Integrand size = 37, antiderivative size = 118 \[ \int \cos (c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{8} a (4 A+4 B+3 C) x+\frac {a (3 A+2 (B+C)) \sin (c+d x)}{3 d}+\frac {a (4 A+4 B+3 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a (B+C) \cos ^2(c+d x) \sin (c+d x)}{3 d}+\frac {a C \cos ^3(c+d x) \sin (c+d x)}{4 d} \]
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Time = 0.16 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {3112, 3102, 2813} \[ \int \cos (c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a (3 A+2 (B+C)) \sin (c+d x)}{3 d}+\frac {a (4 A+4 B+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} a x (4 A+4 B+3 C)+\frac {a (B+C) \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac {a C \sin (c+d x) \cos ^3(c+d x)}{4 d} \]
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Rule 2813
Rule 3102
Rule 3112
Rubi steps \begin{align*} \text {integral}& = \frac {a C \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{4} \int \cos (c+d x) \left (4 a A+a (4 A+4 B+3 C) \cos (c+d x)+4 a (B+C) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {a (B+C) \cos ^2(c+d x) \sin (c+d x)}{3 d}+\frac {a C \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{12} \int \cos (c+d x) (4 a (3 A+2 (B+C))+3 a (4 A+4 B+3 C) \cos (c+d x)) \, dx \\ & = \frac {1}{8} a (4 A+4 B+3 C) x+\frac {a (3 A+2 (B+C)) \sin (c+d x)}{3 d}+\frac {a (4 A+4 B+3 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a (B+C) \cos ^2(c+d x) \sin (c+d x)}{3 d}+\frac {a C \cos ^3(c+d x) \sin (c+d x)}{4 d} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.81 \[ \int \cos (c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a (48 B c+24 c C+48 A d x+48 B d x+36 C d x+24 (4 A+3 (B+C)) \sin (c+d x)+24 (A+B+C) \sin (2 (c+d x))+8 B \sin (3 (c+d x))+8 C \sin (3 (c+d x))+3 C \sin (4 (c+d x)))}{96 d} \]
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Time = 3.40 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.64
method | result | size |
parallelrisch | \(\frac {\left (\frac {\left (A +B +C \right ) \sin \left (2 d x +2 c \right )}{2}+\frac {\left (B +C \right ) \sin \left (3 d x +3 c \right )}{6}+\frac {\sin \left (4 d x +4 c \right ) C}{16}+\left (2 A +\frac {3 B}{2}+\frac {3 C}{2}\right ) \sin \left (d x +c \right )+d x \left (A +B +\frac {3 C}{4}\right )\right ) a}{2 d}\) | \(75\) |
parts | \(\frac {\left (a A +B a \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (B a +a C \right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\sin \left (d x +c \right ) a A}{d}+\frac {a C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(114\) |
derivativedivides | \(\frac {a C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B a \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {a C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a A \sin \left (d x +c \right )}{d}\) | \(141\) |
default | \(\frac {a C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B a \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {a C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a A \sin \left (d x +c \right )}{d}\) | \(141\) |
risch | \(\frac {a x A}{2}+\frac {a B x}{2}+\frac {3 a C x}{8}+\frac {\sin \left (d x +c \right ) a A}{d}+\frac {3 a B \sin \left (d x +c \right )}{4 d}+\frac {3 a C \sin \left (d x +c \right )}{4 d}+\frac {\sin \left (4 d x +4 c \right ) a C}{32 d}+\frac {\sin \left (3 d x +3 c \right ) B a}{12 d}+\frac {\sin \left (3 d x +3 c \right ) a C}{12 d}+\frac {\sin \left (2 d x +2 c \right ) a A}{4 d}+\frac {\sin \left (2 d x +2 c \right ) B a}{4 d}+\frac {\sin \left (2 d x +2 c \right ) a C}{4 d}\) | \(151\) |
norman | \(\frac {\frac {a \left (4 A +4 B +3 C \right ) x}{8}+\frac {a \left (4 A +4 B +3 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {a \left (4 A +4 B +3 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {3 a \left (4 A +4 B +3 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {a \left (4 A +4 B +3 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a \left (4 A +4 B +3 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a \left (12 A +12 B +13 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a \left (60 A +28 B +49 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {a \left (84 A +52 B +31 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\) | \(238\) |
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Time = 0.27 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.74 \[ \int \cos (c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (4 \, A + 4 \, B + 3 \, C\right )} a d x + {\left (6 \, C a \cos \left (d x + c\right )^{3} + 8 \, {\left (B + C\right )} a \cos \left (d x + c\right )^{2} + 3 \, {\left (4 \, A + 4 \, B + 3 \, C\right )} a \cos \left (d x + c\right ) + 8 \, {\left (3 \, A + 2 \, B + 2 \, C\right )} a\right )} \sin \left (d x + c\right )}{24 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (110) = 220\).
Time = 0.19 (sec) , antiderivative size = 320, normalized size of antiderivative = 2.71 \[ \int \cos (c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} \frac {A a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {A a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {A a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {A a \sin {\left (c + d x \right )}}{d} + \frac {B a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {B a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {2 B a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {B a \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {B a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {3 C a x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 C a x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 C a x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 C a \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {2 C a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 C a \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {C a \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + a\right ) \left (A + B \cos {\left (c \right )} + C \cos ^{2}{\left (c \right )}\right ) \cos {\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.12 \[ \int \cos (c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a + 96 \, A a \sin \left (d x + c\right )}{96 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.86 \[ \int \cos (c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{8} \, {\left (4 \, A a + 4 \, B a + 3 \, C a\right )} x + \frac {C a \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {{\left (B a + C a\right )} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {{\left (A a + B a + C a\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (4 \, A a + 3 \, B a + 3 \, C a\right )} \sin \left (d x + c\right )}{4 \, d} \]
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Time = 2.67 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.03 \[ \int \cos (c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\left (A\,a+B\,a+\frac {3\,C\,a}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (5\,A\,a+\frac {7\,B\,a}{3}+\frac {49\,C\,a}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (7\,A\,a+\frac {13\,B\,a}{3}+\frac {31\,C\,a}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (3\,A\,a+3\,B\,a+\frac {13\,C\,a}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,\left (4\,A+4\,B+3\,C\right )}{4\,d}+\frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (4\,A+4\,B+3\,C\right )}{4\,\left (A\,a+B\,a+\frac {3\,C\,a}{4}\right )}\right )\,\left (4\,A+4\,B+3\,C\right )}{4\,d} \]
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