Integrand size = 31, antiderivative size = 134 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {1}{8} a^2 (5 A+2 B) x-\frac {a^2 (5 A+2 B) \cos ^3(c+d x)}{12 d}+\frac {a^2 (5 A+2 B) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))^2}{5 d}-\frac {(5 A+2 B) \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{20 d} \]
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Time = 0.12 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2939, 2757, 2748, 2715, 8} \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=-\frac {a^2 (5 A+2 B) \cos ^3(c+d x)}{12 d}-\frac {(5 A+2 B) \cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{20 d}+\frac {a^2 (5 A+2 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} a^2 x (5 A+2 B)-\frac {B \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d} \]
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Rule 8
Rule 2715
Rule 2748
Rule 2757
Rule 2939
Rubi steps \begin{align*} \text {integral}& = -\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))^2}{5 d}+\frac {1}{5} (5 A+2 B) \int \cos ^2(c+d x) (a+a \sin (c+d x))^2 \, dx \\ & = -\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))^2}{5 d}-\frac {(5 A+2 B) \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{20 d}+\frac {1}{4} (a (5 A+2 B)) \int \cos ^2(c+d x) (a+a \sin (c+d x)) \, dx \\ & = -\frac {a^2 (5 A+2 B) \cos ^3(c+d x)}{12 d}-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))^2}{5 d}-\frac {(5 A+2 B) \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{20 d}+\frac {1}{4} \left (a^2 (5 A+2 B)\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {a^2 (5 A+2 B) \cos ^3(c+d x)}{12 d}+\frac {a^2 (5 A+2 B) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))^2}{5 d}-\frac {(5 A+2 B) \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{20 d}+\frac {1}{8} \left (a^2 (5 A+2 B)\right ) \int 1 \, dx \\ & = \frac {1}{8} a^2 (5 A+2 B) x-\frac {a^2 (5 A+2 B) \cos ^3(c+d x)}{12 d}+\frac {a^2 (5 A+2 B) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))^2}{5 d}-\frac {(5 A+2 B) \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{20 d} \\ \end{align*}
Time = 0.72 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.99 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=-\frac {a^2 \cos (c+d x) \left (80 A+62 B+\frac {60 (5 A+2 B) \arcsin \left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right )}{\sqrt {\cos ^2(c+d x)}}+8 (10 A+7 B) \cos (2 (c+d x))-6 B \cos (4 (c+d x))-135 A \sin (c+d x)-30 B \sin (c+d x)+15 A \sin (3 (c+d x))+30 B \sin (3 (c+d x))\right )}{240 d} \]
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Time = 0.56 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.72
method | result | size |
parallelrisch | \(\frac {\left (\left (-\frac {2 A}{3}-\frac {5 B}{12}\right ) \cos \left (3 d x +3 c \right )+\left (-\frac {A}{8}-\frac {B}{4}\right ) \sin \left (4 d x +4 c \right )+\frac {B \cos \left (5 d x +5 c \right )}{20}+A \sin \left (2 d x +2 c \right )+\left (-2 A -\frac {3 B}{2}\right ) \cos \left (d x +c \right )+\frac {5 d x A}{2}+d x B -\frac {8 A}{3}-\frac {28 B}{15}\right ) a^{2}}{4 d}\) | \(96\) |
risch | \(\frac {5 a^{2} x A}{8}+\frac {a^{2} x B}{4}-\frac {A \,a^{2} \cos \left (d x +c \right )}{2 d}-\frac {3 a^{2} \cos \left (d x +c \right ) B}{8 d}+\frac {a^{2} \cos \left (5 d x +5 c \right ) B}{80 d}-\frac {\sin \left (4 d x +4 c \right ) A \,a^{2}}{32 d}-\frac {\sin \left (4 d x +4 c \right ) B \,a^{2}}{16 d}-\frac {a^{2} \cos \left (3 d x +3 c \right ) A}{6 d}-\frac {5 a^{2} \cos \left (3 d x +3 c \right ) B}{48 d}+\frac {\sin \left (2 d x +2 c \right ) A \,a^{2}}{4 d}\) | \(154\) |
derivativedivides | \(\frac {A \,a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+B \,a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )-\frac {2 A \,a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{3}+2 B \,a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+A \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {B \,a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{3}}{d}\) | \(182\) |
default | \(\frac {A \,a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+B \,a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )-\frac {2 A \,a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{3}+2 B \,a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+A \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {B \,a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{3}}{d}\) | \(182\) |
