Integrand size = 27, antiderivative size = 134 \[ \int \frac {\sec (c+d x) \tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {5 \text {arctanh}(\sin (c+d x))}{128 a d}-\frac {5 \sec (c+d x) \tan (c+d x)}{128 a d}+\frac {5 \sec ^3(c+d x) \tan (c+d x)}{64 a d}-\frac {5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a d}+\frac {\sec ^3(c+d x) \tan ^5(c+d x)}{8 a d}-\frac {\tan ^8(c+d x)}{8 a d} \]
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Time = 0.16 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2914, 2691, 3853, 3855, 2687, 30} \[ \int \frac {\sec (c+d x) \tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {5 \text {arctanh}(\sin (c+d x))}{128 a d}-\frac {\tan ^8(c+d x)}{8 a d}+\frac {\tan ^5(c+d x) \sec ^3(c+d x)}{8 a d}-\frac {5 \tan ^3(c+d x) \sec ^3(c+d x)}{48 a d}+\frac {5 \tan (c+d x) \sec ^3(c+d x)}{64 a d}-\frac {5 \tan (c+d x) \sec (c+d x)}{128 a d} \]
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Rule 30
Rule 2687
Rule 2691
Rule 2914
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^3(c+d x) \tan ^6(c+d x) \, dx}{a}-\frac {\int \sec ^2(c+d x) \tan ^7(c+d x) \, dx}{a} \\ & = \frac {\sec ^3(c+d x) \tan ^5(c+d x)}{8 a d}-\frac {5 \int \sec ^3(c+d x) \tan ^4(c+d x) \, dx}{8 a}-\frac {\text {Subst}\left (\int x^7 \, dx,x,\tan (c+d x)\right )}{a d} \\ & = -\frac {5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a d}+\frac {\sec ^3(c+d x) \tan ^5(c+d x)}{8 a d}-\frac {\tan ^8(c+d x)}{8 a d}+\frac {5 \int \sec ^3(c+d x) \tan ^2(c+d x) \, dx}{16 a} \\ & = \frac {5 \sec ^3(c+d x) \tan (c+d x)}{64 a d}-\frac {5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a d}+\frac {\sec ^3(c+d x) \tan ^5(c+d x)}{8 a d}-\frac {\tan ^8(c+d x)}{8 a d}-\frac {5 \int \sec ^3(c+d x) \, dx}{64 a} \\ & = -\frac {5 \sec (c+d x) \tan (c+d x)}{128 a d}+\frac {5 \sec ^3(c+d x) \tan (c+d x)}{64 a d}-\frac {5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a d}+\frac {\sec ^3(c+d x) \tan ^5(c+d x)}{8 a d}-\frac {\tan ^8(c+d x)}{8 a d}-\frac {5 \int \sec (c+d x) \, dx}{128 a} \\ & = -\frac {5 \text {arctanh}(\sin (c+d x))}{128 a d}-\frac {5 \sec (c+d x) \tan (c+d x)}{128 a d}+\frac {5 \sec ^3(c+d x) \tan (c+d x)}{64 a d}-\frac {5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a d}+\frac {\sec ^3(c+d x) \tan ^5(c+d x)}{8 a d}-\frac {\tan ^8(c+d x)}{8 a d} \\ \end{align*}
Time = 0.65 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.75 \[ \int \frac {\sec (c+d x) \tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {15 \text {arctanh}(\sin (c+d x))+\frac {48+63 \sin (c+d x)-129 \sin ^2(c+d x)-184 \sin ^3(c+d x)+104 \sin ^4(c+d x)+177 \sin ^5(c+d x)-15 \sin ^6(c+d x)}{(-1+\sin (c+d x))^3 (1+\sin (c+d x))^4}}{384 a d} \]
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Time = 0.92 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {7}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {15}{128 \left (\sin \left (d x +c \right )-1\right )}+\frac {5 \ln \left (\sin \left (d x +c \right )-1\right )}{256}-\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {1}{12 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {11}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5}{32 \left (1+\sin \left (d x +c \right )\right )}-\frac {5 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\) | \(115\) |
default | \(\frac {-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {7}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {15}{128 \left (\sin \left (d x +c \right )-1\right )}+\frac {5 \ln \left (\sin \left (d x +c \right )-1\right )}{256}-\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {1}{12 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {11}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5}{32 \left (1+\sin \left (d x +c \right )\right )}-\frac {5 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\) | \(115\) |
risch | \(\frac {i \left (-354 i {\mathrm e}^{12 i \left (d x +c \right )}+15 \,{\mathrm e}^{13 i \left (d x +c \right )}+298 i {\mathrm e}^{10 i \left (d x +c \right )}+326 \,{\mathrm e}^{11 i \left (d x +c \right )}-1140 i {\mathrm e}^{8 i \left (d x +c \right )}+625 \,{\mathrm e}^{9 i \left (d x +c \right )}+1140 i {\mathrm e}^{6 i \left (d x +c \right )}+1140 \,{\mathrm e}^{7 i \left (d x +c \right )}-298 i {\mathrm e}^{4 i \left (d x +c \right )}+625 \,{\mathrm e}^{5 i \left (d x +c \right )}+354 i {\mathrm e}^{2 i \left (d x +c \right )}+326 \,{\mathrm e}^{3 i \left (d x +c \right )}+15 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{192 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{6} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{8} d a}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 a d}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d a}\) | \(231\) |
