Integrand size = 43, antiderivative size = 413 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=-\frac {2 \left (5 a^4 B-30 a^2 b^2 B-3 b^4 B+20 a^3 b (A-C)-4 a b^3 (5 A+3 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (84 a^3 b B+28 a b^3 B+42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}-\frac {2 b^2 \left (350 a A b+105 a^2 B-21 b^2 B-54 a b C\right ) \sin (c+d x)}{105 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 b \left (42 a^3 B-28 a b^2 B+3 a^2 b (49 A-13 C)-b^3 (7 A+5 C)\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}-\frac {2 b (21 A b+7 a B-b C) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt {\sec (c+d x)}}+\frac {2 (8 A b+3 a B) (a+b \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{3 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d} \]
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Time = 1.52 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {4306, 3126, 3128, 3112, 3102, 2827, 2720, 2719} \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=-\frac {2 b^2 \sin (c+d x) \left (105 a^2 B+350 a A b-54 a b C-21 b^2 B\right )}{105 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 b \sin (c+d x) \left (42 a^3 B+3 a^2 b (49 A-13 C)-28 a b^2 B-b^3 (7 A+5 C)\right )}{21 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (7 a^4 (A+3 C)+84 a^3 b B+42 a^2 b^2 (3 A+C)+28 a b^3 B+b^4 (7 A+5 C)\right )}{21 d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (5 a^4 B+20 a^3 b (A-C)-30 a^2 b^2 B-4 a b^3 (5 A+3 C)-3 b^4 B\right )}{5 d}-\frac {2 b \sin (c+d x) (7 a B+21 A b-b C) (a+b \cos (c+d x))^2}{7 d \sqrt {\sec (c+d x)}}+\frac {2 (3 a B+8 A b) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \cos (c+d x))^3}{3 d}+\frac {2 A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{3 d} \]
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Rule 2719
Rule 2720
Rule 2827
Rule 3102
Rule 3112
Rule 3126
Rule 3128
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 {5}{2}}(c+d x)} \, dx \\ & = \frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac {1}{3} \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+3 a B)+\frac {1}{2} (3 b B+a (A+3 C)) \cos (c+d x)-\frac {1}{2} b (7 A-3 C) \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \frac {2 (8 A b+3 a B) (a+b \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{3 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac {1}{3} \left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^2 \left (\frac {1}{4} \left (48 A b^2+21 a b B+a^2 (A+3 C)\right )-\frac {1}{4} \left (14 a A b+3 a^2 B-3 b^2 B-6 a b C\right ) \cos (c+d x)-\frac {3}{4} b (21 A b+7 a B-b C) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx \\ & = -\frac {2 b (21 A b+7 a B-b C) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt {\sec (c+d x)}}+\frac {2 (8 A b+3 a B) (a+b \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{3 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac {1}{21} \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} a \left (126 a b B+3 b^2 (91 A+C)+7 a^2 (A+3 C)\right )-\frac {1}{8} \left (21 a^3 B-63 a b^2 B-3 b^3 (7 A+5 C)+a^2 (91 A b-63 b C)\right ) \cos (c+d x)-\frac {1}{8} b \left (350 a A b+105 a^2 B-21 b^2 B-54 a b C\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx \\ & = -\frac {2 b^2 \left (350 a A b+105 a^2 B-21 b^2 B-54 a b C\right ) \sin (c+d x)}{105 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 b (21 A b+7 a B-b C) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt {\sec (c+d x)}}+\frac {2 (8 A b+3 a B) (a+b \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{3 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac {1}{105} \left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {5}{16} a^2 \left (126 a b B+3 b^2 (91 A+C)+7 a^2 (A+3 C)\right )-\frac {21}{16} \left (5 a^4 B-30 a^2 b^2 B-3 b^4 B+20 a^3 b (A-C)-4 a b^3 (5 A+3 C)\right ) \cos (c+d x)-\frac {15}{16} b \left (42 a^3 B-28 a b^2 B-b^3 (7 A+5 C)+a^2 (147 A b-39 b C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx \\ & = -\frac {2 b^2 \left (350 a A b+105 a^2 B-21 b^2 B-54 a b C\right ) \sin (c+d x)}{105 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 b \left (42 a^3 B-28 a b^2 B+3 a^2 b (49 A-13 C)-b^3 (7 A+5 C)\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}-\frac {2 b (21 A b+7 a B-b C) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt {\sec (c+d x)}}+\frac {2 (8 A b+3 a B) (a+b \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{3 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac {1}{315} \left (32 