \frac {\sec ^{10}(c+d x)}{10 a d}+\frac {7 \sec (c+d x) \tan (c+d x)}{256 a d}+\frac {7 \sec ^3(c+d x) \tan (c+d x)}{384 a d}+\frac {7 \sec ^5(c+d x) \tan (c+d x)}{480 a d}+\frac {\sec ^7(c+d x) \tan (c+d x)}{80 a d}-\frac {\sec ^9(c+d x) \tan (c+d x)}{10 a d}+\frac {7 \int \sec (c+d x) \, dx}{256 a} \\ & = \frac {7 \text {arctanh}(\sin (c+d x))}{256 a d}+\frac {\sec ^{10}(c+d x)}{10 a d}+\frac {7 \sec (c+d x) \tan (c+d x)}{256 a d}+\frac {7 \sec ^3(c+d x) \tan (c+d x)}{384 a d}+\frac {7 \sec ^5(c+d x) \tan (c+d x)}{480 a d}+\frac {\sec ^7(c+d x) \tan (c+d x)}{80 a d}-\frac {\sec ^9(c+d x) \tan (c+d x)}{10 a d} \\ \end{align*}
Time = 3.51 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.75 \[ \int \frac {\sec ^8(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {210 \text {arctanh}(\sin (c+d x))+\frac {30}{(-1+\sin (c+d x))^4}-\frac {80}{(-1+\sin (c+d x))^3}+\frac {135}{(-1+\sin (c+d x))^2}-\frac {210}{-1+\sin (c+d x)}+\frac {48}{(1+\sin (c+d x))^5}+\frac {90}{(1+\sin (c+d x))^4}+\frac {100}{(1+\sin (c+d x))^3}+\frac {75}{(1+\sin (c+d x))^2}}{7680 a d} \]
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Time = 1.76 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {9}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {7}{256 \left (\sin \left (d x +c \right )-1\right )}-\frac {7 \ln \left (\sin \left (d x +c \right )-1\right )}{512}+\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {3}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {5}{384 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {5}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {7 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\) | \(127\) |
default | \(\frac {\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {9}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {7}{256 \left (\sin \left (d x +c \right )-1\right )}-\frac {7 \ln \left (\sin \left (d x +c \right )-1\right )}{512}+\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {3}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {5}{384 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {5}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {7 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\) | \(127\) |
risch | \(-\frac {i \left (1876 \,{\mathrm e}^{5 i \left (d x +c \right )}+1610 i {\mathrm e}^{14 i \left (d x +c \right )}+2372 \,{\mathrm e}^{11 i \left (d x +c \right )}-108410 \,{\mathrm e}^{9 i \left (d x +c \right )}+210 i {\mathrm e}^{16 i \left (d x +c \right )}-10106 i {\mathrm e}^{8 i \left (d x +c \right )}+700 \,{\mathrm e}^{15 i \left (d x +c \right )}+1876 \,{\mathrm e}^{13 i \left (d x +c \right )}+105 \,{\mathrm e}^{i \left (d x +c \right )}-5362 i {\mathrm e}^{6 i \left (d x +c \right )}+5362 i {\mathrm e}^{12 i \left (d x +c \right )}+10106 i {\mathrm e}^{10 i \left (d x +c \right )}-1610 i {\mathrm e}^{4 i \left (d x +c \right )}-210 i {\mathrm e}^{2 i \left (d x +c \right )}+105 \,{\mathrm e}^{17 i \left (d x +c \right )}+700 \,{\mathrm e}^{3 i \left (d x +c \right )}+2372 \,{\mathrm e}^{7 i \left (d x +c \right )}\right )}{1920 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{10} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{8} d a}+\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{256 a d}-\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{256 d a}\) | \(277\) |
parallelrisch | \(\frac {\left (-2940 \sin \left (3 d x +3 c \right )-2100 \sin \left (5 d x +5 c \right )-735 \sin \left (7 d x +7 c \right )-105 \sin \left (9 d x +9 c \right )-11760 \cos \left (2 d x +2 c \right )-5880 \cos \left (4 d x +4 c \right )-1680 \cos \left (6 d x +6 c \right )-210 \cos \left (8 d x +8 c \right )-1470 \sin \left (d x +c \right )-7350\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (2940 \sin \left (3 d x +3 c \right )+2100 \sin \left (5 d x +5 c \right )+735 \sin \left (7 d x +7 c \right )+105 \sin \left (9 d x +9 c \right )+11760 \cos \left (2 d x +2 c \right )+5880 \cos \left (4 d x +4 c \right )+1680 \cos \left (6 d x +6 c \right )+210 \cos \left (8 d x +8 c \right )+1470 \sin \left (d x +c \right )+7350\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-28 \sin \left (3 d x +3 c \right )-4460 \sin \left (5 d x +5 c \right )-2268 \sin \left (7 d x +7 c \right )-384 \sin \left (9 d x +9 c \right )-47752 \cos \left (2 d x +2 c \right )-25256 \cos \left (4 d x +4 c \right )-7544 \cos \left (6 d x +6 c \right )-978 \cos \left (8 d x +8 c \right )+14836 \sin \left (d x +c \right )+81530}{3840 a d \left (70+\sin \left (9 d x +9 c \right )+7 \sin \left (7 d x +7 c \right )+20 \sin \left (5 d x +5 c \right )+28 \sin \left (3 d x +3 c \right )+14 \sin \left (d x +c \right )+2 \cos \left (8 d x +8 c \right )+16 \cos \left (6 d x +6 c \right )+56 \cos \left (4 d x +4 c \right )+112 \cos \left (2 d x +2 c \right )\right )}\) | \(427\) |
