Integrand size = 27, antiderivative size = 97 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {x}{16 a}-\frac {\cos ^5(c+d x)}{5 a d}-\frac {\cos (c+d x) \sin (c+d x)}{16 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{24 a d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a d} \]
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Time = 0.10 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2918, 2645, 30, 2648, 2715, 8} \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\cos ^5(c+d x)}{5 a d}+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 a d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{24 a d}-\frac {\sin (c+d x) \cos (c+d x)}{16 a d}-\frac {x}{16 a} \]
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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 ^4(c+d x) \sin (c+d x) \, dx}{a}-\frac {\int \cos ^4(c+d x) \sin ^2(c+d x) \, dx}{a} \\ & = \frac {\cos ^5(c+d x) \sin (c+d x)}{6 a d}-\frac {\int \cos ^4(c+d x) \, dx}{6 a}-\frac {\text {Subst}\left (\int x^4 \, dx,x,\cos (c+d x)\right )}{a d} \\ & = -\frac {\cos ^5(c+d x)}{5 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{24 a d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a d}-\frac {\int \cos ^2(c+d x) \, dx}{8 a} \\ & = -\frac {\cos ^5(c+d x)}{5 a d}-\frac {\cos (c+d x) \sin (c+d x)}{16 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{24 a d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a d}-\frac {\int 1 \, dx}{16 a} \\ & = -\frac {x}{16 a}-\frac {\cos ^5(c+d x)}{5 a d}-\frac {\cos (c+d x) \sin (c+d x)}{16 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{24 a d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(377\) vs. \(2(97)=194\).
Time = 3.50 (sec) , antiderivative size = 377, normalized size of antiderivative = 3.89 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {-30 (5 c-4 d x) \cos \left (\frac {c}{2}\right )+120 \cos \left (\frac {c}{2}+d x\right )+120 \cos \left (\frac {3 c}{2}+d x\right )+15 \cos \left (\frac {3 c}{2}+2 d x\right )-15 \cos \left (\frac {5 c}{2}+2 d x\right )+60 \cos \left (\frac {5 c}{2}+3 d x\right )+60 \cos \left (\frac {7 c}{2}+3 d x\right )-15 \cos \left (\frac {7 c}{2}+4 d x\right )+15 \cos \left (\frac {9 c}{2}+4 d x\right )+12 \cos \left (\frac {9 c}{2}+5 d x\right )+12 \cos \left (\frac {11 c}{2}+5 d x\right )-5 \cos \left (\frac {11 c}{2}+6 d x\right )+5 \cos \left (\frac {13 c}{2}+6 d x\right )+300 \sin \left (\frac {c}{2}\right )-150 c \sin \left (\frac {c}{2}\right )+120 d x \sin \left (\frac {c}{2}\right )-120 \sin \left (\frac {c}{2}+d x\right )+120 \sin \left (\frac {3 c}{2}+d x\right )+15 \sin \left (\frac {3 c}{2}+2 d x\right )+15 \sin \left (\frac {5 c}{2}+2 d x\right )-60 \sin \left (\frac {5 c}{2}+3 d x\right )+60 \sin \left (\frac {7 c}{2}+3 d x\right )-15 \sin \left (\frac {7 c}{2}+4 d x\right )-15 \sin \left (\frac {9 c}{2}+4 d x\right )-12 \sin \left (\frac {9 c}{2}+5 d x\right )+12 \sin \left (\frac {11 c}{2}+5 d x\right )-5 \sin \left (\frac {11 c}{2}+6 d x\right )-5 \sin \left (\frac {13 c}{2}+6 d x\right )}{1920 a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \]
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Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.80
method | result | size |
parallelrisch | \(\frac {-60 d x -12 \cos \left (5 d x +5 c \right )-60 \cos \left (3 d x +3 c \right )-120 \cos \left (d x +c \right )+5 \sin \left (6 d x +6 c \right )+15 \sin \left (4 d x +4 c \right )-15 \sin \left (2 d x +2 c \right )-192}{960 d a}\) | \(78\) |
risch | \(-\frac {x}{16 a}-\frac {\cos \left (d x +c \right )}{8 a d}+\frac {\sin \left (6 d x +6 c \right )}{192 d a}-\frac {\cos \left (5 d x +5 c \right )}{80 a d}+\frac {\sin \left (4 d x +4 c \right )}{64 d a}-\frac {\cos \left (3 d x +3 c \right )}{16 a d}-\frac {\sin \left (2 d x +2 c \right )}{64 d a}\) | \(107\) |
derivativedivides | \(\frac {\frac {4 \left (-\frac {\left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}-\frac {\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {47 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96}-\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {13 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {13 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {47 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96}-\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}-\frac {1}{10}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d a}\) | \(181\) |
default | \(\frac {\frac {4 \left (-\frac {\left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}-\frac {\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {47 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96}-\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {13 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {13 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {47 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96}-\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}-\frac {1}{10}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d a}\) | \(181\) |
norman | \(\frac {-\frac {21 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {7 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {35 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {7 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {35 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {21 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {35 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {35 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {21 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {21 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {11}{40 a d}-\frac {7 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {7 x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {x}{16 a}+\frac {11 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d a}+\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{5 d a}-\frac {x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {x \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a}-\frac {5 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{20 d a}+\frac {\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}-\frac {29 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}-\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a}-\frac {35 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}-\frac {211 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 d a}-\frac {13 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}-\frac {29 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 d a}-\frac {7 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {433 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 d a}-\frac {8 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {17 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\) | \(562\) |
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Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.62 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {48 \, \cos \left (d x + c\right )^{5} + 15 \, d x - 5 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, a d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2307 vs. \(2 (76) = 152\).
Time = 19.24 (sec) , antiderivative size = 2307, normalized size of antiderivative = 23.78 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (87) = 174\).
Time = 0.31 (sec) , antiderivative size = 379, normalized size of antiderivative = 3.91 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {235 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {480 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {390 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {480 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {390 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {240 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {235 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {240 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {15 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - 48}{a + \frac {6 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {6 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac {15 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{120 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (87) = 174\).
Time = 0.29 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.85 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {15 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 235 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 390 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 390 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 235 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 48\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a}}{240 \, d} \]
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Time = 14.11 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.78 \[ \int \frac {\cos ^6(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {x}{16\,a}-\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {47\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {47\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {2}{5}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \]
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