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {15}{32} \left (84 a^3 b B+28 a b^3 B+42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 C)\right )-\frac {63}{32} \left (5 a^4 B-30 a^2 b^2 B-3 b^4 B+20 a^3 b (A-C)-4 a b^3 (5 A+3 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx \\ & = -\frac {2 b^2 \left (350 a A b+105 a^2 B-21 b^2 B-54 a b C\right ) \sin (c+d x)}{105 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 b \left (42 a^3 B-28 a b^2 B+3 a^2 b (49 A-13 C)-b^3 (7 A+5 C)\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}-\frac {2 b (21 A b+7 a B-b C) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt {\sec (c+d x)}}+\frac {2 (8 A b+3 a B) (a+b \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{3 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}-\frac {1}{5} \left (\left (5 a^4 B-30 a^2 b^2 B-3 b^4 B+20 a^3 b (A-C)-4 a b^3 (5 A+3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (\left (84 a^3 b B+28 a b^3 B+42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 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 (5 a^4 B-30 a^2 b^2 B-3 b^4 B+20 a^3 b (A-C)-4 a b^3 (5 A+3 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (84 a^3 b B+28 a b^3 B+42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}-\frac {2 b^2 \left (350 a A b+105 a^2 B-21 b^2 B-54 a b C\right ) \sin (c+d x)}{105 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 b \left (42 a^3 B-28 a b^2 B+3 a^2 b (49 A-13 C)-b^3 (7 A+5 C)\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}-\frac {2 b (21 A b+7 a B-b C) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt {\sec (c+d x)}}+\frac {2 (8 A b+3 a B) (a+b \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{3 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d} \\ \end{align*}
Time = 6.02 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.77 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=\frac {\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-168 \left (5 a^4 B-30 a^2 b^2 B-3 b^4 B+20 a^3 b (A-C)-4 a b^3 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+40 \left (84 a^3 b B+28 a b^3 B+42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {\left (280 a^4 A+140 A b^4+560 a b^3 B+840 a^2 b^2 C+145 b^4 C+42 \left (80 a^3 A b+20 a^4 B+3 b^4 B+12 a b^3 C\right ) \cos (c+d x)+20 b^2 \left (7 A b^2+28 a b B+42 a^2 C+8 b^2 C\right ) \cos (2 (c+d x))+42 b^4 B \cos (3 (c+d x))+168 a b^3 C \cos (3 (c+d x))+15 b^4 C \cos (4 (c+d x))\right ) \sin (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}\right )}{420 d} \]
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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 = 447, normalized size of antiderivative = 1.08 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=-\frac {5 \, \sqrt {2} {\left (7 i \, {\left (A + 3 \, C\right )} a^{4} + 84 i \, B a^{3} b + 42 i \, {\left (3 \, A + C\right )} a^{2} b^{2} + 28 i \, B a b^{3} + i \, {\left (7 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-7 i \, {\left (A + 3 \, C\right )} a^{4} - 84 i \, B a^{3} b - 42 i \, {\left (3 \, A + C\right )} a^{2} b^{2} - 28 i \, B a b^{3} - i \, {\left (7 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (5 i \, B a^{4} + 20 i \, {\left (A - C\right )} a^{3} b - 30 i \, B a^{2} b^{2} - 4 i \, {\left (5 \, A + 3 \, C\right )} a b^{3} - 3 i \, B b^{4}\right )} \cos \left (d x + c\right ) {\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 ) + 21 \, \sqrt {2} {\left (-5 i \, B a^{4} - 20 i \, {\left (A - C\right )} a^{3} b + 30 i \, B a^{2} b^{2} + 4 i \, {\left (5 \, A + 3 \, C\right )} a b^{3} + 3 i \, B b^{4}\right )} \cos \left (d x + c\right ) {\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 (15 \, C b^{4} \cos \left (d x + c\right )^{4} + 35 \, A a^{4} + 21 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (42 \, C a^{2} b^{2} + 28 \, B a b^{3} + {\left (7 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 105 \, {\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 )}}}{105 \, d \cos \left (d x + c\right )} \]
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Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=\text {Timed out} \]
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\[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {5}{2}} \,d x } \]
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\[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=\int {\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^4\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
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