Optimal. Leaf size=417 \[ \frac {4 a b \left (a^2 (673 A+891 C)+96 A b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{3465 d}+\frac {8 a b \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (3 a^2 (9 A+11 C)+16 A b^2\right ) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^2}{231 d}-\frac {8 a b \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)+64 A b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{693 d}+\frac {2 \left (5 a^4 (9 A+11 C)+66 a^2 b^2 (5 A+7 C)+77 b^4 (A+3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {2 A \sin (c+d x) \sec ^{\frac {11}{2}}(c+d x) (a+b \cos (c+d x))^4}{11 d}+\frac {16 A b \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))^3}{99 d} \]
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Rubi [A] time = 1.38, antiderivative size = 417, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {4221, 3048, 3047, 3031, 3021, 2748, 2636, 2639, 2641} \[ \frac {4 a b \left (a^2 (673 A+891 C)+96 A b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{3465 d}+\frac {2 \left (9 a^2 b^2 (101 A+143 C)+15 a^4 (9 A+11 C)+64 A b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{693 d}+\frac {8 a b \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (3 a^2 (9 A+11 C)+16 A b^2\right ) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^2}{231 d}+\frac {2 \left (66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+77 b^4 (A+3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}-\frac {8 a b \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 A \sin (c+d x) \sec ^{\frac {11}{2}}(c+d x) (a+b \cos (c+d x))^4}{11 d}+\frac {16 A b \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))^3}{99 d} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 2641
Rule 2748
Rule 3021
Rule 3031
Rule 3047
Rule 3048
Rule 4221
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^4 \left (A+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 (4 A b+\frac {1}{2} a (9 A+11 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 {16 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+3 a^2 (9 A+11 C)\right )+\frac {1}{2} a b (73 A+99 C) \cos (c+d x)+\frac {1}{4} b^2 (17 A+99 C) \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\frac {2 \left (16 A b^2+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 {16 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}{4} b \left (96 A b^2+a^2 (673 A+891 C)\right )+\frac {1}{8} a \left (45 a^2 (9 A+11 C)+b^2 (1381 A+2079 C)\right ) \cos (c+d x)+\frac {1}{8} b \left (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 {4 a b \left (96 A b^2+a^2 (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+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 {16 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+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right )-\frac {231}{4} a b \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \cos (c+d x)-\frac {5}{16} b^2 \left (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+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 {4 a b \left (96 A b^2+a^2 (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+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 {16 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}{8} a b \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right )-\frac {45}{32} \left (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+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 {4 a b \left (96 A b^2+a^2 (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+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 {16 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 (4 a b \left (3 b^2 (3 A+5 C)+a^2 (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 (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 (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)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {8 a b \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 \left (64 A b^4+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 {4 a b \left (96 A b^2+a^2 (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+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 {16 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 (4 a b \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {8 a b \left (3 b^2 (3 A+5 C)+a^2 (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 (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)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {8 a b \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 \left (64 A b^4+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 {4 a b \left (96 A b^2+a^2 (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+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 {16 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*}
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Mathematica [A] time = 6.88, size = 425, normalized size = 1.02 \[ \frac {\sqrt {\sec (c+d x)} \left (\frac {2}{11} a^4 A \tan (c+d x) \sec ^4(c+d x)+\frac {8}{45} \sec ^2(c+d x) \left (7 a^3 A b \sin (c+d x)+9 a^3 b C \sin (c+d x)+9 a A b^3 \sin (c+d x)\right )+\frac {8}{9} a^3 A b \tan (c+d x) \sec ^3(c+d x)+\frac {8}{15} a b \left (7 a^2 A+9 a^2 C+9 A b^2+15 b^2 C\right ) \sin (c+d x)+\frac {2}{77} \sec ^3(c+d x) \left (9 a^4 A \sin (c+d x)+11 a^4 C \sin (c+d x)+66 a^2 A b^2 \sin (c+d x)\right )+\frac {2}{231} \sec (c+d x) \left (45 a^4 A \sin (c+d x)+55 a^4 C \sin (c+d x)+330 a^2 A b^2 \sin (c+d x)+462 a^2 b^2 C \sin (c+d x)+77 A b^4 \sin (c+d x)\right )\right )}{d}+\frac {\frac {2 \left (-2156 a^3 A b-2772 a^3 b C-2772 a A b^3-4620 a b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}+2 \left (225 a^4 A+275 a^4 C+1650 a^2 A b^2+2310 a^2 b^2 C+385 A b^4+1155 b^4 C\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{1155 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{4} \cos \left (d x + c\right )^{6} + 4 \, C a b^{3} \cos \left (d x + c\right )^{5} + 4 \, A a^{3} b \cos \left (d x + c\right ) + A a^{4} + {\left (6 \, C a^{2} b^{2} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (C a^{3} b + A a b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{4} + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sec \left (d x + c\right )^{\frac {13}{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {13}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 16.50, size = 1521, normalized size = 3.65 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {13}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{13/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^4 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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