norman | \(\frac {\left (\frac {5}{8} A \,a^{2}+\frac {1}{4} B \,a^{2}\right ) x +\left (\frac {5}{8} A \,a^{2}+\frac {1}{4} B \,a^{2}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {25}{4} A \,a^{2}+\frac {5}{2} B \,a^{2}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {25}{4} A \,a^{2}+\frac {5}{2} B \,a^{2}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {25}{8} A \,a^{2}+\frac {5}{4} B \,a^{2}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {25}{8} A \,a^{2}+\frac {5}{4} B \,a^{2}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {20 A \,a^{2}+14 B \,a^{2}}{15 d}-\frac {\left (4 A \,a^{2}+2 B \,a^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 \left (4 A \,a^{2}+4 B \,a^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 \left (8 A \,a^{2}+2 B \,a^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {\left (8 A \,a^{2}+8 B \,a^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {a^{2} \left (3 A -2 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {a^{2} \left (3 A -2 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {a^{2} \left (6 B +7 A \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {a^{2} \left (6 B +7 A \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}\) | \(399\) |
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Time = 0.26 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.71 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {24 \, B a^{2} \cos \left (d x + c\right )^{5} - 80 \, {\left (A + B\right )} a^{2} \cos \left (d x + c\right )^{3} + 15 \, {\left (5 \, A + 2 \, B\right )} a^{2} d x - 15 \, {\left (2 \, {\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} - {\left (5 \, A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 371 vs. \(2 (117) = 234\).
Time = 0.28 (sec) , antiderivative size = 371, normalized size of antiderivative = 2.77 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\begin {cases} \frac {A a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {A a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {A a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {A a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {A a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {A a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {A a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {A a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} - \frac {2 A a^{2} \cos ^{3}{\left (c + d x \right )}}{3 d} + \frac {B a^{2} x \sin ^{4}{\left (c + d x \right )}}{4} + \frac {B a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2} + \frac {B a^{2} x \cos ^{4}{\left (c + d x \right )}}{4} + \frac {B a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} - \frac {B a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {B a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{4 d} - \frac {2 B a^{2} \cos ^{5}{\left (c + d x \right )}}{15 d} - \frac {B a^{2} \cos ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\left (c \right )}\right ) \left (a \sin {\left (c \right )} + a\right )^{2} \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=-\frac {320 \, A a^{2} \cos \left (d x + c\right )^{3} + 160 \, B a^{2} \cos \left (d x + c\right )^{3} - 15 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} A a^{2} - 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} - 32 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} B a^{2} - 30 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} B a^{2}}{480 \, d} \]
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Time = 0.35 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.97 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {B a^{2} \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {A a^{2} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {1}{8} \, {\left (5 \, A a^{2} + 2 \, B a^{2}\right )} x - \frac {{\left (8 \, A a^{2} + 5 \, B a^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac {{\left (4 \, A a^{2} + 3 \, B a^{2}\right )} \cos \left (d x + c\right )}{8 \, d} - \frac {{\left (A a^{2} + 2 \, B a^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} \]
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Time = 11.17 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.74 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,A+2\,B\right )}{4\,\left (\frac {5\,A\,a^2}{4}+\frac {B\,a^2}{2}\right )}\right )\,\left (5\,A+2\,B\right )}{4\,d}-\frac {a^2\,\left (5\,A+2\,B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{4\,d}-\frac {\frac {4\,A\,a^2}{3}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,A\,a^2}{4}-\frac {B\,a^2}{2}\right )+\frac {14\,B\,a^2}{15}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (4\,A\,a^2+2\,B\,a^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {7\,A\,a^2}{2}+3\,B\,a^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {7\,A\,a^2}{2}+3\,B\,a^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {3\,A\,a^2}{4}-\frac {B\,a^2}{2}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (8\,A\,a^2+8\,B\,a^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {8\,A\,a^2}{3}+\frac {8\,B\,a^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {16\,A\,a^2}{3}+\frac {4\,B\,a^2}{3}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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