Integrand size = 27, antiderivative size = 121 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {5 x}{128 a}-\frac {\cos ^7(c+d x)}{7 a d}-\frac {5 \cos (c+d x) \sin (c+d x)}{128 a d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{192 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{48 a d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{8 a d} \]
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Time = 0.12 (sec) , antiderivative size = 121, 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 = {2918, 2645, 30, 2648, 2715, 8} \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\cos ^7(c+d x)}{7 a d}+\frac {\sin (c+d x) \cos ^7(c+d x)}{8 a d}-\frac {\sin (c+d x) \cos ^5(c+d x)}{48 a d}-\frac {5 \sin (c+d x) \cos ^3(c+d x)}{192 a d}-\frac {5 \sin (c+d x) \cos (c+d x)}{128 a d}-\frac {5 x}{128 a} \]
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Rule 8
Rule 30
Rule 2645
Rule 2648
Rule 2715
Rule 2918
Rubi steps \begin{align*} \text {integral}& = \frac {\int \cos ^6(c+d x) \sin (c+d x) \, dx}{a}-\frac {\int \cos ^6(c+d x) \sin ^2(c+d x) \, dx}{a} \\ & = \frac {\cos ^7(c+d x) \sin (c+d x)}{8 a d}-\frac {\int \cos ^6(c+d x) \, dx}{8 a}-\frac {\text {Subst}\left (\int x^6 \, dx,x,\cos (c+d x)\right )}{a d} \\ & = -\frac {\cos ^7(c+d x)}{7 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{48 a d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{8 a d}-\frac {5 \int \cos ^4(c+d x) \, dx}{48 a} \\ & = -\frac {\cos ^7(c+d x)}{7 a d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{192 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{48 a d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{8 a d}-\frac {5 \int \cos ^2(c+d x) \, dx}{64 a} \\ & = -\frac {\cos ^7(c+d x)}{7 a d}-\frac {5 \cos (c+d x) \sin (c+d x)}{128 a d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{192 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{48 a d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{8 a d}-\frac {5 \int 1 \, dx}{128 a} \\ & = -\frac {5 x}{128 a}-\frac {\cos ^7(c+d x)}{7 a d}-\frac {5 \cos (c+d x) \sin (c+d x)}{128 a d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{192 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{48 a d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{8 a d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(481\) vs. \(2(121)=242\).
Time = 7.30 (sec) , antiderivative size = 481, normalized size of antiderivative = 3.98 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {-336 (7 c-5 d x) \cos \left (\frac {c}{2}\right )+1680 \cos \left (\frac {c}{2}+d x\right )+1680 \cos \left (\frac {3 c}{2}+d x\right )+336 \cos \left (\frac {3 c}{2}+2 d x\right )-336 \cos \left (\frac {5 c}{2}+2 d x\right )+1008 \cos \left (\frac {5 c}{2}+3 d x\right )+1008 \cos \left (\frac {7 c}{2}+3 d x\right )-168 \cos \left (\frac {7 c}{2}+4 d x\right )+168 \cos \left (\frac {9 c}{2}+4 d x\right )+336 \cos \left (\frac {9 c}{2}+5 d x\right )+336 \cos \left (\frac {11 c}{2}+5 d x\right )-112 \cos \left (\frac {11 c}{2}+6 d x\right )+112 \cos \left (\frac {13 c}{2}+6 d x\right )+48 \cos \left (\frac {13 c}{2}+7 d x\right )+48 \cos \left (\frac {15 c}{2}+7 d x\right )-21 \cos \left (\frac {15 c}{2}+8 d x\right )+21 \cos \left (\frac {17 c}{2}+8 d x\right )+4704 \sin \left (\frac {c}{2}\right )-2352 c \sin \left (\frac {c}{2}\right )+1680 d x \sin \left (\frac {c}{2}\right )-1680 \sin \left (\frac {c}{2}+d x\right )+1680 \sin \left (\frac {3 c}{2}+d x\right )+336 \sin \left (\frac {3 c}{2}+2 d x\right )+336 \sin \left (\frac {5 c}{2}+2 d x\right )-1008 \sin \left (\frac {5 c}{2}+3 d x\right )+1008 \sin \left (\frac {7 c}{2}+3 d x\right )-168 \sin \left (\frac {7 c}{2}+4 d x\right )-168 \sin \left (\frac {9 c}{2}+4 d x\right )-336 \sin \left (\frac {9 c}{2}+5 d x\right )+336 \sin \left (\frac {11 c}{2}+5 d x\right )-112 \sin \left (\frac {11 c}{2}+6 d x\right )-112 \sin \left (\frac {13 c}{2}+6 d x\right )-48 \sin \left (\frac {13 c}{2}+7 d x\right )+48 \sin \left (\frac {15 c}{2}+7 d x\right )-21 \sin \left (\frac {15 c}{2}+8 d x\right )-21 \sin \left (\frac {17 c}{2}+8 d x\right )}{43008 a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \]
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Time = 0.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.83
method | result | size |
parallelrisch | \(\frac {-840 d x -48 \cos \left (7 d x +7 c \right )-336 \cos \left (5 d x +5 c \right )-1008 \cos \left (3 d x +3 c \right )-1680 \cos \left (d x +c \right )+21 \sin \left (8 d x +8 c \right )+112 \sin \left (6 d x +6 c \right )+168 \sin \left (4 d x +4 c \right )-336 \sin \left (2 d x +2 c \right )-3072}{21504 d a}\) | \(100\) |
risch | \(-\frac {5 x}{128 a}-\frac {5 \cos \left (d x +c \right )}{64 a d}+\frac {\sin \left (8 d x +8 c \right )}{1024 d a}-\frac {\cos \left (7 d x +7 c \right )}{448 a d}+\frac {\sin \left (6 d x +6 c \right )}{192 d a}-\frac {\cos \left (5 d x +5 c \right )}{64 a d}+\frac {\sin \left (4 d x +4 c \right )}{128 d a}-\frac {3 \cos \left (3 d x +3 c \right )}{64 a d}-\frac {\sin \left (2 d x +2 c \right )}{64 d a}\) | \(141\) |
