Integrand size = 44, antiderivative size = 622 \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\frac {2 (a-b) \sqrt {a+b} \left (3705 a^4 b B+255 a^2 b^3 B+40 b^5 B+1617 a^5 C+3069 a^3 b^2 C-110 a b^4 C\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{3465 a^4 d}+\frac {2 (a-b) \sqrt {a+b} \left (40 b^4 B+3 a^4 (225 B-539 C)-6 a^3 b (505 B-209 C)+15 a^2 b^2 (19 B-121 C)+10 a b^3 (3 B-11 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{3465 a^3 d}+\frac {2 a (14 b B+11 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2 B+113 b^2 B+209 a b C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (1145 a^2 b B+15 b^3 B+539 a^3 C+825 a b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (675 a^4 B+1025 a^2 b^2 B-20 b^4 B+1793 a^3 b C+55 a b^3 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)} \]
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Time = 3.56 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.159, Rules used = {3108, 3068, 3126, 3134, 3077, 2895, 3073} \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\frac {2 \left (81 a^2 B+209 a b C+113 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (539 a^3 C+1145 a^2 b B+825 a b^2 C+15 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3465 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 (a-b) \sqrt {a+b} \left (3 a^4 (225 B-539 C)-6 a^3 b (505 B-209 C)+15 a^2 b^2 (19 B-121 C)+10 a b^3 (3 B-11 C)+40 b^4 B\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{3465 a^3 d}+\frac {2 \left (675 a^4 B+1793 a^3 b C+1025 a^2 b^2 B+55 a b^3 C-20 b^4 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3465 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (a-b) \sqrt {a+b} \left (1617 a^5 C+3705 a^4 b B+3069 a^3 b^2 C+255 a^2 b^3 B-110 a b^4 C+40 b^5 B\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{3465 a^4 d}+\frac {2 a (11 a C+14 b B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)} \]
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Rule 2895
Rule 3068
Rule 3073
Rule 3077
Rule 3108
Rule 3126
Rule 3134
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b \cos (c+d x))^{5/2} (B+C \cos (c+d x))}{\cos ^{\frac {13}{2}}(c+d x)} \, dx \\ & = \frac {2 a B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2}{11} \int \frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{2} a (14 b B+11 a C)+\frac {1}{2} \left (9 a^2 B+11 b^2 B+22 a b C\right ) \cos (c+d x)+\frac {1}{2} b (6 a B+11 b C) \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx \\ & = \frac {2 a (14 b B+11 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {4}{99} \int \frac {\frac {1}{4} a \left (81 a^2 B+113 b^2 B+209 a b C\right )+\frac {1}{4} \left (233 a^2 b B+99 b^3 B+77 a^3 C+297 a b^2 C\right ) \cos (c+d x)+\frac {3}{4} b \left (46 a b B+22 a^2 C+33 b^2 C\right ) \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx \\ & = \frac {2 a (14 b B+11 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2 B+113 b^2 B+209 a b C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {8 \int \frac {\frac {1}{8} a \left (1145 a^2 b B+15 b^3 B+539 a^3 C+825 a b^2 C\right )+\frac {1}{8} a \left (405 a^3 B+1531 a b^2 B+1507 a^2 b C+693 b^3 C\right ) \cos (c+d x)+\frac {1}{2} a b \left (81 a^2 B+113 b^2 B+209 a b C\right ) \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{693 a} \\ & = \frac {2 a (14 b B+11 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2 B+113 b^2 B+209 a b C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (1145 a^2 b B+15 b^3 B+539 a^3 C+825 a b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {16 \int \frac {\frac {3}{16} a \left (675 a^4 B+1025 a^2 b^2 B-20 b^4 B+1793 a^3 b C+55 a b^3 C\right )+\frac {1}{16} a^2 \left (5055 a^2 b B+2305 b^3 B+1617 a^3 C+6655 a b^2 C\right ) \cos (c+d x)+\frac {1}{8} a b \left (1145 a^2 b B+15 b^3 B+539 a^3 C+825 a b^2 C\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{3465 a^2} \\ & = \frac {2 a (14 b B+11 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2 B+113 b^2 B+209 a b C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (1145 a^2 b B+15 b^3 B+539 a^3 C+825 a b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (675 a^4 B+1025 a^2 b^2 B-20 b^4 B+1793 a^3 b C+55 a b^3 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {32 \int \frac {\frac {3}{32} a \left (3705 a^4 b B+255 a^2 b^3 B+40 b^5 B+1617 a^5 C+3069 a^3 b^2 C-110 a b^4 C\right )+\frac {3}{32} a^2 \left (675 a^4 B+3315 a^2 b^2 B+10 b^4 B+2871 a^3 b C+1705 a b^3 