Integrand size = 29, antiderivative size = 150 \[ \int \frac {\sec ^4(c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 \text {arctanh}(\sin (c+d x))}{128 a d}-\frac {\sec ^6(c+d x)}{6 a d}+\frac {\sec ^8(c+d x)}{8 a d}-\frac {3 \sec (c+d x) \tan (c+d x)}{128 a d}-\frac {\sec ^3(c+d x) \tan (c+d x)}{64 a d}+\frac {\sec ^5(c+d x) \tan (c+d x)}{16 a d}-\frac {\sec ^5(c+d x) \tan ^3(c+d x)}{8 a d} \]
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Time = 0.17 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2914, 2686, 14, 2691, 3853, 3855} \[ \int \frac {\sec ^4(c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 \text {arctanh}(\sin (c+d x))}{128 a d}+\frac {\sec ^8(c+d x)}{8 a d}-\frac {\sec ^6(c+d x)}{6 a d}-\frac {\tan ^3(c+d x) \sec ^5(c+d x)}{8 a d}+\frac {\tan (c+d x) \sec ^5(c+d x)}{16 a d}-\frac {\tan (c+d x) \sec ^3(c+d x)}{64 a d}-\frac {3 \tan (c+d x) \sec (c+d x)}{128 a d} \]
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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 ^6(c+d x) \tan ^3(c+d x) \, dx}{a}-\frac {\int \sec ^5(c+d x) \tan ^4(c+d x) \, dx}{a} \\ & = -\frac {\sec ^5(c+d x) \tan ^3(c+d x)}{8 a d}+\frac {3 \int \sec ^5(c+d x) \tan ^2(c+d x) \, dx}{8 a}+\frac {\text {Subst}\left (\int x^5 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{a d} \\ & = \frac {\sec ^5(c+d x) \tan (c+d x)}{16 a d}-\frac {\sec ^5(c+d x) \tan ^3(c+d x)}{8 a d}-\frac {\int \sec ^5(c+d x) \, dx}{16 a}+\frac {\text {Subst}\left (\int \left (-x^5+x^7\right ) \, dx,x,\sec (c+d x)\right )}{a d} \\ & = -\frac {\sec ^6(c+d x)}{6 a d}+\frac {\sec ^8(c+d x)}{8 a d}-\frac {\sec ^3(c+d x) \tan (c+d x)}{64 a d}+\frac {\sec ^5(c+d x) \tan (c+d x)}{16 a d}-\frac {\sec ^5(c+d x) \tan ^3(c+d x)}{8 a d}-\frac {3 \int \sec ^3(c+d x) \, dx}{64 a} \\ & = -\frac {\sec ^6(c+d x)}{6 a d}+\frac {\sec ^8(c+d x)}{8 a d}-\frac {3 \sec (c+d x) \tan (c+d x)}{128 a d}-\frac {\sec ^3(c+d x) \tan (c+d x)}{64 a d}+\frac {\sec ^5(c+d x) \tan (c+d x)}{16 a d}-\frac {\sec ^5(c+d x) \tan ^3(c+d x)}{8 a d}-\frac {3 \int \sec (c+d x) \, dx}{128 a} \\ & = -\frac {3 \text {arctanh}(\sin (c+d x))}{128 a d}-\frac {\sec ^6(c+d x)}{6 a d}+\frac {\sec ^8(c+d x)}{8 a d}-\frac {3 \sec (c+d x) \tan (c+d x)}{128 a d}-\frac {\sec ^3(c+d x) \tan (c+d x)}{64 a d}+\frac {\sec ^5(c+d x) \tan (c+d x)}{16 a d}-\frac {\sec ^5(c+d x) \tan ^3(c+d x)}{8 a d} \\ \end{align*}
Time = 0.98 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.61 \[ \int \frac {\sec ^4(c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {9 \text {arctanh}(\sin (c+d x))+\frac {4}{(-1+\sin (c+d x))^3}+\frac {3}{(-1+\sin (c+d x))^2}-\frac {9}{-1+\sin (c+d x)}-\frac {6}{(1+\sin (c+d x))^4}+\frac {8}{(1+\sin (c+d x))^3}+\frac {6}{(1+\sin (c+d x))^2}}{384 a d} \]
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Time = 0.94 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.69
method | result | size |
derivativedivides | \(\frac {-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {1}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {3}{128 \left (\sin \left (d x +c \right )-1\right )}+\frac {3 \ln \left (\sin \left (d x +c \right )-1\right )}{256}+\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{48 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {3 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\) | \(103\) |
default | \(\frac {-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {1}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {3}{128 \left (\sin \left (d x +c \right )-1\right )}+\frac {3 \ln \left (\sin \left (d x +c \right )-1\right )}{256}+\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{48 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {3 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\) | \(103\) |
risch | \(\frac {i \left (-1161 \,{\mathrm e}^{5 i \left (d x +c \right )}+102 i {\mathrm e}^{10 i \left (d x +c \right )}-102 i {\mathrm e}^{4 i \left (d x +c \right )}+9 \,{\mathrm e}^{13 i \left (d x +c \right )}-172 i {\mathrm e}^{8 i \left (d x +c \right )}-18 i {\mathrm e}^{2 i \left (d x +c \right )}+18 i {\mathrm e}^{12 i \left (d x +c \right )}+9 \,{\mathrm e}^{i \left (d x +c \right )}+42 \,{\mathrm e}^{11 i \left (d x +c \right )}-1161 \,{\mathrm e}^{9 i \left (d x +c \right )}+42 \,{\mathrm e}^{3 i \left (d x +c \right )}+1196 \,{\mathrm e}^{7 i \left (d x +c \right )}+172 i {\mathrm e}^{6 i \left (d x +c \right )}\right )}{192 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{6} d a}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 a d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d a}\) | \(231\) |
