Integrand size = 39, antiderivative size = 143 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{8} a (4 A+3 (B+C)) x+\frac {a (5 A+5 B+4 C) \sin (c+d x)}{5 d}+\frac {a (4 A+3 (B+C)) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a (B+C) \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {a C \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac {a (5 A+5 B+4 C) \sin ^3(c+d x)}{15 d} \]
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Time = 0.27 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3112, 3102, 2827, 2715, 8, 2713} \[ \int \cos ^2(c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {a (5 A+5 B+4 C) \sin ^3(c+d x)}{15 d}+\frac {a (5 A+5 B+4 C) \sin (c+d x)}{5 d}+\frac {a (4 A+3 (B+C)) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} a x (4 A+3 (B+C))+\frac {a (B+C) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {a C \sin (c+d x) \cos ^4(c+d x)}{5 d} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2827
Rule 3102
Rule 3112
Rubi steps \begin{align*} \text {integral}& = \frac {a C \cos ^4(c+d x) \sin (c+d x)}{5 d}+\frac {1}{5} \int \cos ^2(c+d x) \left (5 a A+a (5 A+5 B+4 C) \cos (c+d x)+5 a (B+C) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {a (B+C) \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {a C \cos ^4(c+d x) \sin (c+d x)}{5 d}+\frac {1}{20} \int \cos ^2(c+d x) (5 a (4 A+3 (B+C))+4 a (5 A+5 B+4 C) \cos (c+d x)) \, dx \\ & = \frac {a (B+C) \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {a C \cos ^4(c+d x) \sin (c+d x)}{5 d}+\frac {1}{5} (a (5 A+5 B+4 C)) \int \cos ^3(c+d x) \, dx+\frac {1}{4} (a (4 A+3 (B+C))) \int \cos ^2(c+d x) \, dx \\ & = \frac {a (4 A+3 (B+C)) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a (B+C) \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {a C \cos ^4(c+d x) \sin (c+d x)}{5 d}+\frac {1}{8} (a (4 A+3 (B+C))) \int 1 \, dx-\frac {(a (5 A+5 B+4 C)) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d} \\ & = \frac {1}{8} a (4 A+3 (B+C)) x+\frac {a (5 A+5 B+4 C) \sin (c+d x)}{5 d}+\frac {a (4 A+3 (B+C)) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a (B+C) \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {a C \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac {a (5 A+5 B+4 C) \sin ^3(c+d x)}{15 d} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.65 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a \left (480 (A+B+C) \sin (c+d x)-160 (A+B+2 C) \sin ^3(c+d x)+96 C \sin ^5(c+d x)+15 (4 (4 A+3 (B+C)) (c+d x)+8 (A+B+C) \sin (2 (c+d x))+(B+C) \sin (4 (c+d x)))\right )}{480 d} \]
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Time = 7.22 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.66
method | result | size |
parallelrisch | \(\frac {a \left (3 \left (A +B +C \right ) \sin \left (2 d x +2 c \right )+\left (A +B +\frac {5 C}{4}\right ) \sin \left (3 d x +3 c \right )+\frac {3 \left (B +C \right ) \sin \left (4 d x +4 c \right )}{8}+\frac {3 \sin \left (5 d x +5 c \right ) C}{20}+3 \left (3 A +3 B +\frac {5 C}{2}\right ) \sin \left (d x +c \right )+6 x \left (A +\frac {3 B}{4}+\frac {3 C}{4}\right ) d \right )}{12 d}\) | \(95\) |
parts | \(\frac {\left (a A +B a \right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (B a +a C \right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {a A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {a C \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}\) | \(135\) |
derivativedivides | \(\frac {\frac {a C \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+B a \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {B a \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(173\) |
default | \(\frac {\frac {a C \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+B a \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {B a \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(173\) |
risch | \(\frac {a x A}{2}+\frac {3 a B x}{8}+\frac {3 a C x}{8}+\frac {3 \sin \left (d x +c \right ) a A}{4 d}+\frac {3 a B \sin \left (d x +c \right )}{4 d}+\frac {5 a C \sin \left (d x +c \right )}{8 d}+\frac {a C \sin \left (5 d x +5 c \right )}{80 d}+\frac {\sin \left (4 d x +4 c \right ) B a}{32 d}+\frac {\sin \left (4 d x +4 c \right ) a C}{32 d}+\frac {\sin \left (3 d x +3 c \right ) a A}{12 d}+\frac {\sin \left (3 d x +3 c \right ) B a}{12 d}+\frac {5 \sin \left (3 d x +3 c \right ) a C}{48 d}+\frac {\sin \left (2 d x +2 c \right ) a A}{4 d}+\frac {\sin \left (2 d x +2 c \right ) B a}{4 d}+\frac {\sin \left (2 d x +2 c \right ) a C}{4 d}\) | \(200\) |
