Integrand size = 43, antiderivative size = 515 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=-\frac {2 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d} \]
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Time = 1.62 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {4306, 3126, 3110, 3100, 2827, 2716, 2719, 2720} \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{231 d}+\frac {2 a \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{3465 d}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{693 d}+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{15 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right )}{231 d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{15 d}+\frac {2 (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))^3}{99 d}+\frac {2 A \sin (c+d x) \sec ^{\frac {11}{2}}(c+d x) (a+b \cos (c+d x))^4}{11 d} \]
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Rule 2716
Rule 2719
Rule 2720
Rule 2827
Rule 3100
Rule 3110
Rule 3126
Rule 4306
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx \\ & = \frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {1}{11} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^3 \left (\frac {1}{2} (8 A b+11 a B)+\frac {1}{2} (9 a A+11 b B+11 a C) \cos (c+d x)+\frac {1}{2} b (A+11 C) \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx \\ & = \frac {2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {1}{99} \left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^2 \left (\frac {3}{4} \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right )+\frac {1}{4} \left (146 a A b+77 a^2 B+99 b^2 B+198 a b C\right ) \cos (c+d x)+\frac {1}{4} b (17 A b+11 a B+99 b C) \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx \\ & = \frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {1}{693} \left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x)) \left (\frac {1}{8} \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right )+\frac {1}{8} \left (1441 a^2 b B+693 b^3 B+45 a^3 (9 A+11 C)+a b^2 (1381 A+2079 C)\right ) \cos (c+d x)+\frac {1}{8} b \left (242 a b B+9 a^2 (9 A+11 C)+b^2 (167 A+693 C)\right ) \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}-\frac {\left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {15}{16} \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right )-\frac {231}{16} \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \cos (c+d x)-\frac {5}{16} b^2 \left (242 a b B+9 a^2 (9 A+11 C)+b^2 (167 A+693 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{3465} \\ & = \frac {2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}-\frac {\left (32 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {693}{32} \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right )-\frac {45}{32} \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)} \, dx}{10395} \\ & = \frac {2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {1}{15} \left (\left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)} \, dx+\frac {1}{231} \left (\left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}-\frac {1}{15} \left (\left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = -\frac {2 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d} \\ \end{align*}
Time = 9.54 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.09 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {\frac {2 \left (-2156 a^3 A b-2772 a A b^3-539 a^4 B-4158 a^2 b^2 B-1155 b^4 B-2772 a^3 b C-4620 a b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}+2 \left (225 a^4 A+1650 a^2 A b^2+385 A b^4+1100 a^3 b B+1540 a b^3 B+275 a^4 C+2310 a^2 b^2 C+1155 b^4 C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{1155 d}+\frac {\sqrt {\sec (c+d x)} \left (\frac {2}{15} \left (28 a^3 A b+36 a A b^3+7 a^4 B+54 a^2 b^2 B+15 b^4 B+36 a^3 b C+60 a b^3 C\right ) \sin (c+d x)+\frac {2}{9} \sec ^4(c+d x) \left (4 a^3 A b \sin (c+d x)+a^4 B \sin (c+d x)\right )+\frac {2}{77} \sec ^3(c+d x) \left (9 a^4 A \sin (c+d x)+66 a^2 A b^2 \sin (c+d x)+44 a^3 b B \sin (c+d x)+11 a^4 C \sin (c+d x)\right )+\frac {2}{45} \sec ^2(c+d x) \left (28 a^3 A b \sin (c+d x)+36 a A b^3 \sin (c+d x)+7 a^4 B \sin (c+d x)+54 a^2 b^2 B \sin (c+d x)+36 a^3 b C \sin (c+d x)\right )+\frac {2}{231} \sec (c+d x) \left (45 a^4 A \sin (c+d x)+330 a^2 A b^2 \sin (c+d x)+77 A b^4 \sin (c+d x)+220 a^3 b B \sin (c+d x)+308 a b^3 B \sin (c+d x)+55 a^4 C \sin (c+d x)+462 a^2 b^2 C \sin (c+d x)\right )+\frac {2}{11} a^4 A \sec ^4(c+d x) \tan (c+d x)\right )}{d} \]
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Timed out.
hanged
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 588, normalized size of antiderivative = 1.14 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=-\frac {15 \, \sqrt {2} {\left (5 i \, {\left (9 \, A + 11 \, C\right )} a^{4} + 220 i \, B a^{3} b + 66 i \, {\left (5 \, A + 7 \, C\right )} a^{2} b^{2} + 308 i \, B a b^{3} + 77 i \, {\left (A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-5 i \, {\left (9 \, A + 11 \, C\right )} a^{4} - 220 i \, B a^{3} b - 66 i \, {\left (5 \, A + 7 \, C\right )} a^{2} b^{2} - 308 i \, B a b^{3} - 77 i \, {\left (A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 \, \sqrt {2} {\left (7 i \, B a^{4} + 4 i \, {\left (7 \, A + 9 \, C\right )} a^{3} b + 54 i \, B a^{2} b^{2} + 12 i \, {\left (3 \, A + 5 \, C\right )} a b^{3} + 15 i \, B b^{4}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 \, \sqrt {2} {\left (-7 i \, B a^{4} - 4 i \, {\left (7 \, A + 9 \, C\right )} a^{3} b - 54 i \, B a^{2} b^{2} - 12 i \, {\left (3 \, A + 5 \, C\right )} a b^{3} - 15 i \, B b^{4}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (231 \, {\left (7 \, B a^{4} + 4 \, {\left (7 \, A + 9 \, C\right )} a^{3} b + 54 \, B a^{2} b^{2} + 12 \, {\left (3 \, A + 5 \, C\right )} a b^{3} + 15 \, B b^{4}\right )} \cos \left (d x + c\right )^{5} + 315 \, A a^{4} + 15 \, {\left (5 \, {\left (9 \, A + 11 \, C\right )} a^{4} + 220 \, B a^{3} b + 66 \, {\left (5 \, A + 7 \, C\right )} a^{2} b^{2} + 308 \, B a b^{3} + 77 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} + 77 \, {\left (7 \, B a^{4} + 4 \, {\left (7 \, A + 9 \, C\right )} a^{3} b + 54 \, B a^{2} b^{2} + 36 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 45 \, {\left ({\left (9 \, A + 11 \, C\right )} a^{4} + 44 \, B a^{3} b + 66 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 385 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{3465 \, d \cos \left (d x + c\right )^{5}} \]
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Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\text {Timed out} \]
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\[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {13}{2}} \,d x } \]
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\[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {13}{2}} \,d x } \]
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Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int {\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{13/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^4\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
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