Integrand size = 43, antiderivative size = 193 \[ \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 a (21 A+18 B+16 C) \sin (c+d x)}{45 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (9 B+C) \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}-\frac {4 (21 A+18 B+16 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac {2 C \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{9 d}+\frac {2 (21 A+18 B+16 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 a d} \]
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Time = 0.53 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {3124, 3060, 2838, 2830, 2725} \[ \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 (21 A+18 B+16 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 a d}-\frac {4 (21 A+18 B+16 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{315 d}+\frac {2 a (21 A+18 B+16 C) \sin (c+d x)}{45 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a (9 B+C) \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt {a \cos (c+d x)+a}}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{9 d} \]
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Rule 2725
Rule 2830
Rule 2838
Rule 3060
Rule 3124
Rubi steps \begin{align*} \text {integral}& = \frac {2 C \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{9 d}+\frac {2 \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \left (\frac {3}{2} a (3 A+2 C)+\frac {1}{2} a (9 B+C) \cos (c+d x)\right ) \, dx}{9 a} \\ & = \frac {2 a (9 B+C) \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 C \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{9 d}+\frac {1}{21} (21 A+18 B+16 C) \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \, dx \\ & = \frac {2 a (9 B+C) \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 C \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{9 d}+\frac {2 (21 A+18 B+16 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 a d}+\frac {(2 (21 A+18 B+16 C)) \int \left (\frac {3 a}{2}-a \cos (c+d x)\right ) \sqrt {a+a \cos (c+d x)} \, dx}{105 a} \\ & = \frac {2 a (9 B+C) \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}-\frac {4 (21 A+18 B+16 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac {2 C \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{9 d}+\frac {2 (21 A+18 B+16 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 a d}+\frac {1}{45} (21 A+18 B+16 C) \int \sqrt {a+a \cos (c+d x)} \, dx \\ & = \frac {2 a (21 A+18 B+16 C) \sin (c+d x)}{45 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (9 B+C) \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}-\frac {4 (21 A+18 B+16 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac {2 C \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{9 d}+\frac {2 (21 A+18 B+16 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 a d} \\ \end{align*}
Time = 0.60 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.59 \[ \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\sqrt {a (1+\cos (c+d x))} (1596 A+1368 B+1321 C+(672 A+94 (9 B+8 C)) \cos (c+d x)+4 (63 A+54 B+83 C) \cos (2 (c+d x))+90 B \cos (3 (c+d x))+80 C \cos (3 (c+d x))+35 C \cos (4 (c+d x))) \tan \left (\frac {1}{2} (c+d x)\right )}{1260 d} \]
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Time = 6.61 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.67
method | result | size |
default | \(\frac {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (560 C \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-360 B -1440 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (252 A +756 B +1512 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-420 A -630 B -840 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+315 A +315 B +315 C \right ) \sqrt {2}}{315 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(130\) |
parts | \(\frac {2 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (12 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7\right ) \sqrt {2}}{15 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}+\frac {2 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (40 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-36 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+22 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+9\right ) \sqrt {2}}{35 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}+\frac {2 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (560 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-800 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+552 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-104 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+107\right ) \sqrt {2}}{315 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(254\) |
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Time = 0.27 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.56 \[ \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (35 \, C \cos \left (d x + c\right )^{4} + 5 \, {\left (9 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (21 \, A + 18 \, B + 16 \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (21 \, A + 18 \, B + 16 \, C\right )} \cos \left (d x + c\right ) + 168 \, A + 144 \, B + 128 \, C\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
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Timed out. \[ \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Time = 0.42 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.01 \[ \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {84 \, {\left (3 \, \sqrt {2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 5 \, \sqrt {2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 30 \, \sqrt {2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} A \sqrt {a} + 18 \, {\left (5 \, \sqrt {2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 7 \, \sqrt {2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 35 \, \sqrt {2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 105 \, \sqrt {2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} B \sqrt {a} + {\left (35 \, \sqrt {2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 45 \, \sqrt {2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 252 \, \sqrt {2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 420 \, \sqrt {2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 1890 \, \sqrt {2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} C \sqrt {a}}{2520 \, d} \]
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Time = 0.77 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.16 \[ \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\sqrt {2} {\left (35 \, C \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 45 \, {\left (2 \, B \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + C \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 126 \, {\left (2 \, A \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + B \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 2 \, C \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 210 \, {\left (2 \, A \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 3 \, B \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 2 \, C \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 630 \, {\left (4 \, A \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 3 \, B \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 3 \, C \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{2520 \, d} \]
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Timed out. \[ \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^2\,\sqrt {a+a\,\cos \left (c+d\,x\right )}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
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