parallelrisch | \(\frac {\left (504 \cos \left (2 d x +2 c \right )+252 \cos \left (4 d x +4 c \right )+72 \cos \left (6 d x +6 c \right )+9 \cos \left (8 d x +8 c \right )+315\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (-504 \cos \left (2 d x +2 c \right )-252 \cos \left (4 d x +4 c \right )-72 \cos \left (6 d x +6 c \right )-9 \cos \left (8 d x +8 c \right )-315\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+1998 \sin \left (3 d x +3 c \right )-138 \sin \left (5 d x +5 c \right )-18 \sin \left (7 d x +7 c \right )-3200 \cos \left (2 d x +2 c \right )+448 \cos \left (4 d x +4 c \right )+128 \cos \left (6 d x +6 c \right )+16 \cos \left (8 d x +8 c \right )-4026 \sin \left (d x +c \right )+2608}{384 a d \left (\cos \left (8 d x +8 c \right )+8 \cos \left (6 d x +6 c \right )+28 \cos \left (4 d x +4 c \right )+56 \cos \left (2 d x +2 c \right )+35\right )}\) | \(260\) |
norman | \(\frac {\frac {111 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {111 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d a}+\frac {3 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}+\frac {3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {3 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {277 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d a}+\frac {277 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d a}-\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}-\frac {7 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {125 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}+\frac {125 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}-\frac {43 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{6} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{128 a d}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{128 a d}\) | \(315\) |
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Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.11 \[ \int \frac {\sec ^4(c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {18 \, \cos \left (d x + c\right )^{6} - 6 \, \cos \left (d x + c\right )^{4} - 156 \, \cos \left (d x + c\right )^{2} - 9 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 9 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (9 \, \cos \left (d x + c\right )^{4} + 6 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) + 112}{768 \, {\left (a d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{6}\right )}} \]
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Timed out. \[ \int \frac {\sec ^4(c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.17 \[ \int \frac {\sec ^4(c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {2 \, {\left (9 \, \sin \left (d x + c\right )^{6} + 9 \, \sin \left (d x + c\right )^{5} - 24 \, \sin \left (d x + c\right )^{4} - 24 \, \sin \left (d x + c\right )^{3} - 57 \, \sin \left (d x + c\right )^{2} + 7 \, \sin \left (d x + c\right ) + 16\right )}}{a \sin \left (d x + c\right )^{7} + a \sin \left (d x + c\right )^{6} - 3 \, a \sin \left (d x + c\right )^{5} - 3 \, a \sin \left (d x + c\right )^{4} + 3 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a} - \frac {9 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {9 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{768 \, d} \]
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Time = 0.38 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.91 \[ \int \frac {\sec ^4(c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {36 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {36 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {2 \, {\left (33 \, \sin \left (d x + c\right )^{3} - 135 \, \sin \left (d x + c\right )^{2} + 183 \, \sin \left (d x + c\right ) - 65\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {75 \, \sin \left (d x + c\right )^{4} + 300 \, \sin \left (d x + c\right )^{3} + 402 \, \sin \left (d x + c\right )^{2} + 140 \, \sin \left (d x + c\right ) + 11}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]
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Time = 19.19 (sec) , antiderivative size = 388, normalized size of antiderivative = 2.59 \[ \int \frac {\sec ^4(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 )}^{13}}{64}+\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{32}-\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{32}+\frac {111\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{32}+\frac {125\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}+\frac {277\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{48}-\frac {43\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{48}+\frac {277\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{48}+\frac {125\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{64}+\frac {111\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32}-\frac {7\,{\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}{32}+\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+9\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+30\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-40\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+30\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+9\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-5\,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 )}{64\,a\,d} \]
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