Integrand size = 21, antiderivative size = 157 \[ \int (a+a \sec (c+d x))^2 \sin ^6(c+d x) \, dx=-\frac {25 a^2 x}{16}+\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a^2 \sin (c+d x)}{d}+\frac {7 a^2 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {7 a^2 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {2 a^2 \sin ^3(c+d x)}{3 d}-\frac {2 a^2 \sin ^5(c+d x)}{5 d}+\frac {a^2 \tan (c+d x)}{d} \]
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Time = 0.38 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3957, 2951, 2717, 2713, 2715, 8, 3855, 3852} \[ \int (a+a \sec (c+d x))^2 \sin ^6(c+d x) \, dx=\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a^2 \sin ^5(c+d x)}{5 d}-\frac {2 a^2 \sin ^3(c+d x)}{3 d}-\frac {2 a^2 \sin (c+d x)}{d}+\frac {a^2 \tan (c+d x)}{d}-\frac {a^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {7 a^2 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {7 a^2 \sin (c+d x) \cos (c+d x)}{16 d}-\frac {25 a^2 x}{16} \]
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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 ^4(c+d x) \tan ^2(c+d x) \, dx \\ & = \frac {\int \left (-2 a^8-6 a^8 \cos (c+d x)+6 a^8 \cos ^3(c+d x)+2 a^8 \cos ^4(c+d x)-2 a^8 \cos ^5(c+d x)-a^8 \cos ^6(c+d x)+2 a^8 \sec (c+d x)+a^8 \sec ^2(c+d x)\right ) \, dx}{a^6} \\ & = -2 a^2 x-a^2 \int \cos ^6(c+d x) \, dx+a^2 \int \sec ^2(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^4(c+d x) \, dx-\left (2 a^2\right ) \int \cos ^5(c+d x) \, dx+\left (2 a^2\right ) \int \sec (c+d x) \, dx-\left (6 a^2\right ) \int \cos (c+d x) \, dx+\left (6 a^2\right ) \int \cos ^3(c+d x) \, dx \\ & = -2 a^2 x+\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {6 a^2 \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)}{6 d}-\frac {1}{6} \left (5 a^2\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{2} \left (3 a^2\right ) \int \cos ^2(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 (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac {\left (6 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 {3 a^2 \cos (c+d x) \sin (c+d x)}{4 d}+\frac {7 a^2 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {2 a^2 \sin ^3(c+d x)}{3 d}-\frac {2 a^2 \sin ^5(c+d x)}{5 d}+\frac {a^2 \tan (c+d x)}{d}-\frac {1}{8} \left (5 a^2\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{4} \left (3 a^2\right ) \int 1 \, 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 {7 a^2 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {7 a^2 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {2 a^2 \sin ^3(c+d x)}{3 d}-\frac {2 a^2 \sin ^5(c+d x)}{5 d}+\frac {a^2 \tan (c+d x)}{d}-\frac {1}{16} \left (5 a^2\right ) \int 1 \, dx \\ & = -\frac {25 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 {7 a^2 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {7 a^2 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {2 a^2 \sin ^3(c+d x)}{3 d}-\frac {2 a^2 \sin ^5(c+d x)}{5 d}+\frac {a^2 \tan (c+d x)}{d} \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.79 \[ \int (a+a \sec (c+d x))^2 \sin ^6(c+d x) \, dx=-\frac {a^2 (1+\cos (c+d x))^2 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \left (1080 c+1080 d x+420 \arctan (\tan (c+d x))-1920 \text {arctanh}(\sin (c+d x))+1920 \sin (c+d x)+640 \sin ^3(c+d x)+384 \sin ^5(c+d x)-255 \sin (2 (c+d x))-15 \sin (4 (c+d x))+5 \sin (6 (c+d x))-960 \tan (c+d x)\right )}{3840 d} \]
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Time = 2.13 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.90
method | result | size |
parallelrisch | \(\frac {a^{2} \left (-3000 d x \cos \left (d x +c \right )-3840 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )+3840 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )+2175 \sin \left (d x +c \right )-2360 \sin \left (2 d x +2 c \right )-24 \sin \left (6 d x +6 c \right )+256 \sin \left (4 d x +4 c \right )-5 \sin \left (7 d x +7 c \right )+10 \sin \left (5 d x +5 c \right )+270 \sin \left (3 d x +3 c \right )\right )}{1920 d \cos \left (d x +c \right )}\) | \(142\) |
derivativedivides | \(\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{7}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(164\) |
default | \(\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{7}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(164\) |
parts | \(\frac {a^{2} \left (-\frac {\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{7}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )}{d}+\frac {2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}\) | \(169\) |
