Integrand size = 29, antiderivative size = 172 \[ \int \frac {\sec ^7(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {7 \text {arctanh}(\sin (c+d x))}{256 a d}+\frac {\sec ^8(c+d x)}{8 a d}-\frac {\sec ^{10}(c+d x)}{10 a d}-\frac {7 \sec (c+d x) \tan (c+d x)}{256 a d}-\frac {7 \sec ^3(c+d x) \tan (c+d x)}{384 a d}-\frac {7 \sec ^5(c+d x) \tan (c+d x)}{480 a d}-\frac {\sec ^7(c+d x) \tan (c+d x)}{80 a d}+\frac {\sec ^9(c+d x) \tan (c+d x)}{10 a d} \]
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Time = 0.16 (sec) , antiderivative size = 172, 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, 2686, 14} \[ \int \frac {\sec ^7(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {7 \text {arctanh}(\sin (c+d x))}{256 a d}-\frac {\sec ^{10}(c+d x)}{10 a d}+\frac {\sec ^8(c+d x)}{8 a d}+\frac {\tan (c+d x) \sec ^9(c+d x)}{10 a d}-\frac {\tan (c+d x) \sec ^7(c+d x)}{80 a d}-\frac {7 \tan (c+d x) \sec ^5(c+d x)}{480 a d}-\frac {7 \tan (c+d x) \sec ^3(c+d x)}{384 a d}-\frac {7 \tan (c+d x) \sec (c+d x)}{256 a d} \]
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Rule 14
Rule 2686
Rule 2691
Rule 2914
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^9(c+d x) \tan ^2(c+d x) \, dx}{a}-\frac {\int \sec ^8(c+d x) \tan ^3(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^7 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{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 {\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 {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 ^8(c+d x)}{8 a d}-\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 ^8(c+d x)}{8 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}-\frac {7 \int \sec (c+d x) \, dx}{256 a} \\ & = -\frac {7 \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 {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 = 1.71 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.71 \[ \int \frac {\sec ^7(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {210 \text {arctanh}(\sin (c+d x))-\frac {2 \left (96+201 \sin (c+d x)-279 \sin ^2(c+d x)+511 \sin ^3(c+d x)+511 \sin ^4(c+d x)-385 \sin ^5(c+d x)-385 \sin ^6(c+d x)+105 \sin ^7(c+d x)+105 \sin ^8(c+d x)\right )}{(-1+\sin (c+d x))^4 (1+\sin (c+d x))^5}}{7680 a d} \]
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Time = 1.69 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}-\frac {1}{192 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {1}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {1}{128 \sin \left (d x +c \right )-128}+\frac {7 \ln \left (\sin \left (d x +c \right )-1\right )}{512}-\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {1}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {1}{384 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {5}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5}{256 \left (1+\sin \left (d x +c \right )\right )}-\frac {7 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\) | \(139\) |
default | \(\frac {\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}-\frac {1}{192 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {1}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {1}{128 \sin \left (d x +c \right )-128}+\frac {7 \ln \left (\sin \left (d x +c \right )-1\right )}{512}-\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {1}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {1}{384 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {5}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5}{256 \left (1+\sin \left (d x +c \right )\right )}-\frac {7 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\) | \(139\) |
risch | \(\frac {i \left (1610 i {\mathrm e}^{14 i \left (d x +c \right )}+105 \,{\mathrm e}^{17 i \left (d x +c \right )}+210 i {\mathrm e}^{16 i \left (d x +c \right )}+700 \,{\mathrm e}^{15 i \left (d x +c \right )}+51334 i {\mathrm e}^{8 i \left (d x +c \right )}+1876 \,{\mathrm e}^{13 i \left (d x +c \right )}-5362 i {\mathrm e}^{6 i \left (d x +c \right )}+2372 \,{\mathrm e}^{11 i \left (d x +c \right )}+5362 i {\mathrm e}^{12 i \left (d x +c \right )}+14470 \,{\mathrm e}^{9 i \left (d x +c \right )}-51334 i {\mathrm e}^{10 i \left (d x +c \right )}+2372 \,{\mathrm e}^{7 i \left (d x +c \right )}-1610 i {\mathrm e}^{4 i \left (d x +c \right )}+1876 \,{\mathrm e}^{5 i \left (d x +c \right )}-210 i {\mathrm e}^{2 i \left (d x +c \right )}+700 \,{\mathrm e}^{3 i \left (d x +c \right )}+105 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{1920 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{10} d a}-\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{256 a d}+\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{256 d a}\) | \(277\) |
