Integrand size = 21, antiderivative size = 104 \[ \int \frac {\sin ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {3 x}{16 a^2}-\frac {3 \cos (c+d x) \sin (c+d x)}{16 a^2 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{8 a^2 d}-\frac {(a-a \cos (c+d x))^3 \sin ^3(c+d x)}{6 a^5 d}-\frac {\sin ^5(c+d x)}{10 a^2 d} \]
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Time = 0.37 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3957, 2954, 2949, 2748, 2715, 8} \[ \int \frac {\sin ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\sin ^3(c+d x) (a-a \cos (c+d x))^3}{6 a^5 d}-\frac {\sin ^5(c+d x)}{10 a^2 d}-\frac {\sin ^3(c+d x) \cos (c+d x)}{8 a^2 d}-\frac {3 \sin (c+d x) \cos (c+d x)}{16 a^2 d}+\frac {3 x}{16 a^2} \]
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Rule 8
Rule 2715
Rule 2748
Rule 2949
Rule 2954
Rule 3957
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos ^2(c+d x) \sin ^6(c+d x)}{(-a-a \cos (c+d x))^2} \, dx \\ & = \frac {\int \cos ^2(c+d x) (-a+a \cos (c+d x))^2 \sin ^2(c+d x) \, dx}{a^4} \\ & = -\frac {(a-a \cos (c+d x))^3 \sin ^3(c+d x)}{6 a^5 d}-\frac {\int (-a+a \cos (c+d x)) \sin ^4(c+d x) \, dx}{2 a^3} \\ & = -\frac {(a-a \cos (c+d x))^3 \sin ^3(c+d x)}{6 a^5 d}-\frac {\sin ^5(c+d x)}{10 a^2 d}+\frac {\int \sin ^4(c+d x) \, dx}{2 a^2} \\ & = -\frac {\cos (c+d x) \sin ^3(c+d x)}{8 a^2 d}-\frac {(a-a \cos (c+d x))^3 \sin ^3(c+d x)}{6 a^5 d}-\frac {\sin ^5(c+d x)}{10 a^2 d}+\frac {3 \int \sin ^2(c+d x) \, dx}{8 a^2} \\ & = -\frac {3 \cos (c+d x) \sin (c+d x)}{16 a^2 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{8 a^2 d}-\frac {(a-a \cos (c+d x))^3 \sin ^3(c+d x)}{6 a^5 d}-\frac {\sin ^5(c+d x)}{10 a^2 d}+\frac {3 \int 1 \, dx}{16 a^2} \\ & = \frac {3 x}{16 a^2}-\frac {3 \cos (c+d x) \sin (c+d x)}{16 a^2 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{8 a^2 d}-\frac {(a-a \cos (c+d x))^3 \sin ^3(c+d x)}{6 a^5 d}-\frac {\sin ^5(c+d x)}{10 a^2 d} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.07 \[ \int \frac {\sin ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (360 d x-480 \sin (c+d x)+30 \sin (2 (c+d x))+80 \sin (3 (c+d x))-90 \sin (4 (c+d x))+48 \sin (5 (c+d x))-10 \sin (6 (c+d x))+25 \tan \left (\frac {c}{2}\right )\right )}{480 a^2 d (1+\sec (c+d x))^2} \]
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Time = 0.93 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.74
method | result | size |
parallelrisch | \(\frac {180 d x -240 \sin \left (d x +c \right )-5 \sin \left (6 d x +6 c \right )-45 \sin \left (4 d x +4 c \right )+15 \sin \left (2 d x +2 c \right )+24 \sin \left (5 d x +5 c \right )+40 \sin \left (3 d x +3 c \right )}{960 a^{2} d}\) | \(77\) |
risch | \(\frac {3 x}{16 a^{2}}-\frac {\sin \left (d x +c \right )}{4 a^{2} d}-\frac {\sin \left (6 d x +6 c \right )}{192 a^{2} d}+\frac {\sin \left (5 d x +5 c \right )}{40 a^{2} d}-\frac {3 \sin \left (4 d x +4 c \right )}{64 a^{2} d}+\frac {\sin \left (3 d x +3 c \right )}{24 a^{2} d}+\frac {\sin \left (2 d x +2 c \right )}{64 a^{2} d}\) | \(107\) |
derivativedivides | \(\frac {\frac {32 \left (\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{256}-\frac {205 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{768}-\frac {29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{640}-\frac {99 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{640}-\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{256}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{256}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}+\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{a^{2} d}\) | \(115\) |
default | \(\frac {\frac {32 \left (\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{256}-\frac {205 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{768}-\frac {29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{640}-\frac {99 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{640}-\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{256}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{256}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}+\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{a^{2} d}\) | \(115\) |
norman | \(\frac {\frac {3 x}{16 a}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}-\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 a d}-\frac {99 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 a d}-\frac {29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 a d}-\frac {205 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 a d}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 a d}+\frac {9 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a}+\frac {45 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16 a}+\frac {15 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4 a}+\frac {45 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{16 a}+\frac {9 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8 a}+\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{16 a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6} a}\) | \(241\) |
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Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.68 \[ \int \frac {\sin ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {45 \, d x - {\left (40 \, \cos \left (d x + c\right )^{5} - 96 \, \cos \left (d x + c\right )^{4} + 50 \, \cos \left (d x + c\right )^{3} + 32 \, \cos \left (d x + c\right )^{2} - 45 \, \cos \left (d x + c\right ) + 64\right )} \sin \left (d x + c\right )}{240 \, a^{2} d} \]
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\[ \int \frac {\sin ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {\sin ^{6}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (95) = 190\).
Time = 0.29 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.81 \[ \int \frac {\sin ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {\frac {45 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {255 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {594 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {174 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {1025 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {45 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{a^{2} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac {45 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{120 \, d} \]
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Time = 0.34 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.09 \[ \int \frac {\sin ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\frac {45 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 1025 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 174 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 594 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 255 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a^{2}}}{240 \, d} \]
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Time = 16.58 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.03 \[ \int \frac {\sin ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {3\,x}{16\,a^2}-\frac {-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+\frac {205\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+\frac {29\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{20}+\frac {99\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{20}+\frac {17\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8}+\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \]
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