C\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{10395 a^3} \\ & = \frac {2 a (14 b B+11 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2 B+113 b^2 B+209 a b C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (1145 a^2 b B+15 b^3 B+539 a^3 C+825 a b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (675 a^4 B+1025 a^2 b^2 B-20 b^4 B+1793 a^3 b C+55 a b^3 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {\left ((a-b) \left (40 b^4 B+3 a^4 (225 B-539 C)-6 a^3 b (505 B-209 C)+15 a^2 b^2 (19 B-121 C)+10 a b^3 (3 B-11 C)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{3465 a^2}+\frac {\left (3705 a^4 b B+255 a^2 b^3 B+40 b^5 B+1617 a^5 C+3069 a^3 b^2 C-110 a b^4 C\right ) \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{3465 a^2} \\ & = \frac {2 (a-b) \sqrt {a+b} \left (3705 a^4 b B+255 a^2 b^3 B+40 b^5 B+1617 a^5 C+3069 a^3 b^2 C-110 a b^4 C\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{3465 a^4 d}+\frac {2 (a-b) \sqrt {a+b} \left (40 b^4 B+3 a^4 (225 B-539 C)-6 a^3 b (505 B-209 C)+15 a^2 b^2 (19 B-121 C)+10 a b^3 (3 B-11 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{3465 a^3 d}+\frac {2 a (14 b B+11 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2 B+113 b^2 B+209 a b C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (1145 a^2 b B+15 b^3 B+539 a^3 C+825 a b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (675 a^4 B+1025 a^2 b^2 B-20 b^4 B+1793 a^3 b C+55 a b^3 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)} \\ \end{align*}
Result contains complex when optimal does not.
Time = 7.95 (sec) , antiderivative size = 1640, normalized size of antiderivative = 2.64 \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\frac {-\frac {4 a \left (675 a^6 B-390 a^4 b^2 B-245 a^2 b^4 B-40 b^6 B+1254 a^5 b C-1364 a^3 b^3 C+110 a b^5 C\right ) \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}-4 a \left (-3705 a^5 b B-255 a^3 b^3 B-40 a b^5 B-1617 a^6 C-3069 a^4 b^2 C+110 a^2 b^4 C\right ) \left (\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}-\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{b \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}\right )+2 \left (-3705 a^4 b^2 B-255 a^2 b^4 B-40 b^6 B-1617 a^5 b C-3069 a^3 b^3 C+110 a b^5 C\right ) \left (\frac {i \cos \left (\frac {1}{2} (c+d x)\right ) \sqrt {a+b \cos (c+d x)} E\left (i \text {arcsinh}\left (\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\cos (c+d x)}}\right )|-\frac {2 a}{-a-b}\right ) \sec (c+d x)}{b \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)} \sqrt {\frac {(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac {2 a \left (\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}-\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{b \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}\right )}{b}+\frac {\sqrt {a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt {\cos (c+d x)}}\right )}{3465 a^3 d}+\frac {\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \left (\frac {2}{99} \sec ^5(c+d x) \left (23 a b B \sin (c+d x)+11 a^2 C \sin (c+d x)\right )+\frac {2}{693} \sec ^4(c+d x) \left (81 a^2 B \sin (c+d x)+113 b^2 B \sin (c+d x)+209 a b C \sin (c+d x)\right )+\frac {2 \sec ^3(c+d x) \left (1145 a^2 b B \sin (c+d x)+15 b^3 B \sin (c+d x)+539 a^3 C \sin (c+d x)+825 a b^2 C \sin (c+d x)\right )}{3465 a}+\frac {2 \sec ^2(c+d x) \left (675 a^4 B \sin (c+d x)+1025 a^2 b^2 B \sin (c+d x)-20 b^4 B \sin (c+d x)+1793 a^3 b C \sin (c+d x)+55 a b^3 C \sin (c+d x)\right )}{3465 a^2}+\frac {2 \sec (c+d x) \left (3705 a^4 b B \sin (c+d x)+255 a^2 b^3 B \sin (c+d x)+40 b^5 B \sin (c+d x)+1617 a^5 C \sin (c+d x)+3069 a^3 b^2 C \sin (c+d x)-110 a b^4 C \sin (c+d x)\right )}{3465 a^3}+\frac {2}{11} a^2 B \sec ^5(c+d x) \tan (c+d x)\right )}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(7344\) vs. \(2(572)=1144\).
Time = 59.36 (sec) , antiderivative size = 7345, normalized size of antiderivative = 11.81
method | result | size |
parts | \(\text {Expression too large to display}\) | \(7345\) |
default | \(\text {Expression too large to display}\) | \(7451\) |
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\[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {15}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {15}{2}}} \,d x } \]
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\[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {15}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^{15/2}} \,d x \]
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