Integrand size = 43, antiderivative size = 515 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (308 a^3 b B+220 a b^3 B+77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 \left (682 a^3 b B+660 a b^3 B+64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 b \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d}+\frac {2 (11 b B+8 a C) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d} \]
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Time = 1.47 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {4181, 4161, 4132, 3853, 3856, 2719, 4131, 2720} \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \sec (c+d x))^2}{231 d}+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{3465 d}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{693 d}+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right )}{15 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (77 a^4 (3 A+C)+308 a^3 b B+66 a^2 b^2 (7 A+5 C)+220 a b^3 B+5 b^4 (11 A+9 C)\right )}{231 d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right )}{15 d}+\frac {2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{99 d}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d} \]
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Rule 2719
Rule 2720
Rule 3853
Rule 3856
Rule 4131
Rule 4132
Rule 4161
Rule 4181
Rubi steps \begin{align*} \text {integral}& = \frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d}+\frac {2}{11} \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \left (\frac {1}{2} a (11 A+C)+\frac {1}{2} (11 A b+11 a B+9 b C) \sec (c+d x)+\frac {1}{2} (11 b B+8 a C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {2 (11 b B+8 a C) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d}+\frac {4}{99} \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \left (\frac {1}{4} a (99 a A+11 b B+17 a C)+\frac {1}{4} \left (198 a A b+99 a^2 B+77 b^2 B+146 a b C\right ) \sec (c+d x)+\frac {3}{4} \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d}+\frac {2 (11 b B+8 a C) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d}+\frac {8}{693} \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \left (\frac {1}{8} a \left (242 a b B+9 b^2 (11 A+9 C)+a^2 (693 A+167 C)\right )+\frac {1}{8} \left (693 a^3 B+1441 a b^2 B+45 b^3 (11 A+9 C)+a^2 b (2079 A+1381 C)\right ) \sec (c+d x)+\frac {1}{8} \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {2 b \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d}+\frac {2 (11 b B+8 a C) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d}+\frac {16 \int \sqrt {\sec (c+d x)} \left (\frac {5}{16} a^2 \left (242 a b B+9 b^2 (11 A+9 C)+a^2 (693 A+167 C)\right )+\frac {231}{16} \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \sec (c+d x)+\frac {15}{16} \left (682 a^3 b B+660 a b^3 B+64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sec ^2(c+d x)\right ) \, dx}{3465} \\ & = \frac {2 b \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d}+\frac {2 (11 b B+8 a C) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d}+\frac {16 \int \sqrt {\sec (c+d x)} \left (\frac {5}{16} a^2 \left (242 a b B+9 b^2 (11 A+9 C)+a^2 (693 A+167 C)\right )+\frac {15}{16} \left (682 a^3 b B+660 a b^3 B+64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sec ^2(c+d x)\right ) \, dx}{3465}+\frac {1}{15} \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx \\ & = \frac {2 \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 \left (682 a^3 b B+660 a b^3 B+64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 b \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d}+\frac {2 (11 b B+8 a C) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d}+\frac {1}{15} \left (-15 a^4 B-54 a^2 b^2 B-7 b^4 B-12 a^3 b (5 A+3 C)-4 a b^3 (9 A+7 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{231} \left (308 a^3 b B+220 a b^3 B+77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right ) \int \sqrt {\sec (c+d x)} \, dx \\ & = \frac {2 \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 \left (682 a^3 b B+660 a b^3 B+64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 b \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d}+\frac {2 (11 b B+8 a C) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d}+\frac {1}{15} \left (\left (-15 a^4 B-54 a^2 b^2 B-7 b^4 B-12 a^3 b (5 A+3 C)-4 a b^3 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{231} \left (\left (308 a^3 b B+220 a b^3 B+77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 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 (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 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 (308 a^3 b B+220 a b^3 B+77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 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 (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 \left (682 a^3 b B+660 a b^3 B+64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 b \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d}+\frac {2 (11 b B+8 a C) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d} \\ \end{align*}
