Integrand size = 29, antiderivative size = 84 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {1}{8} a (4 A+B) x-\frac {a (4 A+B) \cos ^3(c+d x)}{12 d}+\frac {a (4 A+B) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))}{4 d} \]
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Time = 0.06 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2939, 2748, 2715, 8} \[ \int \cos ^2(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {a (4 A+B) \cos ^3(c+d x)}{12 d}+\frac {a (4 A+B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} a x (4 A+B)-\frac {B \cos ^3(c+d x) (a \sin (c+d x)+a)}{4 d} \]
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Rule 8
Rule 2715
Rule 2748
Rule 2939
Rubi steps \begin{align*} \text {integral}& = -\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))}{4 d}+\frac {1}{4} (4 A+B) \int \cos ^2(c+d x) (a+a \sin (c+d x)) \, dx \\ & = -\frac {a (4 A+B) \cos ^3(c+d x)}{12 d}-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))}{4 d}+\frac {1}{4} (a (4 A+B)) \int \cos ^2(c+d x) \, dx \\ & = -\frac {a (4 A+B) \cos ^3(c+d x)}{12 d}+\frac {a (4 A+B) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))}{4 d}+\frac {1}{8} (a (4 A+B)) \int 1 \, dx \\ & = \frac {1}{8} a (4 A+B) x-\frac {a (4 A+B) \cos ^3(c+d x)}{12 d}+\frac {a (4 A+B) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))}{4 d} \\ \end{align*}
Time = 0.86 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.18 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {a \cos (c+d x) \left (-8 (A+B)-\frac {6 (4 A+B) \arcsin \left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right )}{\sqrt {\cos ^2(c+d x)}}+3 (4 A-B) \sin (c+d x)+8 (A+B) \sin ^2(c+d x)+6 B \sin ^3(c+d x)\right )}{24 d} \]
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Time = 0.35 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \(\frac {a \left (8 \left (-A -B \right ) \cos \left (3 d x +3 c \right )+48 d x A +12 d x B -24 A \cos \left (d x +c \right )+24 A \sin \left (2 d x +2 c \right )-3 B \sin \left (4 d x +4 c \right )-24 B \cos \left (d x +c \right )-32 A -32 B \right )}{96 d}\) | \(84\) |
derivativedivides | \(\frac {B a \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-\frac {A \left (\cos ^{3}\left (d x +c \right )\right ) a}{3}-\frac {B a \left (\cos ^{3}\left (d x +c \right )\right )}{3}+a A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(96\) |
default | \(\frac {B a \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-\frac {A \left (\cos ^{3}\left (d x +c \right )\right ) a}{3}-\frac {B a \left (\cos ^{3}\left (d x +c \right )\right )}{3}+a A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(96\) |
risch | \(\frac {a x A}{2}+\frac {a B x}{8}-\frac {a A \cos \left (d x +c \right )}{4 d}-\frac {a \cos \left (d x +c \right ) B}{4 d}-\frac {\sin \left (4 d x +4 c \right ) B a}{32 d}-\frac {a \cos \left (3 d x +3 c \right ) A}{12 d}-\frac {a \cos \left (3 d x +3 c \right ) B}{12 d}+\frac {\sin \left (2 d x +2 c \right ) a A}{4 d}\) | \(102\) |
norman | \(\frac {\left (\frac {1}{2} a A +\frac {1}{8} B a \right ) x +\left (2 a A +\frac {1}{2} B a \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2 a A +\frac {1}{2} B a \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3 a A +\frac {3}{4} B a \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} a A +\frac {1}{8} B a \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2 a A +2 B a}{3 d}-\frac {2 \left (a A +B a \right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {2 \left (a A +B a \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 \left (a A +B a \right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a \left (4 A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {a \left (4 A -B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {a \left (4 A +7 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {a \left (4 A +7 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\) | \(294\) |
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Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.77 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {8 \, {\left (A + B\right )} a \cos \left (d x + c\right )^{3} - 3 \, {\left (4 \, A + B\right )} a d x + 3 \, {\left (2 \, B a \cos \left (d x + c\right )^{3} - {\left (4 \, A + B\right )} a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (75) = 150\).
Time = 0.17 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.37 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\begin {cases} \frac {A a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {A a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {A a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} - \frac {A a \cos ^{3}{\left (c + d x \right )}}{3 d} + \frac {B a x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {B a x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {B a x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {B a \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {B a \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {B a \cos ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\left (c \right )}\right ) \left (a \sin {\left (c \right )} + a\right ) \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.88 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {32 \, A a \cos \left (d x + c\right )^{3} + 32 \, B a \cos \left (d x + c\right )^{3} - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a - 3 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} B a}{96 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.99 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {1}{8} \, {\left (4 \, A a + B a\right )} x - \frac {B a \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {A a \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} - \frac {{\left (A a + B a\right )} \cos \left (3 \, d x + 3 \, c\right )}{12 \, d} - \frac {{\left (A a + B a\right )} \cos \left (d x + c\right )}{4 \, d} \]
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Time = 11.07 (sec) , antiderivative size = 276, normalized size of antiderivative = 3.29 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (4\,A+B\right )}{4\,\left (A\,a+\frac {B\,a}{4}\right )}\right )\,\left (4\,A+B\right )}{4\,d}-\frac {a\,\left (4\,A+B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{4\,d}-\frac {\left (A\,a-\frac {B\,a}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (2\,A\,a+2\,B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (A\,a+\frac {7\,B\,a}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (2\,A\,a+2\,B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (-A\,a-\frac {7\,B\,a}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {2\,A\,a}{3}+\frac {2\,B\,a}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\left (\frac {B\,a}{4}-A\,a\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {2\,A\,a}{3}+\frac {2\,B\,a}{3}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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