Integrand size = 29, antiderivative size = 87 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 x}{8 a}-\frac {\cos (c+d x)}{a d}+\frac {\cos ^3(c+d x)}{3 a d}+\frac {3 \cos (c+d x) \sin (c+d x)}{8 a d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{4 a d} \]
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Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2918, 2713, 2715, 8} \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\cos ^3(c+d x)}{3 a d}-\frac {\cos (c+d x)}{a d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{4 a d}+\frac {3 \sin (c+d x) \cos (c+d x)}{8 a d}-\frac {3 x}{8 a} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2918
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sin ^3(c+d x) \, dx}{a}-\frac {\int \sin ^4(c+d x) \, dx}{a} \\ & = \frac {\cos (c+d x) \sin ^3(c+d x)}{4 a d}-\frac {3 \int \sin ^2(c+d x) \, dx}{4 a}-\frac {\text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a d} \\ & = -\frac {\cos (c+d x)}{a d}+\frac {\cos ^3(c+d x)}{3 a d}+\frac {3 \cos (c+d x) \sin (c+d x)}{8 a d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{4 a d}-\frac {3 \int 1 \, dx}{8 a} \\ & = -\frac {3 x}{8 a}-\frac {\cos (c+d x)}{a d}+\frac {\cos ^3(c+d x)}{3 a d}+\frac {3 \cos (c+d x) \sin (c+d x)}{8 a d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{4 a d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(271\) vs. \(2(87)=174\).
Time = 1.43 (sec) , antiderivative size = 271, normalized size of antiderivative = 3.11 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {24 (c-3 d x) \cos \left (\frac {c}{2}\right )-72 \cos \left (\frac {c}{2}+d x\right )-72 \cos \left (\frac {3 c}{2}+d x\right )+24 \cos \left (\frac {3 c}{2}+2 d x\right )-24 \cos \left (\frac {5 c}{2}+2 d x\right )+8 \cos \left (\frac {5 c}{2}+3 d x\right )+8 \cos \left (\frac {7 c}{2}+3 d x\right )-3 \cos \left (\frac {7 c}{2}+4 d x\right )+3 \cos \left (\frac {9 c}{2}+4 d x\right )-48 \sin \left (\frac {c}{2}\right )+24 c \sin \left (\frac {c}{2}\right )-72 d x \sin \left (\frac {c}{2}\right )+72 \sin \left (\frac {c}{2}+d x\right )-72 \sin \left (\frac {3 c}{2}+d x\right )+24 \sin \left (\frac {3 c}{2}+2 d x\right )+24 \sin \left (\frac {5 c}{2}+2 d x\right )-8 \sin \left (\frac {5 c}{2}+3 d x\right )+8 \sin \left (\frac {7 c}{2}+3 d x\right )-3 \sin \left (\frac {7 c}{2}+4 d x\right )-3 \sin \left (\frac {9 c}{2}+4 d x\right )}{192 a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \]
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Time = 0.21 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64
method | result | size |
parallelrisch | \(\frac {-36 d x -72 \cos \left (d x +c \right )-3 \sin \left (4 d x +4 c \right )+8 \cos \left (3 d x +3 c \right )+24 \sin \left (2 d x +2 c \right )-64}{96 d a}\) | \(56\) |
risch | \(-\frac {3 x}{8 a}-\frac {3 \cos \left (d x +c \right )}{4 a d}-\frac {\sin \left (4 d x +4 c \right )}{32 d a}+\frac {\cos \left (3 d x +3 c \right )}{12 a d}+\frac {\sin \left (2 d x +2 c \right )}{4 d a}\) | \(73\) |
derivativedivides | \(\frac {\frac {16 \left (-\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}-\frac {11 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}-\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}-\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}-\frac {1}{12}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\) | \(116\) |
default | \(\frac {\frac {16 \left (-\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}-\frac {11 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}-\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}-\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}-\frac {1}{12}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\) | \(116\) |
norman | \(\frac {\frac {3 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {3 x}{8 a}-\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}-\frac {15 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {15 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {15 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}-\frac {15 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}-\frac {15 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}-\frac {15 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}-\frac {15 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {15 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {3 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {3 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {7}{12 a d}+\frac {3 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{6 d a}+\frac {5 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d a}+\frac {7 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}-\frac {13 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}-\frac {11 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}+\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\) | \(399\) |
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Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {8 \, \cos \left (d x + c\right )^{3} - 9 \, d x - 3 \, {\left (2 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 24 \, \cos \left (d x + c\right )}{24 \, a d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1049 vs. \(2 (70) = 140\).
Time = 6.70 (sec) , antiderivative size = 1049, normalized size of antiderivative = 12.06 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (79) = 158\).
Time = 0.29 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.72 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {64 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {33 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {48 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {33 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {9 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 16}{a + \frac {4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {9 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{12 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.31 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {9 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 33 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 33 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 64 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 16\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a}}{24 \, d} \]
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Time = 9.62 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.91 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {{\cos \left (c+d\,x\right )}^3}{3\,a\,d}-\frac {\cos \left (c+d\,x\right )}{a\,d}-\frac {3\,x}{8\,a}-\frac {{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,a\,d}+\frac {5\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,a\,d} \]
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