Integrand size = 25, antiderivative size = 123 \[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{8} a^2 (12 A+7 C) x+\frac {a^2 (12 A+7 C) \sin (c+d x)}{6 d}+\frac {a^2 (12 A+7 C) \cos (c+d x) \sin (c+d x)}{24 d}-\frac {C (a+a \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 a d} \]
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Time = 0.15 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3103, 2830, 2723} \[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {a^2 (12 A+7 C) \sin (c+d x)}{6 d}+\frac {a^2 (12 A+7 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac {1}{8} a^2 x (12 A+7 C)+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}-\frac {C \sin (c+d x) (a \cos (c+d x)+a)^2}{12 d} \]
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Rule 2723
Rule 2830
Rule 3103
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 (a (4 A+3 C)-a C \cos (c+d x)) \, dx}{4 a} \\ & = -\frac {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} (12 A+7 C) \int (a+a \cos (c+d x))^2 \, dx \\ & = \frac {1}{8} a^2 (12 A+7 C) x+\frac {a^2 (12 A+7 C) \sin (c+d x)}{6 d}+\frac {a^2 (12 A+7 C) \cos (c+d x) \sin (c+d x)}{24 d}-\frac {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 = 0.35 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.59 \[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {a^2 (144 A d x+84 C d x+48 (4 A+3 C) \sin (c+d x)+24 (A+2 C) \sin (2 (c+d x))+16 C \sin (3 (c+d x))+3 C \sin (4 (c+d x)))}{96 d} \]
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Time = 5.12 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.58
method | result | size |
parallelrisch | \(\frac {3 \left (\left (\frac {A}{6}+\frac {C}{3}\right ) \sin \left (2 d x +2 c \right )+\frac {\sin \left (3 d x +3 c \right ) C}{9}+\frac {\sin \left (4 d x +4 c \right ) C}{48}+\left (\frac {4 A}{3}+C \right ) \sin \left (d x +c \right )+d x \left (A +\frac {7 C}{12}\right )\right ) a^{2}}{2 d}\) | \(71\) |
risch | \(\frac {3 a^{2} x A}{2}+\frac {7 a^{2} C x}{8}+\frac {2 \sin \left (d x +c \right ) A \,a^{2}}{d}+\frac {3 \sin \left (d x +c \right ) a^{2} C}{2 d}+\frac {\sin \left (4 d x +4 c \right ) a^{2} C}{32 d}+\frac {\sin \left (3 d x +3 c \right ) a^{2} C}{6 d}+\frac {\sin \left (2 d x +2 c \right ) A \,a^{2}}{4 d}+\frac {\sin \left (2 d x +2 c \right ) a^{2} C}{2 d}\) | \(118\) |
parts | \(a^{2} x A +\frac {\left (A \,a^{2}+a^{2} C \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {a^{2} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {2 \sin \left (d x +c \right ) A \,a^{2}}{d}+\frac {2 a^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}\) | \(126\) |
derivativedivides | \(\frac {A \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 A \,a^{2} \sin \left (d x +c \right )+\frac {2 a^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+A \,a^{2} \left (d x +c \right )+a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(142\) |
default | \(\frac {A \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 A \,a^{2} \sin \left (d x +c \right )+\frac {2 a^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+A \,a^{2} \left (d x +c \right )+a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(142\) |
norman | \(\frac {\frac {a^{2} \left (12 A +7 C \right ) x}{8}+\frac {5 a^{2} \left (4 A +5 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {11 a^{2} \left (12 A +7 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {a^{2} \left (12 A +7 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {a^{2} \left (12 A +7 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {3 a^{2} \left (12 A +7 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {a^{2} \left (12 A +7 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a^{2} \left (12 A +7 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a^{2} \left (156 A +83 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\) | \(229\) |
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Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.70 \[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (12 \, A + 7 \, C\right )} a^{2} d x + {\left (6 \, C a^{2} \cos \left (d x + c\right )^{3} + 16 \, C a^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (4 \, A + 7 \, C\right )} a^{2} \cos \left (d x + c\right ) + 16 \, {\left (3 \, A + 2 \, C\right )} a^{2}\right )} \sin \left (d x + c\right )}{24 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 309 vs. \(2 (109) = 218\).
Time = 0.18 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.51 \[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} \frac {A a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {A a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + A a^{2} x + \frac {A a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {2 A a^{2} \sin {\left (c + d x \right )}}{d} + \frac {3 C a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 C a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {C a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {3 C a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {C a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 C a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {4 C a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 C a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {2 C a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {C a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\left (c \right )}\right ) \left (a \cos {\left (c \right )} + a\right )^{2} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.07 \[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} + 96 \, {\left (d x + c\right )} A a^{2} - 64 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} + 192 \, A a^{2} \sin \left (d x + c\right )}{96 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.84 \[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {C a^{2} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {C a^{2} \sin \left (3 \, d x + 3 \, c\right )}{6 \, d} + \frac {1}{8} \, {\left (12 \, A a^{2} + 7 \, C a^{2}\right )} x + \frac {{\left (A a^{2} + 2 \, C a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (4 \, A a^{2} + 3 \, C a^{2}\right )} \sin \left (d x + c\right )}{2 \, d} \]
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Time = 1.09 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.95 \[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {3\,A\,a^2\,x}{2}+\frac {7\,C\,a^2\,x}{8}+\frac {2\,A\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {3\,C\,a^2\,\sin \left (c+d\,x\right )}{2\,d}+\frac {A\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {C\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{6\,d}+\frac {C\,a^2\,\sin \left (4\,c+4\,d\,x\right )}{32\,d} \]
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