parallelrisch | \(\frac {\left (840 \cos \left (2 d x +2 c \right )+420 \cos \left (4 d x +4 c \right )+120 \cos \left (6 d x +6 c \right )+15 \cos \left (8 d x +8 c \right )+525\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (-840 \cos \left (2 d x +2 c \right )-420 \cos \left (4 d x +4 c \right )-120 \cos \left (6 d x +6 c \right )-15 \cos \left (8 d x +8 c \right )-525\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-1790 \sin \left (3 d x +3 c \right )+794 \sin \left (5 d x +5 c \right )-30 \sin \left (7 d x +7 c \right )+2688 \cos \left (2 d x +2 c \right )-1344 \cos \left (4 d x +4 c \right )+384 \cos \left (6 d x +6 c \right )-48 \cos \left (8 d x +8 c \right )+3530 \sin \left (d x +c \right )-1680}{384 a d \left (\cos \left (8 d x +8 c \right )+8 \cos \left (6 d x +6 c \right )+28 \cos \left (4 d x +4 c \right )+56 \cos \left (2 d x +2 c \right )+35\right )}\) | \(260\) |
norman | \(\frac {\frac {5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {5 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {33 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}+\frac {33 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d a}+\frac {5 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}+\frac {289 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}-\frac {85 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}-\frac {85 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}-\frac {35 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}-\frac {35 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}+\frac {113 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}+\frac {113 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{6}}+\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{128 a d}-\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{128 a d}\) | \(315\) |
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Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.25 \[ \int \frac {\sec (c+d x) \tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {30 \, \cos \left (d x + c\right )^{6} + 118 \, \cos \left (d x + c\right )^{4} - 68 \, \cos \left (d x + c\right )^{2} - 15 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (177 \, \cos \left (d x + c\right )^{4} - 170 \, \cos \left (d x + c\right )^{2} + 56\right )} \sin \left (d x + c\right ) + 16}{768 \, {\left (a d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{6}\right )}} \]
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Timed out. \[ \int \frac {\sec (c+d x) \tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.31 \[ \int \frac {\sec (c+d x) \tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{6} - 177 \, \sin \left (d x + c\right )^{5} - 104 \, \sin \left (d x + c\right )^{4} + 184 \, \sin \left (d x + c\right )^{3} + 129 \, \sin \left (d x + c\right )^{2} - 63 \, \sin \left (d x + c\right ) - 48\right )}}{a \sin \left (d x + c\right )^{7} + a \sin \left (d x + c\right )^{6} - 3 \, a \sin \left (d x + c\right )^{5} - 3 \, a \sin \left (d x + c\right )^{4} + 3 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a} - \frac {15 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {15 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{768 \, d} \]
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Time = 0.40 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.01 \[ \int \frac {\sec (c+d x) \tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {2 \, {\left (55 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )^{2} - 111 \, \sin \left (d x + c\right ) + 57\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {125 \, \sin \left (d x + c\right )^{4} + 980 \, \sin \left (d x + c\right )^{3} + 1662 \, \sin \left (d x + c\right )^{2} + 1140 \, \sin \left (d x + c\right ) + 285}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]
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Time = 21.58 (sec) , antiderivative size = 388, normalized size of antiderivative = 2.90 \[ \int \frac {\sec (c+d x) \tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}+\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{32}-\frac {35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{96}-\frac {85\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{96}+\frac {113\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{192}+\frac {33\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{16}+\frac {289\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{16}+\frac {33\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{16}+\frac {113\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{192}-\frac {85\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{96}-\frac {35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96}+\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{32}+\frac {5\,\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 {5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{64\,a\,d} \]
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