risch | \(-\frac {25 a^{2} x}{16}+\frac {17 i a^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}-\frac {17 i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{128 d}-\frac {11 i a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {11 i a^{2} {\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {2 i a^{2}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {a^{2} \sin \left (6 d x +6 c \right )}{192 d}-\frac {a^{2} \sin \left (5 d x +5 c \right )}{40 d}+\frac {a^{2} \sin \left (4 d x +4 c \right )}{64 d}+\frac {7 a^{2} \sin \left (3 d x +3 c \right )}{24 d}\) | \(212\) |
norman | \(\frac {\frac {25 a^{2} x}{16}+\frac {7 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {27 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 d}+\frac {797 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{40 d}-\frac {91 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{2 d}-\frac {8041 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{120 d}-\frac {431 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{12 d}+\frac {125 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{16}+\frac {225 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16}+\frac {125 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{16}-\frac {125 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{16}-\frac {225 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{16}-\frac {125 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{16}-\frac {25 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{16}-\frac {57 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{8 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}+\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(330\) |
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Time = 0.29 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.01 \[ \int (a+a \sec (c+d x))^2 \sin ^6(c+d x) \, dx=-\frac {375 \, a^{2} d x \cos \left (d x + c\right ) - 240 \, a^{2} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) + 240 \, a^{2} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (40 \, a^{2} \cos \left (d x + c\right )^{6} + 96 \, a^{2} \cos \left (d x + c\right )^{5} - 70 \, a^{2} \cos \left (d x + c\right )^{4} - 352 \, a^{2} \cos \left (d x + c\right )^{3} - 105 \, a^{2} \cos \left (d x + c\right )^{2} + 736 \, a^{2} \cos \left (d x + c\right ) - 240 \, a^{2}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )} \]
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Timed out. \[ \int (a+a \sec (c+d x))^2 \sin ^6(c+d x) \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.11 \[ \int (a+a \sec (c+d x))^2 \sin ^6(c+d x) \, dx=-\frac {64 \, {\left (6 \, \sin \left (d x + c\right )^{5} + 10 \, \sin \left (d x + c\right )^{3} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 30 \, \sin \left (d x + c\right )\right )} a^{2} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 60 \, d x + 60 \, c + 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} + 120 \, {\left (15 \, d x + 15 \, c - \frac {9 \, \tan \left (d x + c\right )^{3} + 7 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1} - 8 \, \tan \left (d x + c\right )\right )} a^{2}}{960 \, d} \]
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Time = 0.35 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.23 \[ \int (a+a \sec (c+d x))^2 \sin ^6(c+d x) \, dx=-\frac {375 \, {\left (d x + c\right )} a^{2} - 480 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 480 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {480 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + \frac {2 \, {\left (615 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 3485 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 7926 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 8586 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2595 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 345 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]
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Time = 14.71 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.50 \[ \int (a+a \sec (c+d x))^2 \sin ^6(c+d x) \, dx=\frac {\frac {57\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{8}+\frac {431\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{12}+\frac {8041\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{120}+\frac {91\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2}-\frac {797\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{40}-\frac {27\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}-\frac {7\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {25\,a^2\,x}{16}+\frac {4\,a^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]
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