parallelrisch | \(\frac {\left (105 \cos \left (10 d x +10 c \right )+22050 \cos \left (2 d x +2 c \right )+12600 \cos \left (4 d x +4 c \right )+4725 \cos \left (6 d x +6 c \right )+1050 \cos \left (8 d x +8 c \right )+13230\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (-105 \cos \left (10 d x +10 c \right )-22050 \cos \left (2 d x +2 c \right )-12600 \cos \left (4 d x +4 c \right )-4725 \cos \left (6 d x +6 c \right )-1050 \cos \left (8 d x +8 c \right )-13230\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-22440 \sin \left (3 d x +3 c \right )-8792 \sin \left (5 d x +5 c \right )-2030 \sin \left (7 d x +7 c \right )-210 \sin \left (9 d x +9 c \right )-96 \cos \left (10 d x +10 c \right )+102720 \cos \left (2 d x +2 c \right )-11520 \cos \left (4 d x +4 c \right )-4320 \cos \left (6 d x +6 c \right )-960 \cos \left (8 d x +8 c \right )+181140 \sin \left (d x +c \right )-85824}{3840 a d \left (\cos \left (10 d x +10 c \right )+10 \cos \left (8 d x +8 c \right )+45 \cos \left (6 d x +6 c \right )+120 \cos \left (4 d x +4 c \right )+210 \cos \left (2 d x +2 c \right )+126\right )}\) | \(315\) |
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Time = 0.29 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.09 \[ \int \frac {\sec ^7(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {210 \, \cos \left (d x + c\right )^{8} - 70 \, \cos \left (d x + c\right )^{6} - 28 \, \cos \left (d x + c\right )^{4} - 16 \, \cos \left (d x + c\right )^{2} - 105 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (105 \, \cos \left (d x + c\right )^{6} + 70 \, \cos \left (d x + c\right )^{4} + 56 \, \cos \left (d x + c\right )^{2} - 432\right )} \sin \left (d x + c\right ) + 96}{7680 \, {\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\right )}} \]
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Timed out. \[ \int \frac {\sec ^7(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.24 \[ \int \frac {\sec ^7(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {2 \, {\left (105 \, \sin \left (d x + c\right )^{8} + 105 \, \sin \left (d x + c\right )^{7} - 385 \, \sin \left (d x + c\right )^{6} - 385 \, \sin \left (d x + c\right )^{5} + 511 \, \sin \left (d x + c\right )^{4} + 511 \, \sin \left (d x + c\right )^{3} - 279 \, \sin \left (d x + c\right )^{2} + 201 \, \sin \left (d x + c\right ) + 96\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} - \frac {105 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {105 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{7680 \, d} \]
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Time = 0.57 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.91 \[ \int \frac {\sec ^7(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {420 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {420 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {5 \, {\left (175 \, \sin \left (d x + c\right )^{4} - 748 \, \sin \left (d x + c\right )^{3} + 1182 \, \sin \left (d x + c\right )^{2} - 788 \, \sin \left (d x + c\right ) + 155\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac {959 \, \sin \left (d x + c\right )^{5} + 5395 \, \sin \left (d x + c\right )^{4} + 12290 \, \sin \left (d x + c\right )^{3} + 14170 \, \sin \left (d x + c\right )^{2} + 8135 \, \sin \left (d x + c\right ) + 1627}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{30720 \, d} \]
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Time = 17.96 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.88 \[ \int \frac {\sec ^7(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}+\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{64}+\frac {221\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{96}+\frac {95\,{\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 {889\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{960}+\frac {7343\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{480}+\frac {1603\,{\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 {1603\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{960}+\frac {7343\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{480}+\frac {889\,{\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 {95\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{192}+\frac {221\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96}+\frac {7\,{\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 )}-\frac {7\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,a\,d} \]
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