Integrand size = 21, antiderivative size = 199 \[ \int (a+a \sec (c+d x))^2 \sin ^8(c+d x) \, dx=-\frac {245 a^2 x}{128}+\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a^2 \sin (c+d x)}{d}+\frac {139 a^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {11 a^2 \cos ^3(c+d x) \sin (c+d x)}{192 d}-\frac {17 a^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {2 a^2 \sin ^3(c+d x)}{3 d}-\frac {2 a^2 \sin ^5(c+d x)}{5 d}-\frac {2 a^2 \sin ^7(c+d x)}{7 d}+\frac {a^2 \tan (c+d x)}{d} \]
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Time = 0.46 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3957, 2951, 2717, 2715, 8, 2713, 3855, 3852} \[ \int (a+a \sec (c+d x))^2 \sin ^8(c+d x) \, dx=\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a^2 \sin ^7(c+d x)}{7 d}-\frac {2 a^2 \sin ^5(c+d x)}{5 d}-\frac {2 a^2 \sin ^3(c+d x)}{3 d}-\frac {2 a^2 \sin (c+d x)}{d}+\frac {a^2 \tan (c+d x)}{d}+\frac {a^2 \sin (c+d x) \cos ^7(c+d x)}{8 d}-\frac {17 a^2 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {11 a^2 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {139 a^2 \sin (c+d x) \cos (c+d x)}{128 d}-\frac {245 a^2 x}{128} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2717
Rule 2951
Rule 3852
Rule 3855
Rule 3957
Rubi steps \begin{align*} \text {integral}& = \int (-a-a \cos (c+d x))^2 \sin ^6(c+d x) \tan ^2(c+d x) \, dx \\ & = \frac {\int \left (-3 a^{10}-8 a^{10} \cos (c+d x)+2 a^{10} \cos ^2(c+d x)+12 a^{10} \cos ^3(c+d x)+2 a^{10} \cos ^4(c+d x)-8 a^{10} \cos ^5(c+d x)-3 a^{10} \cos ^6(c+d x)+2 a^{10} \cos ^7(c+d x)+a^{10} \cos ^8(c+d x)+2 a^{10} \sec (c+d x)+a^{10} \sec ^2(c+d x)\right ) \, dx}{a^8} \\ & = -3 a^2 x+a^2 \int \cos ^8(c+d x) \, dx+a^2 \int \sec ^2(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^2(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^4(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^7(c+d x) \, dx+\left (2 a^2\right ) \int \sec (c+d x) \, dx-\left (3 a^2\right ) \int \cos ^6(c+d x) \, dx-\left (8 a^2\right ) \int \cos (c+d x) \, dx-\left (8 a^2\right ) \int \cos ^5(c+d x) \, dx+\left (12 a^2\right ) \int \cos ^3(c+d x) \, dx \\ & = -3 a^2 x+\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {8 a^2 \sin (c+d x)}{d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{2 d}-\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{2 d}+\frac {a^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{8} \left (7 a^2\right ) \int \cos ^6(c+d x) \, dx+a^2 \int 1 \, dx+\frac {1}{2} \left (3 a^2\right ) \int \cos ^2(c+d x) \, dx-\frac {1}{2} \left (5 a^2\right ) \int \cos ^4(c+d x) \, dx-\frac {a^2 \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{d}+\frac {\left (8 a^2\right ) \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac {\left (12 a^2\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d} \\ & = -2 a^2 x+\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a^2 \sin (c+d x)}{d}+\frac {7 a^2 \cos (c+d x) \sin (c+d x)}{4 d}-\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{8 d}-\frac {17 a^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {2 a^2 \sin ^3(c+d x)}{3 d}-\frac {2 a^2 \sin ^5(c+d x)}{5 d}-\frac {2 a^2 \sin ^7(c+d x)}{7 d}+\frac {a^2 \tan (c+d x)}{d}+\frac {1}{48} \left (35 a^2\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{4} \left (3 a^2\right ) \int 1 \, dx-\frac {1}{8} \left (15 a^2\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {5 a^2 x}{4}+\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a^2 \sin (c+d x)}{d}+\frac {13 a^2 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {11 a^2 \cos ^3(c+d x) \sin (c+d x)}{192 d}-\frac {17 a^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {2 a^2 \sin ^3(c+d x)}{3 d}-\frac {2 a^2 \sin ^5(c+d x)}{5 d}-\frac {2 a^2 \sin ^7(c+d x)}{7 d}+\frac {a^2 \tan (c+d x)}{d}+\frac {1}{64} \left (35 a^2\right ) \int \cos ^2(c+d x) \, dx-\frac {1}{16} \left (15 a^2\right ) \int 1 \, dx \\ & = -\frac {35 a^2 x}{16}+\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a^2 \sin (c+d x)}{d}+\frac {139 a^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {11 a^2 \cos ^3(c+d x) \sin (c+d x)}{192 d}-\frac {17 a^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {2 a^2 \sin ^3(c+d x)}{3 d}-\frac {2 a^2 \sin ^5(c+d x)}{5 d}-\frac {2 a^2 \sin ^7(c+d x)}{7 d}+\frac {a^2 \tan (c+d x)}{d}+\frac {1}{128} \left (35 a^2\right ) \int 1 \, dx \\ & = -\frac {245 a^2 x}{128}+\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a^2 \sin (c+d x)}{d}+\frac {139 a^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {11 a^2 \cos ^3(c+d x) \sin (c+d x)}{192 d}-\frac {17 a^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {2 a^2 \sin ^3(c+d x)}{3 d}-\frac {2 a^2 \sin ^5(c+d x)}{5 d}-\frac {2 a^2 \sin ^7(c+d x)}{7 d}+\frac {a^2 \tan (c+d x)}{d} \\ \end{align*}
Time = 1.07 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.72 \[ \int (a+a \sec (c+d x))^2 \sin ^8(c+d x) \, dx=-\frac {a^2 (1+\cos (c+d x))^2 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \left (168000 c+168000 d x+37800 \arctan (\tan (c+d x))-215040 \text {arctanh}(\sin (c+d x))+215040 \sin (c+d x)+71680 \sin ^3(c+d x)+43008 \sin ^5(c+d x)+30720 \sin ^7(c+d x)-55440 \sin (2 (c+d x))+2520 \sin (4 (c+d x))+560 \sin (6 (c+d x))-105 \sin (8 (c+d x))-107520 \tan (c+d x)\right )}{430080 d} \]
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Time = 2.29 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.82
method | result | size |