Time = 13.20 (sec) , antiderivative size = 713, normalized size of antiderivative = 1.38 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \cos ^6(c+d x) \left (\frac {2 \left (-4620 a^3 A b-2772 a A b^3-1155 a^4 B-4158 a^2 b^2 B-539 b^4 B-2772 a^3 b C-2156 a b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}+2 \left (1155 a^4 A+2310 a^2 A b^2+275 A b^4+1540 a^3 b B+1100 a b^3 B+385 a^4 C+1650 a^2 b^2 C+225 b^4 C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}\right ) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{1155 d (b+a \cos (c+d x))^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {4}{15} \left (60 a^3 A b+36 a A b^3+15 a^4 B+54 a^2 b^2 B+7 b^4 B+36 a^3 b C+28 a b^3 C\right ) \sin (c+d x)+\frac {4}{9} \sec ^4(c+d x) \left (b^4 B \sin (c+d x)+4 a b^3 C \sin (c+d x)\right )+\frac {4}{45} \sec ^2(c+d x) \left (36 a A b^3 \sin (c+d x)+54 a^2 b^2 B \sin (c+d x)+7 b^4 B \sin (c+d x)+36 a^3 b C \sin (c+d x)+28 a b^3 C \sin (c+d x)\right )+\frac {4}{77} \sec ^3(c+d x) \left (11 A b^4 \sin (c+d x)+44 a b^3 B \sin (c+d x)+66 a^2 b^2 C \sin (c+d x)+9 b^4 C \sin (c+d x)\right )+\frac {4}{231} \sec (c+d x) \left (462 a^2 A b^2 \sin (c+d x)+55 A b^4 \sin (c+d x)+308 a^3 b B \sin (c+d x)+220 a b^3 B \sin (c+d x)+77 a^4 C \sin (c+d x)+330 a^2 b^2 C \sin (c+d x)+45 b^4 C \sin (c+d x)\right )+\frac {4}{11} b^4 C \sec ^4(c+d x) \tan (c+d x)\right )}{d (b+a \cos (c+d x))^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {11}{2}}(c+d x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1522\) vs. \(2(531)=1062\).
Time = 11.84 (sec) , antiderivative size = 1523, normalized size of antiderivative = 2.96
method | result | size |
default | \(\text {Expression too large to display}\) | \(1523\) |
parts | \(\text {Expression too large to display}\) | \(1905\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 588, normalized size of antiderivative = 1.14 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {15 \, \sqrt {2} {\left (77 i \, {\left (3 \, A + C\right )} a^{4} + 308 i \, B a^{3} b + 66 i \, {\left (7 \, A + 5 \, C\right )} a^{2} b^{2} + 220 i \, B a b^{3} + 5 i \, {\left (11 \, A + 9 \, 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 (-77 i \, {\left (3 \, A + C\right )} a^{4} - 308 i \, B a^{3} b - 66 i \, {\left (7 \, A + 5 \, C\right )} a^{2} b^{2} - 220 i \, B a b^{3} - 5 i \, {\left (11 \, A + 9 \, 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 (15 i \, B a^{4} + 12 i \, {\left (5 \, A + 3 \, C\right )} a^{3} b + 54 i \, B a^{2} b^{2} + 4 i \, {\left (9 \, A + 7 \, C\right )} a b^{3} + 7 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 (-15 i \, B a^{4} - 12 i \, {\left (5 \, A + 3 \, C\right )} a^{3} b - 54 i \, B a^{2} b^{2} - 4 i \, {\left (9 \, A + 7 \, C\right )} a b^{3} - 7 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 (15 \, B a^{4} + 12 \, {\left (5 \, A + 3 \, C\right )} a^{3} b + 54 \, B a^{2} b^{2} + 4 \, {\left (9 \, A + 7 \, C\right )} a b^{3} + 7 \, B b^{4}\right )} \cos \left (d x + c\right )^{5} + 315 \, C b^{4} + 15 \, {\left (77 \, C a^{4} + 308 \, B a^{3} b + 66 \, {\left (7 \, A + 5 \, C\right )} a^{2} b^{2} + 220 \, B a b^{3} + 5 \, {\left (11 \, A + 9 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 77 \, {\left (36 \, C a^{3} b + 54 \, B a^{2} b^{2} + 4 \, {\left (9 \, A + 7 \, C\right )} a b^{3} + 7 \, B b^{4}\right )} \cos \left (d x + c\right )^{3} + 45 \, {\left (66 \, C a^{2} b^{2} + 44 \, B a b^{3} + {\left (11 \, A + 9 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 385 \, {\left (4 \, C a b^{3} + B b^{4}\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. \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Timed out. \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{4} \sqrt {\sec \left (d x + c\right )} \,d x } \]
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Timed out. \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^4\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]
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