norman | \(\frac {\frac {a \left (4 A +3 B +3 C \right ) x}{8}+\frac {a \left (4 A +3 B +3 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {5 a \left (4 A +3 B +3 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {5 a \left (4 A +3 B +3 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {5 a \left (4 A +3 B +3 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {5 a \left (4 A +3 B +3 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a \left (4 A +3 B +3 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a \left (12 A +13 B +13 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a \left (20 A +29 B +13 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {4 a \left (25 A +25 B +29 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {a \left (44 A +35 B +19 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}\) | \(290\) |
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Time = 0.29 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.76 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (4 \, A + 3 \, B + 3 \, C\right )} a d x + {\left (24 \, C a \cos \left (d x + c\right )^{4} + 30 \, {\left (B + C\right )} a \cos \left (d x + c\right )^{3} + 8 \, {\left (5 \, A + 5 \, B + 4 \, C\right )} a \cos \left (d x + c\right )^{2} + 15 \, {\left (4 \, A + 3 \, B + 3 \, C\right )} a \cos \left (d x + c\right ) + 16 \, {\left (5 \, A + 5 \, B + 4 \, C\right )} a\right )} \sin \left (d x + c\right )}{120 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 428 vs. \(2 (134) = 268\).
Time = 0.27 (sec) , antiderivative size = 428, normalized size of antiderivative = 2.99 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} \frac {A a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {A a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {2 A a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {A a \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {A a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {3 B a x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 B a x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 B a x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 B a \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {2 B a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 B a \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {B a \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 C a x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 C a x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 C a x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {8 C a \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 C a \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {3 C a \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {C a \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 C a \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + a\right ) \left (A + B \cos {\left (c \right )} + C \cos ^{2}{\left (c \right )}\right ) \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.16 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a - 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a + 160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a - 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a - 32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a - 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a}{480 \, d} \]
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Time = 0.33 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.90 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{8} \, {\left (4 \, A a + 3 \, B a + 3 \, C a\right )} x + \frac {C a \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {{\left (B a + C a\right )} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {{\left (4 \, A a + 4 \, B a + 5 \, C a\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (A a + B a + C a\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (6 \, A a + 6 \, B a + 5 \, C a\right )} \sin \left (d x + c\right )}{8 \, d} \]
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Time = 2.79 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.95 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\left (A\,a+\frac {3\,B\,a}{4}+\frac {3\,C\,a}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {10\,A\,a}{3}+\frac {29\,B\,a}{6}+\frac {13\,C\,a}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {20\,A\,a}{3}+\frac {20\,B\,a}{3}+\frac {116\,C\,a}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {22\,A\,a}{3}+\frac {35\,B\,a}{6}+\frac {19\,C\,a}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (3\,A\,a+\frac {13\,B\,a}{4}+\frac {13\,C\,a}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,\left (4\,A+3\,B+3\,C\right )}{4\,d}+\frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (4\,A+3\,B+3\,C\right )}{4\,\left (A\,a+\frac {3\,B\,a}{4}+\frac {3\,C\,a}{4}\right )}\right )\,\left (4\,A+3\,B+3\,C\right )}{4\,d} \]
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