parallelrisch | \(\frac {a^{2} \left (-411600 d x \cos \left (d x +c \right )-430080 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )+430080 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )+270480 \sin \left (d x +c \right )+35392 \sin \left (4 d x +4 c \right )+105 \sin \left (9 d x +9 c \right )-271040 \sin \left (2 d x +2 c \right )+480 \sin \left (8 d x +8 c \right )-5568 \sin \left (6 d x +6 c \right )-455 \sin \left (7 d x +7 c \right )-3080 \sin \left (5 d x +5 c \right )+52920 \sin \left (3 d x +3 c \right )\right )}{215040 d \cos \left (d x +c \right )}\) | \(164\) |
derivativedivides | \(\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{9}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )-\frac {35 d x}{16}-\frac {35 c}{16}\right )+2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{7}}{7}-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}\) | \(194\) |
default | \(\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{9}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )-\frac {35 d x}{16}-\frac {35 c}{16}\right )+2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{7}}{7}-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}\) | \(194\) |
parts | \(\frac {a^{2} \left (-\frac {\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}+\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{9}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )-\frac {35 d x}{16}-\frac {35 c}{16}\right )}{d}+\frac {2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{7}}{7}-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}\) | \(199\) |
risch | \(-\frac {245 a^{2} x}{128}+\frac {33 i a^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}-\frac {93 i a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{64 d}+\frac {93 i a^{2} {\mathrm e}^{i \left (d x +c \right )}}{64 d}-\frac {33 i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{128 d}+\frac {2 i a^{2}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {a^{2} \sin \left (8 d x +8 c \right )}{1024 d}+\frac {a^{2} \sin \left (7 d x +7 c \right )}{224 d}-\frac {a^{2} \sin \left (6 d x +6 c \right )}{192 d}-\frac {9 a^{2} \sin \left (5 d x +5 c \right )}{160 d}-\frac {3 a^{2} \sin \left (4 d x +4 c \right )}{128 d}+\frac {37 a^{2} \sin \left (3 d x +3 c \right )}{96 d}\) | \(246\) |
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Time = 0.28 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.93 \[ \int (a+a \sec (c+d x))^2 \sin ^8(c+d x) \, dx=-\frac {25725 \, a^{2} d x \cos \left (d x + c\right ) - 13440 \, a^{2} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) + 13440 \, a^{2} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) - {\left (1680 \, a^{2} \cos \left (d x + c\right )^{8} + 3840 \, a^{2} \cos \left (d x + c\right )^{7} - 4760 \, a^{2} \cos \left (d x + c\right )^{6} - 16896 \, a^{2} \cos \left (d x + c\right )^{5} + 770 \, a^{2} \cos \left (d x + c\right )^{4} + 31232 \, a^{2} \cos \left (d x + c\right )^{3} + 14595 \, a^{2} \cos \left (d x + c\right )^{2} - 45056 \, a^{2} \cos \left (d x + c\right ) + 13440 \, a^{2}\right )} \sin \left (d x + c\right )}{13440 \, d \cos \left (d x + c\right )} \]
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Timed out. \[ \int (a+a \sec (c+d x))^2 \sin ^8(c+d x) \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.08 \[ \int (a+a \sec (c+d x))^2 \sin ^8(c+d x) \, dx=-\frac {1024 \, {\left (30 \, \sin \left (d x + c\right )^{7} + 42 \, \sin \left (d x + c\right )^{5} + 70 \, \sin \left (d x + c\right )^{3} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 210 \, \sin \left (d x + c\right )\right )} a^{2} - 35 \, {\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 840 \, d x + 840 \, c + 3 \, \sin \left (8 \, d x + 8 \, c\right ) + 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} + 2240 \, {\left (105 \, d x + 105 \, c - \frac {87 \, \tan \left (d x + c\right )^{5} + 136 \, \tan \left (d x + c\right )^{3} + 57 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1} - 48 \, \tan \left (d x + c\right )\right )} a^{2}}{107520 \, d} \]
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Time = 0.39 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.13 \[ \int (a+a \sec (c+d x))^2 \sin ^8(c+d x) \, dx=-\frac {25725 \, {\left (d x + c\right )} a^{2} - 26880 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 26880 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {26880 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + \frac {2 \, {\left (39165 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 300265 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 989261 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1791073 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1814943 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 670131 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 147735 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 14595 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{8}}}{13440 \, d} \]
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Time = 15.70 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.47 \[ \int (a+a \sec (c+d x))^2 \sin ^8(c+d x) \, dx=\frac {4\,a^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {245\,a^2\,x}{128}+\frac {\frac {501\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{64}+\frac {2633\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{48}+\frac {38047\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{240}+\frac {388613\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{1680}+\frac {13781\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{96}-\frac {32681\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{560}-\frac {1739\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{80}-\frac {61\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{16}-\frac {11\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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