Integrand size = 21, antiderivative size = 125 \[ \int \frac {\sin ^8(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {5 x}{128 a}-\frac {5 \cos (c+d x) \sin (c+d x)}{128 a d}+\frac {5 \cos ^3(c+d x) \sin (c+d x)}{64 a d}+\frac {5 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a d}+\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a d}+\frac {\sin ^7(c+d x)}{7 a d} \]
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Time = 0.30 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3957, 2918, 2644, 30, 2648, 2715, 8} \[ \int \frac {\sin ^8(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\sin ^7(c+d x)}{7 a d}+\frac {\sin ^5(c+d x) \cos ^3(c+d x)}{8 a d}+\frac {5 \sin ^3(c+d x) \cos ^3(c+d x)}{48 a d}+\frac {5 \sin (c+d x) \cos ^3(c+d x)}{64 a d}-\frac {5 \sin (c+d x) \cos (c+d x)}{128 a d}-\frac {5 x}{128 a} \]
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Rule 8
Rule 30
Rule 2644
Rule 2648
Rule 2715
Rule 2918
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos (c+d x) \sin ^8(c+d x)}{-a-a \cos (c+d x)} \, dx \\ & = \frac {\int \cos (c+d x) \sin ^6(c+d x) \, dx}{a}-\frac {\int \cos ^2(c+d x) \sin ^6(c+d x) \, dx}{a} \\ & = \frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a d}-\frac {5 \int \cos ^2(c+d x) \sin ^4(c+d x) \, dx}{8 a}+\frac {\text {Subst}\left (\int x^6 \, dx,x,\sin (c+d x)\right )}{a d} \\ & = \frac {5 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a d}+\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a d}+\frac {\sin ^7(c+d x)}{7 a d}-\frac {5 \int \cos ^2(c+d x) \sin ^2(c+d x) \, dx}{16 a} \\ & = \frac {5 \cos ^3(c+d x) \sin (c+d x)}{64 a d}+\frac {5 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a d}+\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a d}+\frac {\sin ^7(c+d x)}{7 a d}-\frac {5 \int \cos ^2(c+d x) \, dx}{64 a} \\ & = -\frac {5 \cos (c+d x) \sin (c+d x)}{128 a d}+\frac {5 \cos ^3(c+d x) \sin (c+d x)}{64 a d}+\frac {5 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a d}+\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a d}+\frac {\sin ^7(c+d x)}{7 a d}-\frac {5 \int 1 \, dx}{128 a} \\ & = -\frac {5 x}{128 a}-\frac {5 \cos (c+d x) \sin (c+d x)}{128 a d}+\frac {5 \cos ^3(c+d x) \sin (c+d x)}{64 a d}+\frac {5 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a d}+\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a d}+\frac {\sin ^7(c+d x)}{7 a d} \\ \end{align*}
Time = 1.61 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.06 \[ \int \frac {\sin ^8(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (1176 c-840 d x+1680 \sin (c+d x)+336 \sin (2 (c+d x))-1008 \sin (3 (c+d x))+168 \sin (4 (c+d x))+336 \sin (5 (c+d x))-112 \sin (6 (c+d x))-48 \sin (7 (c+d x))+21 \sin (8 (c+d x))-1176 \tan \left (\frac {c}{2}\right )\right )}{10752 a d (1+\sec (c+d x))} \]
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Time = 0.67 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.79
method | result | size |
parallelrisch | \(\frac {-840 d x +1680 \sin \left (d x +c \right )-48 \sin \left (7 d x +7 c \right )+336 \sin \left (5 d x +5 c \right )-1008 \sin \left (3 d x +3 c \right )+21 \sin \left (8 d x +8 c \right )-112 \sin \left (6 d x +6 c \right )+168 \sin \left (4 d x +4 c \right )+336 \sin \left (2 d x +2 c \right )}{21504 d a}\) | \(99\) |
risch | \(-\frac {5 x}{128 a}+\frac {5 \sin \left (d x +c \right )}{64 a d}+\frac {\sin \left (8 d x +8 c \right )}{1024 d a}-\frac {\sin \left (7 d x +7 c \right )}{448 d a}-\frac {\sin \left (6 d x +6 c \right )}{192 d a}+\frac {\sin \left (5 d x +5 c \right )}{64 d a}+\frac {\sin \left (4 d x +4 c \right )}{128 d a}-\frac {3 \sin \left (3 d x +3 c \right )}{64 d a}+\frac {\sin \left (2 d x +2 c \right )}{64 d a}\) | \(141\) |
derivativedivides | \(\frac {-\frac {256 \left (-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16384}-\frac {115 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{49152}-\frac {383 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{49152}-\frac {5053 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{344064}-\frac {44099 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{344064}+\frac {383 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{49152}+\frac {115 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{49152}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{16384}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d a}\) | \(142\) |
default | \(\frac {-\frac {256 \left (-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16384}-\frac {115 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{49152}-\frac {383 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{49152}-\frac {5053 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{344064}-\frac {44099 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{344064}+\frac {383 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{49152}+\frac {115 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{49152}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{16384}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d a}\) | \(142\) |
norman | \(\frac {-\frac {5 x}{128 a}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 a d}+\frac {115 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{192 a d}+\frac {383 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{192 a d}+\frac {5053 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{1344 a d}+\frac {44099 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{1344 a d}-\frac {383 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{192 a d}-\frac {115 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{192 a d}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{64 a d}-\frac {5 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{16 a}-\frac {35 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{32 a}-\frac {35 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{16 a}-\frac {175 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64 a}-\frac {35 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{16 a}-\frac {35 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{32 a}-\frac {5 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{16 a}-\frac {5 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{128 a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8}}\) | \(310\) |
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Time = 0.26 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.73 \[ \int \frac {\sin ^8(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {105 \, d x - {\left (336 \, \cos \left (d x + c\right )^{7} - 384 \, \cos \left (d x + c\right )^{6} - 952 \, \cos \left (d x + c\right )^{5} + 1152 \, \cos \left (d x + c\right )^{4} + 826 \, \cos \left (d x + c\right )^{3} - 1152 \, \cos \left (d x + c\right )^{2} - 105 \, \cos \left (d x + c\right ) + 384\right )} \sin \left (d x + c\right )}{2688 \, a d} \]
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\[ \int \frac {\sin ^8(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\sin ^{8}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (113) = 226\).
Time = 0.29 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.88 \[ \int \frac {\sin ^8(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2681 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5053 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {44099 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {2681 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {805 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {105 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}}}{a + \frac {8 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {28 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {56 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {70 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {56 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {28 \, a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {8 \, a \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {a \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}}} - \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{1344 \, d} \]
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Time = 0.34 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.11 \[ \int \frac {\sin ^8(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {105 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 805 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 2681 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 44099 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 5053 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 2681 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 805 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{8} a}}{2688 \, d} \]
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Time = 16.33 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.06 \[ \int \frac {\sin ^8(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}-\frac {115\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{192}-\frac {383\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{192}+\frac {44099\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{1344}+\frac {5053\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{1344}+\frac {383\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{192}+\frac {115\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192}+\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^8}-\frac {5\,x}{128\,a} \]
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