derivativedivides | \(\frac {\frac {4 \left (-\frac {1}{14}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{256}-\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{14}-\frac {397 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {895 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}-\frac {3 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {1765 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}-\frac {5 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {1765 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}-\frac {5 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {895 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}-\frac {\left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {397 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}-\frac {\left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {5 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d a}\) | \(233\) |
default | \(\frac {\frac {4 \left (-\frac {1}{14}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{256}-\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{14}-\frac {397 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {895 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}-\frac {3 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {1765 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}-\frac {5 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {1765 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}-\frac {5 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {895 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}-\frac {\left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {397 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}-\frac {\left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {5 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d a}\) | \(233\) |
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Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.58 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {384 \, \cos \left (d x + c\right )^{7} + 105 \, d x - 7 \, {\left (48 \, \cos \left (d x + c\right )^{7} - 8 \, \cos \left (d x + c\right )^{5} - 10 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2688 \, a d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 3888 vs. \(2 (102) = 204\).
Time = 50.20 (sec) , antiderivative size = 3888, normalized size of antiderivative = 32.13 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 501 vs. \(2 (109) = 218\).
Time = 0.30 (sec) , antiderivative size = 501, normalized size of antiderivative = 4.14 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {384 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {2779 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {8064 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {6265 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {8064 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {12355 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {13440 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {12355 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {13440 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {6265 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {2688 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {2779 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {2688 \, \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} - \frac {105 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} - 384}{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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Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (109) = 218\).
Time = 0.30 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.91 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {105 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 2688 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} - 2779 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 2688 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 6265 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 13440 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 12355 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 13440 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 12355 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 8064 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 6265 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 8064 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2779 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 384 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 384\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{8} a}}{2688 \, d} \]
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Time = 13.60 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.86 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {5\,x}{128\,a}-\frac {\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-\frac {397\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{192}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+\frac {895\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{192}+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {1765\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{192}+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {1765\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{192}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {895\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{192}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {397\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{7}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {2}{7}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^8} \]
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