Integrand size = 39, antiderivative size = 110 \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=\frac {(2 A-2 B+3 C) x}{2 a}-\frac {(A-2 B+2 C) \sin (c+d x)}{a d}+\frac {(2 A-2 B+3 C) \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))} \]
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Time = 0.13 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {3120, 2813} \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=-\frac {(A-2 B+2 C) \sin (c+d x)}{a d}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}+\frac {(2 A-2 B+3 C) \sin (c+d x) \cos (c+d x)}{2 a d}+\frac {x (2 A-2 B+3 C)}{2 a} \]
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Rule 2813
Rule 3120
Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac {\int \cos (c+d x) (-a (A-2 B+2 C)+a (2 A-2 B+3 C) \cos (c+d x)) \, dx}{a^2} \\ & = \frac {(2 A-2 B+3 C) x}{2 a}-\frac {(A-2 B+2 C) \sin (c+d x)}{a d}+\frac {(2 A-2 B+3 C) \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))} \\ \end{align*}
Time = 1.62 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.94 \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (4 (2 A-2 B+3 C) d x \cos \left (\frac {d x}{2}\right )+4 (2 A-2 B+3 C) d x \cos \left (c+\frac {d x}{2}\right )-16 A \sin \left (\frac {d x}{2}\right )+20 B \sin \left (\frac {d x}{2}\right )-20 C \sin \left (\frac {d x}{2}\right )+4 B \sin \left (c+\frac {d x}{2}\right )-4 C \sin \left (c+\frac {d x}{2}\right )+4 B \sin \left (c+\frac {3 d x}{2}\right )-3 C \sin \left (c+\frac {3 d x}{2}\right )+4 B \sin \left (2 c+\frac {3 d x}{2}\right )-3 C \sin \left (2 c+\frac {3 d x}{2}\right )+C \sin \left (2 c+\frac {5 d x}{2}\right )+C \sin \left (3 c+\frac {5 d x}{2}\right )\right )}{8 a d (1+\cos (c+d x))} \]
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Time = 1.81 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.61
method | result | size |
parallelrisch | \(\frac {\left (C \cos \left (2 d x +2 c \right )+\left (4 B -2 C \right ) \cos \left (d x +c \right )-4 A +8 B -7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+4 x \left (-B +\frac {3 C}{2}+A \right ) d}{4 a d}\) | \(67\) |
derivativedivides | \(\frac {-A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {2 \left (-\frac {3 C}{2}+B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (B -\frac {C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (2 A -2 B +3 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\) | \(115\) |
default | \(\frac {-A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {2 \left (-\frac {3 C}{2}+B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (B -\frac {C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (2 A -2 B +3 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\) | \(115\) |
risch | \(\frac {x A}{a}-\frac {B x}{a}+\frac {3 C x}{2 a}-\frac {i {\mathrm e}^{i \left (d x +c \right )} B}{2 a d}+\frac {i {\mathrm e}^{i \left (d x +c \right )} C}{2 a d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} B}{2 a d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )} C}{2 a d}-\frac {2 i A}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}+\frac {2 i B}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}-\frac {2 i C}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}+\frac {\sin \left (2 d x +2 c \right ) C}{4 a d}\) | \(185\) |
norman | \(\frac {\frac {\left (2 A -2 B +3 C \right ) x}{2 a}-\frac {\left (A -3 B +2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {\left (A -B +C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}+\frac {3 \left (2 A -2 B +3 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {3 \left (2 A -2 B +3 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {\left (2 A -2 B +3 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {\left (3 A -7 B +7 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {\left (3 A -5 B +6 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}\) | \(223\) |
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Time = 0.26 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.83 \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=\frac {{\left (2 \, A - 2 \, B + 3 \, C\right )} d x \cos \left (d x + c\right ) + {\left (2 \, A - 2 \, B + 3 \, C\right )} d x + {\left (C \cos \left (d x + c\right )^{2} + {\left (2 \, B - C\right )} \cos \left (d x + c\right ) - 2 \, A + 4 \, B - 4 \, C\right )} \sin \left (d x + c\right )}{2 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 993 vs. \(2 (87) = 174\).
Time = 1.11 (sec) , antiderivative size = 993, normalized size of antiderivative = 9.03 \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (106) = 212\).
Time = 0.30 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.48 \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=-\frac {C {\left (\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a + \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} + B {\left (\frac {2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {2 \, \sin \left (d x + c\right )}{{\left (a + \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - A {\left (\frac {2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{d} \]
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Time = 0.30 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.25 \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=\frac {\frac {{\left (d x + c\right )} {\left (2 \, A - 2 \, B + 3 \, C\right )}}{a} - \frac {2 \, {\left (A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a} + \frac {2 \, {\left (2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a}}{2 \, d} \]
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Time = 1.92 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.02 \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=\frac {\left (2\,B-3\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,B-C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )}+\frac {x\,\left (2\,A-2\,B+3\,C\right )}{2\,a}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A-B+C\right )}{a\,d} \]
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