Integrand size = 40, antiderivative size = 147 \[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\frac {(B-3 C) x}{a^3}-\frac {(7 B-27 C) \sin (c+d x)}{15 a^3 d}+\frac {(B-C) \cos ^3(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {(4 B-9 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {(B-3 C) \sin (c+d x)}{d \left (a^3+a^3 \cos (c+d x)\right )} \]
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Time = 0.60 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {3108, 3056, 3047, 3102, 12, 2814, 2727} \[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=-\frac {(7 B-27 C) \sin (c+d x)}{15 a^3 d}-\frac {(B-3 C) \sin (c+d x)}{d \left (a^3 \cos (c+d x)+a^3\right )}+\frac {x (B-3 C)}{a^3}+\frac {(B-C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac {(4 B-9 C) \sin (c+d x) \cos ^2(c+d x)}{15 a d (a \cos (c+d x)+a)^2} \]
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Rule 12
Rule 2727
Rule 2814
Rule 3047
Rule 3056
Rule 3102
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))^3} \, dx \\ & = \frac {(B-C) \cos ^3(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {\int \frac {\cos ^2(c+d x) (3 a (B-C)-a (B-6 C) \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx}{5 a^2} \\ & = \frac {(B-C) \cos ^3(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {(4 B-9 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac {\int \frac {\cos (c+d x) \left (2 a^2 (4 B-9 C)-a^2 (7 B-27 C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{15 a^4} \\ & = \frac {(B-C) \cos ^3(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {(4 B-9 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac {\int \frac {2 a^2 (4 B-9 C) \cos (c+d x)-a^2 (7 B-27 C) \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx}{15 a^4} \\ & = -\frac {(7 B-27 C) \sin (c+d x)}{15 a^3 d}+\frac {(B-C) \cos ^3(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {(4 B-9 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac {\int \frac {15 a^3 (B-3 C) \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{15 a^5} \\ & = -\frac {(7 B-27 C) \sin (c+d x)}{15 a^3 d}+\frac {(B-C) \cos ^3(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {(4 B-9 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac {(B-3 C) \int \frac {\cos (c+d x)}{a+a \cos (c+d x)} \, dx}{a^2} \\ & = \frac {(B-3 C) x}{a^3}-\frac {(7 B-27 C) \sin (c+d x)}{15 a^3 d}+\frac {(B-C) \cos ^3(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {(4 B-9 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {(B-3 C) \int \frac {1}{a+a \cos (c+d x)} \, dx}{a^2} \\ & = \frac {(B-3 C) x}{a^3}-\frac {(7 B-27 C) \sin (c+d x)}{15 a^3 d}+\frac {(B-C) \cos ^3(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {(4 B-9 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {(B-3 C) \sin (c+d x)}{d \left (a^3+a^3 \cos (c+d x)\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(361\) vs. \(2(147)=294\).
Time = 2.12 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.46 \[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (300 (B-3 C) d x \cos \left (\frac {d x}{2}\right )+300 (B-3 C) d x \cos \left (c+\frac {d x}{2}\right )+150 B d x \cos \left (c+\frac {3 d x}{2}\right )-450 C d x \cos \left (c+\frac {3 d x}{2}\right )+150 B d x \cos \left (2 c+\frac {3 d x}{2}\right )-450 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+30 B d x \cos \left (2 c+\frac {5 d x}{2}\right )-90 C d x \cos \left (2 c+\frac {5 d x}{2}\right )+30 B d x \cos \left (3 c+\frac {5 d x}{2}\right )-90 C d x \cos \left (3 c+\frac {5 d x}{2}\right )-740 B \sin \left (\frac {d x}{2}\right )+1755 C \sin \left (\frac {d x}{2}\right )+540 B \sin \left (c+\frac {d x}{2}\right )-1125 C \sin \left (c+\frac {d x}{2}\right )-460 B \sin \left (c+\frac {3 d x}{2}\right )+1215 C \sin \left (c+\frac {3 d x}{2}\right )+180 B \sin \left (2 c+\frac {3 d x}{2}\right )-225 C \sin \left (2 c+\frac {3 d x}{2}\right )-128 B \sin \left (2 c+\frac {5 d x}{2}\right )+363 C \sin \left (2 c+\frac {5 d x}{2}\right )+75 C \sin \left (3 c+\frac {5 d x}{2}\right )+15 C \sin \left (3 c+\frac {7 d x}{2}\right )+15 C \sin \left (4 c+\frac {7 d x}{2}\right )\right )}{120 a^3 d (1+\cos (c+d x))^3} \]
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Time = 1.71 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.61
method | result | size |
parallelrisch | \(\frac {-204 \left (\left (\frac {16 B}{51}-\frac {39 C}{34}\right ) \cos \left (2 d x +2 c \right )-\frac {5 C \cos \left (3 d x +3 c \right )}{68}+\left (B -\frac {243 C}{68}\right ) \cos \left (d x +c \right )+\frac {38 B}{51}-\frac {87 C}{34}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sec ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+240 d x \left (B -3 C \right )}{240 a^{3} d}\) | \(89\) |
derivativedivides | \(\frac {-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{5}+\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{5}+\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{3}-2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C -7 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {8 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+8 \left (B -3 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(134\) |
default | \(\frac {-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{5}+\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{5}+\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{3}-2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C -7 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {8 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+8 \left (B -3 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(134\) |
