Integrand size = 39, antiderivative size = 156 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{8} (4 a A+3 b B+3 a C) x+\frac {(5 A b+5 a B+4 b C) \sin (c+d x)}{5 d}+\frac {(4 a A+3 b B+3 a C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {(b B+a C) \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {b C \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac {(5 A b+5 a B+4 b C) \sin ^3(c+d x)}{15 d} \]
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Time = 0.25 (sec) , antiderivative size = 156, 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} \[ \int \cos ^2(c+d x) (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {\sin ^3(c+d x) (5 a B+5 A b+4 b C)}{15 d}+\frac {\sin (c+d x) (5 a B+5 A b+4 b C)}{5 d}+\frac {\sin (c+d x) \cos (c+d x) (4 a A+3 a C+3 b B)}{8 d}+\frac {1}{8} x (4 a A+3 a C+3 b B)+\frac {(a C+b B) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {b C \sin (c+d x) \cos ^4(c+d x)}{5 d} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2827
Rule 3102
Rule 3112
Rubi steps \begin{align*} \text {integral}& = \frac {b 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+(5 A b+5 a B+4 b C) \cos (c+d x)+5 (b B+a C) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {(b B+a C) \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {b C \cos ^4(c+d x) \sin (c+d x)}{5 d}+\frac {1}{20} \int \cos ^2(c+d x) (5 (4 a A+3 b B+3 a C)+4 (5 A b+5 a B+4 b C) \cos (c+d x)) \, dx \\ & = \frac {(b B+a C) \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {b C \cos ^4(c+d x) \sin (c+d x)}{5 d}+\frac {1}{4} (4 a A+3 b B+3 a C) \int \cos ^2(c+d x) \, dx+\frac {1}{5} (5 A b+5 a B+4 b C) \int \cos ^3(c+d x) \, dx \\ & = \frac {(4 a A+3 b B+3 a C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {(b B+a C) \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {b C \cos ^4(c+d x) \sin (c+d x)}{5 d}+\frac {1}{8} (4 a A+3 b B+3 a C) \int 1 \, dx-\frac {(5 A b+5 a B+4 b C) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d} \\ & = \frac {1}{8} (4 a A+3 b B+3 a C) x+\frac {(5 A b+5 a B+4 b C) \sin (c+d x)}{5 d}+\frac {(4 a A+3 b B+3 a C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {(b B+a C) \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {b C \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac {(5 A b+5 a B+4 b C) \sin ^3(c+d x)}{15 d} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.75 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {480 (A b+a B+b C) \sin (c+d x)-160 (A b+a B+2 b C) \sin ^3(c+d x)+96 b C \sin ^5(c+d x)+15 (4 (4 a A+3 b B+3 a C) (c+d x)+8 (b B+a (A+C)) \sin (2 (c+d x))+(b B+a C) \sin (4 (c+d x)))}{480 d} \]
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Time = 0.39 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.78
method | result | size |
parallelrisch | \(\frac {120 \left (B b +a \left (A +C \right )\right ) \sin \left (2 d x +2 c \right )+10 \left (\left (4 A +5 C \right ) b +4 B a \right ) \sin \left (3 d x +3 c \right )+15 \left (B b +C a \right ) \sin \left (4 d x +4 c \right )+6 C b \sin \left (5 d x +5 c \right )+60 \left (\left (6 A +5 C \right ) b +6 B a \right ) \sin \left (d x +c \right )+240 \left (\frac {3 B b}{4}+a \left (A +\frac {3 C}{4}\right )\right ) x d}{480 d}\) | \(121\) |
parts | \(\frac {\left (A b +B a \right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (B b +C a \right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {a A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {C b \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}\) | \(135\) |
derivativedivides | \(\frac {\frac {C b \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+B b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+C a \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {A b \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {B a \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(173\) |
default | \(\frac {\frac {C b \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+B b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+C a \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {A b \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {B a \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(173\) |
risch | \(\frac {x a A}{2}+\frac {3 x B b}{8}+\frac {3 a C x}{8}+\frac {3 \sin \left (d x +c \right ) A b}{4 d}+\frac {3 \sin \left (d x +c \right ) B a}{4 d}+\frac {5 b C \sin \left (d x +c \right )}{8 d}+\frac {C b \sin \left (5 d x +5 c \right )}{80 d}+\frac {\sin \left (4 d x +4 c \right ) B b}{32 d}+\frac {\sin \left (4 d x +4 c \right ) C a}{32 d}+\frac {\sin \left (3 d x +3 c \right ) A b}{12 d}+\frac {\sin \left (3 d x +3 c \right ) B a}{12 d}+\frac {5 \sin \left (3 d x +3 c \right ) C b}{48 d}+\frac {\sin \left (2 d x +2 c \right ) a A}{4 d}+\frac {\sin \left (2 d x +2 c \right ) B b}{4 d}+\frac {\sin \left (2 d x +2 c \right ) C a}{4 d}\) | \(200\) |