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Time = 0.30 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.21 \[ \int \frac {\sec ^8(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {210 \, \cos \left (d x + c\right )^{8} - 70 \, \cos \left (d x + c\right )^{6} - 28 \, \cos \left (d x + c\right )^{4} - 16 \, \cos \left (d x + c\right )^{2} - 105 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (105 \, \cos \left (d x + c\right )^{6} + 70 \, \cos \left (d x + c\right )^{4} + 56 \, \cos \left (d x + c\right )^{2} + 48\right )} \sin \left (d x + c\right ) - 864}{7680 \, {\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\right )}} \]
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Timed out. \[ \int \frac {\sec ^8(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.39 \[ \int \frac {\sec ^8(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {2 \, {\left (105 \, \sin \left (d x + c\right )^{8} + 105 \, \sin \left (d x + c\right )^{7} - 385 \, \sin \left (d x + c\right )^{6} - 385 \, \sin \left (d x + c\right )^{5} + 511 \, \sin \left (d x + c\right )^{4} + 511 \, \sin \left (d x + c\right )^{3} - 279 \, \sin \left (d x + c\right )^{2} - 279 \, \sin \left (d x + c\right ) - 384\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} - \frac {105 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {105 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{7680 \, d} \]
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Time = 0.55 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.01 \[ \int \frac {\sec ^8(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {420 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {420 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {5 \, {\left (175 \, \sin \left (d x + c\right )^{4} - 868 \, \sin \left (d x + c\right )^{3} + 1662 \, \sin \left (d x + c\right )^{2} - 1484 \, \sin \left (d x + c\right ) + 539\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac {959 \, \sin \left (d x + c\right )^{5} + 4795 \, \sin \left (d x + c\right )^{4} + 9290 \, \sin \left (d x + c\right )^{3} + 8290 \, \sin \left (d x + c\right )^{2} + 2735 \, \sin \left (d x + c\right ) - 293}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{30720 \, d} \]
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Time = 18.39 (sec) , antiderivative size = 496, normalized size of antiderivative = 3.22 \[ \int \frac {\sec ^8(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {7\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,a\,d}+\frac {-\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}+\frac {121\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{64}+\frac {163\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{96}+\frac {289\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{192}-\frac {2261\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{480}+\frac {12551\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{960}+\frac {6097\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{480}+\frac {11837\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{960}-\frac {2471\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{192}+\frac {11837\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{960}+\frac {6097\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{480}+\frac {12551\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{960}-\frac {2261\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{480}+\frac {289\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{192}+\frac {163\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96}+\frac {121\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64}-\frac {7\,\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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