risch | \(\frac {B x}{a^{3}}-\frac {3 C x}{a^{3}}-\frac {i C \,{\mathrm e}^{i \left (d x +c \right )}}{2 a^{3} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} C}{2 a^{3} d}-\frac {2 i \left (45 B \,{\mathrm e}^{4 i \left (d x +c \right )}-90 C \,{\mathrm e}^{4 i \left (d x +c \right )}+135 B \,{\mathrm e}^{3 i \left (d x +c \right )}-300 C \,{\mathrm e}^{3 i \left (d x +c \right )}+185 B \,{\mathrm e}^{2 i \left (d x +c \right )}-420 C \,{\mathrm e}^{2 i \left (d x +c \right )}+115 B \,{\mathrm e}^{i \left (d x +c \right )}-270 C \,{\mathrm e}^{i \left (d x +c \right )}+32 B -72 C \right )}{15 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}\) | \(178\) |
norman | \(\frac {\frac {\left (B -3 C \right ) x}{a}+\frac {\left (B -3 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {4 \left (B -3 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {6 \left (B -3 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {4 \left (B -3 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {\left (B -C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 a d}+\frac {\left (4 B -9 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{30 a d}-\frac {\left (7 B -25 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}-\frac {5 \left (8 B -27 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 a d}-\frac {\left (26 B -81 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a d}-\frac {\left (43 B -153 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 a d}-\frac {\left (553 B -1773 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} a^{2}}\) | \(296\) |
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Time = 0.28 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.12 \[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\frac {15 \, {\left (B - 3 \, C\right )} d x \cos \left (d x + c\right )^{3} + 45 \, {\left (B - 3 \, C\right )} d x \cos \left (d x + c\right )^{2} + 45 \, {\left (B - 3 \, C\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (B - 3 \, C\right )} d x + {\left (15 \, C \cos \left (d x + c\right )^{3} - {\left (32 \, B - 117 \, C\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (17 \, B - 57 \, C\right )} \cos \left (d x + c\right ) - 22 \, B + 72 \, C\right )} \sin \left (d x + c\right )}{15 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (134) = 268\).
Time = 3.13 (sec) , antiderivative size = 502, normalized size of antiderivative = 3.41 \[ \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))^3} \, dx=\begin {cases} \frac {60 B d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{60 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 60 a^{3} d} + \frac {60 B d x}{60 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 60 a^{3} d} - \frac {3 B \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{60 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 60 a^{3} d} + \frac {17 B \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{60 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 60 a^{3} d} - \frac {85 B \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{60 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 60 a^{3} d} - \frac {105 B \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{60 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 60 a^{3} d} - \frac {180 C d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{60 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 60 a^{3} d} - \frac {180 C d x}{60 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 60 a^{3} d} + \frac {3 C \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{60 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 60 a^{3} d} - \frac {27 C \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{60 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 60 a^{3} d} + \frac {225 C \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{60 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 60 a^{3} d} + \frac {375 C \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{60 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 60 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \left (B \cos {\left (c \right )} + C \cos ^{2}{\left (c \right )}\right ) \cos ^{2}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{3}} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.57 \[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\frac {3 \, C {\left (\frac {40 \, \sin \left (d x + c\right )}{{\left (a^{3} + \frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} - B {\left (\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{60 \, d} \]
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Time = 0.33 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.05 \[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\frac {\frac {60 \, {\left (d x + c\right )} {\left (B - 3 \, C\right )}}{a^{3}} + \frac {120 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{3}} - \frac {3 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 20 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 255 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \]
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Time = 1.25 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.03 \[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {B-C}{6\,a^3}+\frac {2\,B-4\,C}{12\,a^3}\right )}{d}+\frac {x\,\left (B-3\,C\right )}{a^3}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,\left (B-C\right )}{4\,a^3}-\frac {3\,C}{2\,a^3}+\frac {2\,B-4\,C}{2\,a^3}\right )}{d}+\frac {2\,C\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (B-C\right )}{20\,a^3\,d} \]
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