Integrand size = 29, antiderivative size = 125 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx=\frac {3}{8} a (A+B) x+\frac {a (5 A+4 B) \sin (c+d x)}{5 d}+\frac {3 a (A+B) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a (A+B) \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {a B \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac {a (5 A+4 B) \sin ^3(c+d x)}{15 d} \]
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Time = 0.18 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {3047, 3102, 2827, 2713, 2715, 8} \[ \int \cos ^3(c+d x) (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx=-\frac {a (5 A+4 B) \sin ^3(c+d x)}{15 d}+\frac {a (5 A+4 B) \sin (c+d x)}{5 d}+\frac {a (A+B) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3 a (A+B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3}{8} a x (A+B)+\frac {a B \sin (c+d x) \cos ^4(c+d x)}{5 d} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2827
Rule 3047
Rule 3102
Rubi steps \begin{align*} \text {integral}& = \int \cos ^3(c+d x) \left (a A+(a A+a B) \cos (c+d x)+a B \cos ^2(c+d x)\right ) \, dx \\ & = \frac {a B \cos ^4(c+d x) \sin (c+d x)}{5 d}+\frac {1}{5} \int \cos ^3(c+d x) (a (5 A+4 B)+5 a (A+B) \cos (c+d x)) \, dx \\ & = \frac {a B \cos ^4(c+d x) \sin (c+d x)}{5 d}+(a (A+B)) \int \cos ^4(c+d x) \, dx+\frac {1}{5} (a (5 A+4 B)) \int \cos ^3(c+d x) \, dx \\ & = \frac {a (A+B) \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {a B \cos ^4(c+d x) \sin (c+d x)}{5 d}+\frac {1}{4} (3 a (A+B)) \int \cos ^2(c+d x) \, dx-\frac {(a (5 A+4 B)) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d} \\ & = \frac {a (5 A+4 B) \sin (c+d x)}{5 d}+\frac {3 a (A+B) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a (A+B) \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {a B \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac {a (5 A+4 B) \sin ^3(c+d x)}{15 d}+\frac {1}{8} (3 a (A+B)) \int 1 \, dx \\ & = \frac {3}{8} a (A+B) x+\frac {a (5 A+4 B) \sin (c+d x)}{5 d}+\frac {3 a (A+B) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a (A+B) \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {a B \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac {a (5 A+4 B) \sin ^3(c+d x)}{15 d} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.62 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx=\frac {a \left (480 (A+B) \sin (c+d x)-160 (A+2 B) \sin ^3(c+d x)+96 B \sin ^5(c+d x)+15 (A+B) (12 (c+d x)+8 \sin (2 (c+d x))+\sin (4 (c+d x)))\right )}{480 d} \]
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Time = 3.46 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.70
method | result | size |
parallelrisch | \(\frac {\left (8 \left (A +B \right ) \sin \left (2 d x +2 c \right )+\frac {2 \left (4 A +5 B \right ) \sin \left (3 d x +3 c \right )}{3}+\left (A +B \right ) \sin \left (4 d x +4 c \right )+\frac {2 B \sin \left (5 d x +5 c \right )}{5}+4 \left (6 A +5 B \right ) \sin \left (d x +c \right )+12 \left (A +B \right ) x d \right ) a}{32 d}\) | \(87\) |
parts | \(\frac {\left (a A +B a \right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {a A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {B a \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}\) | \(102\) |
derivativedivides | \(\frac {\frac {B a \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+a A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+B a \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(128\) |
default | \(\frac {\frac {B a \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+a A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+B a \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(128\) |
risch | \(\frac {3 a x A}{8}+\frac {3 a B x}{8}+\frac {3 \sin \left (d x +c \right ) a A}{4 d}+\frac {5 a B \sin \left (d x +c \right )}{8 d}+\frac {B a \sin \left (5 d x +5 c \right )}{80 d}+\frac {\sin \left (4 d x +4 c \right ) a A}{32 d}+\frac {\sin \left (4 d x +4 c \right ) B a}{32 d}+\frac {\sin \left (3 d x +3 c \right ) a A}{12 d}+\frac {5 \sin \left (3 d x +3 c \right ) B a}{48 d}+\frac {\sin \left (2 d x +2 c \right ) a A}{4 d}+\frac {\sin \left (2 d x +2 c \right ) B a}{4 d}\) | \(150\) |
norman | \(\frac {\frac {3 a \left (A +B \right ) x}{8}+\frac {13 a \left (A +B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {3 a \left (A +B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {15 a \left (A +B \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {15 a \left (A +B \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {15 a \left (A +B \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {15 a \left (A +B \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {3 a \left (A +B \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {4 a \left (25 A +29 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {a \left (29 A +13 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {a \left (35 A +19 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}\) | \(225\) |
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Time = 0.33 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.70 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx=\frac {45 \, {\left (A + B\right )} a d x + {\left (24 \, B a \cos \left (d x + c\right )^{4} + 30 \, {\left (A + B\right )} a \cos \left (d x + c\right )^{3} + 8 \, {\left (5 \, A + 4 \, B\right )} a \cos \left (d x + c\right )^{2} + 45 \, {\left (A + B\right )} a \cos \left (d x + c\right ) + 16 \, {\left (5 \, A + 4 \, B\right )} a\right )} \sin \left (d x + c\right )}{120 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (117) = 234\).
Time = 0.27 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.66 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx=\begin {cases} \frac {3 A a x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 A a x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 A a x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 A a \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {2 A a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 A a \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {A a \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 B a x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 B a x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 B a x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {8 B a \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 B a \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {3 B a \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {B a \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 B a \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (A + B \cos {\left (c \right )}\right ) \left (a \cos {\left (c \right )} + a\right ) \cos ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.99 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx=-\frac {160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a - 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a - 32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a - 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a}{480 \, d} \]
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Time = 0.32 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.90 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx=\frac {3}{8} \, {\left (A a + B a\right )} x + \frac {B a \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {{\left (A a + B a\right )} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {{\left (4 \, A a + 5 \, B a\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (A a + B a\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (6 \, A a + 5 \, B a\right )} \sin \left (d x + c\right )}{8 \, d} \]
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Time = 0.97 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.89 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx=\frac {\left (\frac {3\,A\,a}{4}+\frac {3\,B\,a}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {29\,A\,a}{6}+\frac {13\,B\,a}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {20\,A\,a}{3}+\frac {116\,B\,a}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {35\,A\,a}{6}+\frac {19\,B\,a}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {13\,A\,a}{4}+\frac {13\,B\,a}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {3\,a\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,\left (A+B\right )}{4\,d}+\frac {3\,a\,\mathrm {atan}\left (\frac {3\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A+B\right )}{4\,\left (\frac {3\,A\,a}{4}+\frac {3\,B\,a}{4}\right )}\right )\,\left (A+B\right )}{4\,d} \]
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