Integrand size = 32, antiderivative size = 129 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{8} a^2 (8 B+7 C) x+\frac {a^2 (8 B+7 C) \sin (c+d x)}{6 d}+\frac {a^2 (8 B+7 C) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(4 B-C) (a+a \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 a d} \]
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Time = 0.16 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {3102, 2830, 2723} \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a^2 (8 B+7 C) \sin (c+d x)}{6 d}+\frac {a^2 (8 B+7 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac {1}{8} a^2 x (8 B+7 C)+\frac {(4 B-C) \sin (c+d x) (a \cos (c+d x)+a)^2}{12 d}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d} \]
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Rule 2723
Rule 2830
Rule 3102
Rubi steps \begin{align*} \text {integral}& = \frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 a d}+\frac {\int (a+a \cos (c+d x))^2 (3 a C+a (4 B-C) \cos (c+d x)) \, dx}{4 a} \\ & = \frac {(4 B-C) (a+a \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 a d}+\frac {1}{12} (8 B+7 C) \int (a+a \cos (c+d x))^2 \, dx \\ & = \frac {1}{8} a^2 (8 B+7 C) x+\frac {a^2 (8 B+7 C) \sin (c+d x)}{6 d}+\frac {a^2 (8 B+7 C) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(4 B-C) (a+a \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 a d} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.67 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a^2 (84 c C+96 B d x+84 C d x+24 (7 B+6 C) \sin (c+d x)+48 (B+C) \sin (2 (c+d x))+8 B \sin (3 (c+d x))+16 C \sin (3 (c+d x))+3 C \sin (4 (c+d x)))}{96 d} \]
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Time = 4.73 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.57
method | result | size |
parallelrisch | \(\frac {\left (\frac {\left (B +C \right ) \sin \left (2 d x +2 c \right )}{2}+\frac {\left (\frac {B}{2}+C \right ) \sin \left (3 d x +3 c \right )}{6}+\frac {\sin \left (4 d x +4 c \right ) C}{32}+\frac {\left (\frac {7 B}{2}+3 C \right ) \sin \left (d x +c \right )}{2}+d x \left (B +\frac {7 C}{8}\right )\right ) a^{2}}{d}\) | \(74\) |
parts | \(\frac {\left (B \,a^{2}+2 a^{2} C \right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (2 B \,a^{2}+a^{2} C \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {a^{2} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\sin \left (d x +c \right ) B \,a^{2}}{d}\) | \(128\) |
risch | \(a^{2} B x +\frac {7 a^{2} C x}{8}+\frac {7 \sin \left (d x +c \right ) B \,a^{2}}{4 d}+\frac {3 \sin \left (d x +c \right ) a^{2} C}{2 d}+\frac {\sin \left (4 d x +4 c \right ) a^{2} C}{32 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{2}}{12 d}+\frac {\sin \left (3 d x +3 c \right ) a^{2} C}{6 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{2}}{2 d}+\frac {\sin \left (2 d x +2 c \right ) a^{2} C}{2 d}\) | \(135\) |
derivativedivides | \(\frac {a^{2} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,a^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {2 a^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+2 B \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \sin \left (d x +c \right ) a^{2}}{d}\) | \(154\) |
default | \(\frac {a^{2} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,a^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {2 a^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+2 B \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \sin \left (d x +c \right ) a^{2}}{d}\) | \(154\) |
norman | \(\frac {\frac {a^{2} \left (8 B +7 C \right ) x}{8}+\frac {11 a^{2} \left (8 B +7 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {a^{2} \left (8 B +7 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {a^{2} \left (8 B +7 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {3 a^{2} \left (8 B +7 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {a^{2} \left (8 B +7 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a^{2} \left (8 B +7 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a^{2} \left (24 B +25 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{2} \left (136 B +83 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\) | \(229\) |
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Time = 0.25 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.70 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (8 \, B + 7 \, C\right )} a^{2} d x + {\left (6 \, C a^{2} \cos \left (d x + c\right )^{3} + 8 \, {\left (B + 2 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (8 \, B + 7 \, C\right )} a^{2} \cos \left (d x + c\right ) + 8 \, {\left (5 \, B + 4 \, C\right )} a^{2}\right )} \sin \left (d x + c\right )}{24 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (112) = 224\).
Time = 0.19 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.64 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} B a^{2} x \sin ^{2}{\left (c + d x \right )} + B a^{2} x \cos ^{2}{\left (c + d x \right )} + \frac {2 B a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {B a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {B a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {B a^{2} \sin {\left (c + d x \right )}}{d} + \frac {3 C a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 C a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {C a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {3 C a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {C a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 C a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {4 C a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 C a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {2 C a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {C a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (B \cos {\left (c \right )} + C \cos ^{2}{\left (c \right )}\right ) \left (a \cos {\left (c \right )} + a\right )^{2} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.12 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} - 48 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} + 64 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} - 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} - 96 \, B a^{2} \sin \left (d x + c\right )}{96 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.85 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {C a^{2} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {1}{8} \, {\left (8 \, B a^{2} + 7 \, C a^{2}\right )} x + \frac {{\left (B a^{2} + 2 \, C a^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {{\left (B a^{2} + C a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{2 \, d} + \frac {{\left (7 \, B a^{2} + 6 \, C a^{2}\right )} \sin \left (d x + c\right )}{4 \, d} \]
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Time = 1.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.04 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=B\,a^2\,x+\frac {7\,C\,a^2\,x}{8}+\frac {7\,B\,a^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {3\,C\,a^2\,\sin \left (c+d\,x\right )}{2\,d}+\frac {B\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {B\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {C\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {C\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{6\,d}+\frac {C\,a^2\,\sin \left (4\,c+4\,d\,x\right )}{32\,d} \]
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