Integrand size = 27, antiderivative size = 73 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {x}{8 a}-\frac {\cos ^3(c+d x)}{3 a d}-\frac {\cos (c+d x) \sin (c+d x)}{8 a d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d} \]
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Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2918, 2645, 30, 2648, 2715, 8} \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\cos ^3(c+d x)}{3 a d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac {\sin (c+d x) \cos (c+d x)}{8 a d}-\frac {x}{8 a} \]
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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 ^2(c+d x) \sin (c+d x) \, dx}{a}-\frac {\int \cos ^2(c+d x) \sin ^2(c+d x) \, dx}{a} \\ & = \frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\int \cos ^2(c+d x) \, dx}{4 a}-\frac {\text {Subst}\left (\int x^2 \, dx,x,\cos (c+d x)\right )}{a d} \\ & = -\frac {\cos ^3(c+d x)}{3 a d}-\frac {\cos (c+d x) \sin (c+d x)}{8 a d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\int 1 \, dx}{8 a} \\ & = -\frac {x}{8 a}-\frac {\cos ^3(c+d x)}{3 a d}-\frac {\cos (c+d x) \sin (c+d x)}{8 a d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(219\) vs. \(2(73)=146\).
Time = 1.12 (sec) , antiderivative size = 219, normalized size of antiderivative = 3.00 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {-24 (c-d x) \cos \left (\frac {c}{2}\right )+24 \cos \left (\frac {c}{2}+d x\right )+24 \cos \left (\frac {3 c}{2}+d x\right )+8 \cos \left (\frac {5 c}{2}+3 d x\right )+8 \cos \left (\frac {7 c}{2}+3 d x\right )-3 \cos \left (\frac {7 c}{2}+4 d x\right )+3 \cos \left (\frac {9 c}{2}+4 d x\right )+48 \sin \left (\frac {c}{2}\right )-24 c \sin \left (\frac {c}{2}\right )+24 d x \sin \left (\frac {c}{2}\right )-24 \sin \left (\frac {c}{2}+d x\right )+24 \sin \left (\frac {3 c}{2}+d x\right )-8 \sin \left (\frac {5 c}{2}+3 d x\right )+8 \sin \left (\frac {7 c}{2}+3 d x\right )-3 \sin \left (\frac {7 c}{2}+4 d x\right )-3 \sin \left (\frac {9 c}{2}+4 d x\right )}{192 a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \]
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Time = 0.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.62
method | result | size |
parallelrisch | \(\frac {-12 d x -8 \cos \left (3 d x +3 c \right )-24 \cos \left (d x +c \right )+3 \sin \left (4 d x +4 c \right )-32}{96 d a}\) | \(45\) |
risch | \(-\frac {x}{8 a}-\frac {\cos \left (d x +c \right )}{4 a d}+\frac {\sin \left (4 d x +4 c \right )}{32 d a}-\frac {\cos \left (3 d x +3 c \right )}{12 a d}\) | \(56\) |
derivativedivides | \(\frac {\frac {4 \left (-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16}-\frac {1}{6}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\) | \(129\) |
default | \(\frac {\frac {4 \left (-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16}-\frac {1}{6}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\) | \(129\) |
norman | \(\frac {-\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}-\frac {x}{8 a}-\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}-\frac {5 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {5 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {5 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}-\frac {5 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}-\frac {5 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}-\frac {5 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}-\frac {5 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {5 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {5}{12 a d}+\frac {\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d a}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{6 d a}-\frac {5 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {3 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {3 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{6 d a}-\frac {19 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}-\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{6 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\) | \(399\) |
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Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.68 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {8 \, \cos \left (d x + c\right )^{3} + 3 \, d x - 3 \, {\left (2 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, a d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1134 vs. \(2 (56) = 112\).
Time = 6.42 (sec) , antiderivative size = 1134, normalized size of antiderivative = 15.53 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (65) = 130\).
Time = 0.32 (sec) , antiderivative size = 257, normalized size of antiderivative = 3.52 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {8 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {21 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {24 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {24 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 8}{a + \frac {4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{12 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.74 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {3 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a}}{24 \, d} \]
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Time = 9.61 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.59 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {6\,\cos \left (c+d\,x\right )+2\,\cos \left (3\,c+3\,d\,x\right )-\frac {3\,\sin \left (4\,c+4\,d\,x\right )}{4}+3\,d\,x}{24\,a\,d} \]
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