norman | \(\frac {\left (\frac {1}{2} a A +\frac {3}{8} B b +\frac {3}{8} C a \right ) x +\left (5 a A +\frac {15}{4} B b +\frac {15}{4} C a \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (5 a A +\frac {15}{4} B b +\frac {15}{4} C a \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} a A +\frac {3}{8} B b +\frac {3}{8} C a \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{2} a A +\frac {15}{8} B b +\frac {15}{8} C a \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{2} a A +\frac {15}{8} B b +\frac {15}{8} C a \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4 \left (25 A b +25 B a +29 C b \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}-\frac {\left (4 a A -8 A b -8 B a +5 B b +5 C a -8 C b \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {\left (4 a A +8 A b +8 B a +5 B b +5 C a +8 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {\left (12 a A -32 A b -32 B a +3 B b +3 C a -16 C b \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {\left (12 a A +32 A b +32 B a +3 B b +3 C a +16 C b \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}\) | \(354\) |
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Time = 0.28 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.78 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {15 \, {\left ({\left (4 \, A + 3 \, C\right )} a + 3 \, B b\right )} d x + {\left (24 \, C b \cos \left (d x + c\right )^{4} + 30 \, {\left (C a + B b\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (5 \, B a + {\left (5 \, A + 4 \, C\right )} b\right )} \cos \left (d x + c\right )^{2} + 80 \, B a + 16 \, {\left (5 \, A + 4 \, C\right )} b + 15 \, {\left ({\left (4 \, A + 3 \, C\right )} a + 3 \, B b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 428 vs. \(2 (151) = 302\).
Time = 0.28 (sec) , antiderivative size = 428, normalized size of antiderivative = 2.74 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} \frac {A a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {A a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {A a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {2 A b \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {A b \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {2 B a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {B a \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 B b x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 B b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 B b x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 B b \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 B b \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {3 C a x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 C a x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 C a x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 C a \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 C a \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {8 C b \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 C b \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {C b \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\left (c \right )}\right ) \left (A + B \cos {\left (c \right )} + C \cos ^{2}{\left (c \right )}\right ) \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.35 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.06 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a - 160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a - 160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B b + 32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C b}{480 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.83 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{8} \, {\left (4 \, A a + 3 \, C a + 3 \, B b\right )} x + \frac {C b \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {{\left (C a + B b\right )} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {{\left (4 \, B a + 4 \, A b + 5 \, C b\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (A a + C a + B b\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (6 \, B a + 6 \, A b + 5 \, C b\right )} \sin \left (d x + c\right )}{8 \, d} \]
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Time = 5.73 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.66 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {x\,\left (A\,a+\frac {3\,B\,b}{4}+\frac {3\,C\,a}{4}\right )}{2}+\frac {\left (2\,A\,b-A\,a+2\,B\,a-\frac {5\,B\,b}{4}-\frac {5\,C\,a}{4}+2\,C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {16\,A\,b}{3}-2\,A\,a+\frac {16\,B\,a}{3}-\frac {B\,b}{2}-\frac {C\,a}{2}+\frac {8\,C\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {20\,A\,b}{3}+\frac {20\,B\,a}{3}+\frac {116\,C\,b}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (2\,A\,a+\frac {16\,A\,b}{3}+\frac {16\,B\,a}{3}+\frac {B\,b}{2}+\frac {C\,a}{2}+\frac {8\,C\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (A\,a+2\,A\,b+2\,B\,a+\frac {5\,B\,b}{4}+\frac {5\,C\,a}{4}+2\,C\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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