Integrand size = 23, antiderivative size = 81 \[ \int (a+a \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{2} a (2 A+C) x+\frac {a (3 A+C) \sin (c+d x)}{3 d}-\frac {a C \cos (c+d x) \sin (c+d x)}{6 d}+\frac {C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 a d} \]
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Time = 0.12 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3103, 2813} \[ \int (a+a \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {a (3 A+C) \sin (c+d x)}{3 d}+\frac {1}{2} a x (2 A+C)+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 a d}-\frac {a C \sin (c+d x) \cos (c+d x)}{6 d} \]
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Rule 2813
Rule 3103
Rubi steps \begin{align*} \text {integral}& = \frac {C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 a d}+\frac {\int (a+a \cos (c+d x)) (a (3 A+2 C)-a C \cos (c+d x)) \, dx}{3 a} \\ & = \frac {1}{2} a (2 A+C) x+\frac {a (3 A+C) \sin (c+d x)}{3 d}-\frac {a C \cos (c+d x) \sin (c+d x)}{6 d}+\frac {C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 a d} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.73 \[ \int (a+a \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {a (6 c C+12 A d x+6 C d x+3 (4 A+3 C) \sin (c+d x)+3 C \sin (2 (c+d x))+C \sin (3 (c+d x)))}{12 d} \]
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Time = 3.63 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.63
method | result | size |
parallelrisch | \(\frac {a \left (\frac {\sin \left (2 d x +2 c \right ) C}{4}+\frac {\sin \left (3 d x +3 c \right ) C}{12}+\left (A +\frac {3 C}{4}\right ) \sin \left (d x +c \right )+d x \left (A +\frac {C}{2}\right )\right )}{d}\) | \(51\) |
derivativedivides | \(\frac {\frac {a C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a A \sin \left (d x +c \right )+a A \left (d x +c \right )}{d}\) | \(68\) |
default | \(\frac {\frac {a C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a A \sin \left (d x +c \right )+a A \left (d x +c \right )}{d}\) | \(68\) |
risch | \(a x A +\frac {a C x}{2}+\frac {\sin \left (d x +c \right ) a A}{d}+\frac {3 a C \sin \left (d x +c \right )}{4 d}+\frac {\sin \left (3 d x +3 c \right ) a C}{12 d}+\frac {\sin \left (2 d x +2 c \right ) a C}{4 d}\) | \(68\) |
parts | \(a x A +\frac {\sin \left (d x +c \right ) a A}{d}+\frac {a C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {a C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}\) | \(69\) |
norman | \(\frac {\frac {a \left (2 A +C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a \left (2 A +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {a \left (2 A +C \right ) x}{2}+\frac {3 a \left (2 A +C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {3 a \left (2 A +C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a \left (2 A +C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {4 a \left (3 A +C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}\) | \(151\) |
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Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.69 \[ \int (a+a \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (2 \, A + C\right )} a d x + {\left (2 \, C a \cos \left (d x + c\right )^{2} + 3 \, C a \cos \left (d x + c\right ) + 2 \, {\left (3 \, A + 2 \, C\right )} a\right )} \sin \left (d x + c\right )}{6 \, d} \]
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Time = 0.12 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.49 \[ \int (a+a \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} A a x + \frac {A a \sin {\left (c + d x \right )}}{d} + \frac {C a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {C a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {2 C a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {C a \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {C a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\left (c \right )}\right ) \left (a \cos {\left (c \right )} + a\right ) & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.83 \[ \int (a+a \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {12 \, {\left (d x + c\right )} A a - 4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a + 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a + 12 \, A a \sin \left (d x + c\right )}{12 \, d} \]
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Time = 0.27 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.79 \[ \int (a+a \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{2} \, {\left (2 \, A a + C a\right )} x + \frac {C a \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {C a \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (4 \, A a + 3 \, C a\right )} \sin \left (d x + c\right )}{4 \, d} \]
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Time = 1.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.83 \[ \int (a+a \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx=A\,a\,x+\frac {C\,a\,x}{2}+\frac {A\,a\,\sin \left (c+d\,x\right )}{d}+\frac {3\,C\,a\,\sin \left (c+d\,x\right )}{4\,d}+\frac {C\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a\,\sin \left (3\,c+3\,d\,x\right )}{12\,d} \]
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