Integrand size = 40, antiderivative size = 147 \[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx=-\frac {(4 B-7 C) x}{2 a^2}+\frac {2 (5 B-8 C) \sin (c+d x)}{3 a^2 d}-\frac {(4 B-7 C) \cos (c+d x) \sin (c+d x)}{2 a^2 d}+\frac {(5 B-8 C) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}+\frac {(B-C) \cos ^3(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2} \]
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Time = 0.60 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {3108, 3056, 2813} \[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx=\frac {2 (5 B-8 C) \sin (c+d x)}{3 a^2 d}+\frac {(5 B-8 C) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac {(4 B-7 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac {x (4 B-7 C)}{2 a^2}+\frac {(B-C) \sin (c+d x) \cos ^3(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
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Rule 2813
Rule 3056
Rule 3108
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos ^3(c+d x) (B+C \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx \\ & = \frac {(B-C) \cos ^3(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {\int \frac {\cos ^2(c+d x) (3 a (B-C)-a (2 B-5 C) \cos (c+d x))}{a+a \cos (c+d x)} \, dx}{3 a^2} \\ & = \frac {(5 B-8 C) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}+\frac {(B-C) \cos ^3(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {\int \cos (c+d x) \left (2 a^2 (5 B-8 C)-3 a^2 (4 B-7 C) \cos (c+d x)\right ) \, dx}{3 a^4} \\ & = -\frac {(4 B-7 C) x}{2 a^2}+\frac {2 (5 B-8 C) \sin (c+d x)}{3 a^2 d}-\frac {(4 B-7 C) \cos (c+d x) \sin (c+d x)}{2 a^2 d}+\frac {(5 B-8 C) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}+\frac {(B-C) \cos ^3(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(315\) vs. \(2(147)=294\).
Time = 1.81 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.14 \[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (-36 (4 B-7 C) d x \cos \left (\frac {d x}{2}\right )-36 (4 B-7 C) d x \cos \left (c+\frac {d x}{2}\right )-48 B d x \cos \left (c+\frac {3 d x}{2}\right )+84 C d x \cos \left (c+\frac {3 d x}{2}\right )-48 B d x \cos \left (2 c+\frac {3 d x}{2}\right )+84 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+264 B \sin \left (\frac {d x}{2}\right )-381 C \sin \left (\frac {d x}{2}\right )-120 B \sin \left (c+\frac {d x}{2}\right )+147 C \sin \left (c+\frac {d x}{2}\right )+164 B \sin \left (c+\frac {3 d x}{2}\right )-239 C \sin \left (c+\frac {3 d x}{2}\right )+36 B \sin \left (2 c+\frac {3 d x}{2}\right )-63 C \sin \left (2 c+\frac {3 d x}{2}\right )+12 B \sin \left (2 c+\frac {5 d x}{2}\right )-15 C \sin \left (2 c+\frac {5 d x}{2}\right )+12 B \sin \left (3 c+\frac {5 d x}{2}\right )-15 C \sin \left (3 c+\frac {5 d x}{2}\right )+3 C \sin \left (3 c+\frac {7 d x}{2}\right )+3 C \sin \left (4 c+\frac {7 d x}{2}\right )\right )}{48 a^2 d (1+\cos (c+d x))^2} \]
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Time = 1.56 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.60
method | result | size |
parallelrisch | \(\frac {7 \left (\frac {3 \left (B -C \right ) \cos \left (2 d x +2 c \right )}{28}+\frac {3 C \cos \left (3 d x +3 c \right )}{112}+\left (B -\frac {163 C}{112}\right ) \cos \left (d x +c \right )+\frac {23 B}{28}-\frac {5 C}{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 \left (B -\frac {7 C}{4}\right ) x d}{3 a^{2} d}\) | \(88\) |
derivativedivides | \(\frac {-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{3}+\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{3}+5 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {4 \left (\left (\frac {5 C}{2}-B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3 C}{2}-B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-2 \left (4 B -7 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}}\) | \(135\) |
default | \(\frac {-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{3}+\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{3}+5 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {4 \left (\left (\frac {5 C}{2}-B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3 C}{2}-B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-2 \left (4 B -7 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}}\) | \(135\) |
risch | \(-\frac {2 B x}{a^{2}}+\frac {7 C x}{2 a^{2}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} C}{8 a^{2} d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} B}{2 a^{2} d}+\frac {i C \,{\mathrm e}^{i \left (d x +c \right )}}{a^{2} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} B}{2 a^{2} d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )} C}{a^{2} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} C}{8 a^{2} d}+\frac {2 i \left (9 B \,{\mathrm e}^{2 i \left (d x +c \right )}-12 C \,{\mathrm e}^{2 i \left (d x +c \right )}+15 B \,{\mathrm e}^{i \left (d x +c \right )}-21 C \,{\mathrm e}^{i \left (d x +c \right )}+8 B -11 C \right )}{3 d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3}}\) | \(207\) |
norman | \(\frac {\frac {\left (11 B -18 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {\left (4 B -7 C \right ) x}{2 a}-\frac {\left (B -C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 a d}-\frac {2 \left (4 B -7 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {3 \left (4 B -7 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {2 \left (4 B -7 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {\left (4 B -7 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {\left (9 B -13 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a d}+\frac {\left (11 B -17 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 a d}+\frac {\left (61 B -100 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}+\frac {\left (95 B -149 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 a d}}{a \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\) | \(281\) |
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Time = 0.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.94 \[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx=-\frac {3 \, {\left (4 \, B - 7 \, C\right )} d x \cos \left (d x + c\right )^{2} + 6 \, {\left (4 \, B - 7 \, C\right )} d x \cos \left (d x + c\right ) + 3 \, {\left (4 \, B - 7 \, C\right )} d x - {\left (3 \, C \cos \left (d x + c\right )^{3} + 6 \, {\left (B - C\right )} \cos \left (d x + c\right )^{2} + {\left (28 \, B - 43 \, C\right )} \cos \left (d x + c\right ) + 20 \, B - 32 \, C\right )} \sin \left (d x + c\right )}{6 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 848 vs. \(2 (136) = 272\).
Time = 2.32 (sec) , antiderivative size = 848, normalized size of antiderivative = 5.77 \[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (137) = 274\).
Time = 0.30 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.93 \[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx=-\frac {C {\left (\frac {6 \, {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2} + \frac {2 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {42 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )} - B {\left (\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {24 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac {12 \, \sin \left (d x + c\right )}{{\left (a^{2} + \frac {a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{6 \, d} \]
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Time = 0.38 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.12 \[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx=-\frac {\frac {3 \, {\left (d x + c\right )} {\left (4 \, B - 7 \, C\right )}}{a^{2}} - \frac {6 \, {\left (2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{2}} + \frac {B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \]
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Time = 1.20 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.03 \[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,\left (B-C\right )}{2\,a^2}+\frac {2\,B-4\,C}{2\,a^2}\right )}{d}-\frac {x\,\left (4\,B-7\,C\right )}{2\,a^2}+\frac {\left (2\,B-5\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,B-3\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (B-C\right )}{6\,a^2\,d} \]
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