Optimal. Leaf size=622 \[ \frac {2 (a-b) \sqrt {a+b} \left (3705 a^4 A b+255 a^2 A b^3+40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B\right ) \cot (c+d x) E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{3465 a^4 d}+\frac {2 (a-b) \sqrt {a+b} \left (40 A b^4+3 a^4 (225 A-539 B)-6 a^3 b (505 A-209 B)+15 a^2 b^2 (19 A-121 B)+10 a b^3 (3 A-11 B)\right ) \cot (c+d x) F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{3465 a^3 d}+\frac {2 a (14 A b+11 a B) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (675 a^4 A+1025 a^2 A b^2-20 A b^4+1793 a^3 b B+55 a b^3 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a A (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)} \]
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Rubi [A]
time = 1.66, antiderivative size = 622, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3068, 3126,
3134, 3077, 2895, 3073} \begin {gather*} \frac {2 \left (81 a^2 A+209 a b B+113 A b^2\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 B+1145 a^2 A b+825 a b^2 B+15 A b^3\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 A-539 B)-6 a^3 b (505 A-209 B)+15 a^2 b^2 (19 A-121 B)+10 a b^3 (3 A-11 B)+40 A b^4\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}} F\left (\text {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 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\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 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\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 (\text {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 B+14 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2895
Rule 3068
Rule 3073
Rule 3077
Rule 3126
Rule 3134
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac {13}{2}}(c+d x)} \, dx &=\frac {2 a A (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 A b+11 a B)+\frac {1}{2} \left (9 a^2 A+11 A b^2+22 a b B\right ) \cos (c+d x)+\frac {1}{2} b (6 a A+11 b B) \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx\\ &=\frac {2 a (14 A b+11 a B) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 a A (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 A+113 A b^2+209 a b B\right )+\frac {1}{4} \left (233 a^2 A b+99 A b^3+77 a^3 B+297 a b^2 B\right ) \cos (c+d x)+\frac {3}{4} b \left (46 a A b+22 a^2 B+33 b^2 B\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 A b+11 a B) \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 A+113 A b^2+209 a b B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a A (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 A b+15 A b^3+539 a^3 B+825 a b^2 B\right )+\frac {1}{8} a \left (405 a^3 A+1531 a A b^2+1507 a^2 b B+693 b^3 B\right ) \cos (c+d x)+\frac {1}{2} a b \left (81 a^2 A+113 A b^2+209 a b B\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 A b+11 a B) \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 A+113 A b^2+209 a b B\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 A b+15 A b^3+539 a^3 B+825 a b^2 B\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 A (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 A+1025 a^2 A b^2-20 A b^4+1793 a^3 b B+55 a b^3 B\right )+\frac {1}{16} a^2 \left (5055 a^2 A b+2305 A b^3+1617 a^3 B+6655 a b^2 B\right ) \cos (c+d x)+\frac {1}{8} a b \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\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 A b+11 a B) \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 A+113 A b^2+209 a b B\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 A b+15 A b^3+539 a^3 B+825 a b^2 B\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 A+1025 a^2 A b^2-20 A b^4+1793 a^3 b B+55 a b^3 B\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 A (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 A b+255 a^2 A b^3+40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B\right )+\frac {3}{32} a^2 \left (675 a^4 A+3315 a^2 A b^2+10 A b^4+2871 a^3 b B+1705 a b^3 B\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 A b+11 a B) \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 A+113 A b^2+209 a b B\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 A b+15 A b^3+539 a^3 B+825 a b^2 B\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 A+1025 a^2 A b^2-20 A b^4+1793 a^3 b B+55 a b^3 B\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 A (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 A b^4+3 a^4 (225 A-539 B)-6 a^3 b (505 A-209 B)+15 a^2 b^2 (19 A-121 B)+10 a b^3 (3 A-11 B)\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 A b+255 a^2 A b^3+40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B\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 A b+255 a^2 A b^3+40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B\right ) \cot (c+d x) E\left (\sin ^{-1}\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 A b^4+3 a^4 (225 A-539 B)-6 a^3 b (505 A-209 B)+15 a^2 b^2 (19 A-121 B)+10 a b^3 (3 A-11 B)\right ) \cot (c+d x) F\left (\sin ^{-1}\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 A b+11 a B) \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 A+113 A b^2+209 a b B\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 A b+15 A b^3+539 a^3 B+825 a b^2 B\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 A+1025 a^2 A b^2-20 A b^4+1793 a^3 b B+55 a b^3 B\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 A (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 6.87, size = 1640, normalized size = 2.64 \begin {gather*} \frac {-\frac {4 a \left (675 a^6 A-390 a^4 A b^2-245 a^2 A b^4-40 A b^6+1254 a^5 b B-1364 a^3 b^3 B+110 a b^5 B\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) F\left (\text {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 A b-255 a^3 A b^3-40 a A b^5-1617 a^6 B-3069 a^4 b^2 B+110 a^2 b^4 B\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) F\left (\text {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) \Pi \left (-\frac {a}{b};\text {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 A b^2-255 a^2 A b^4-40 A b^6-1617 a^5 b B-3069 a^3 b^3 B+110 a b^5 B\right ) \left (\frac {i \cos \left (\frac {1}{2} (c+d x)\right ) \sqrt {a+b \cos (c+d x)} E\left (i \sinh ^{-1}\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) F\left (\text {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) \Pi \left (-\frac {a}{b};\text {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 A b \sin (c+d x)+11 a^2 B \sin (c+d x)\right )+\frac {2}{693} \sec ^4(c+d x) \left (81 a^2 A \sin (c+d x)+113 A b^2 \sin (c+d x)+209 a b B \sin (c+d x)\right )+\frac {2 \sec ^3(c+d x) \left (1145 a^2 A b \sin (c+d x)+15 A b^3 \sin (c+d x)+539 a^3 B \sin (c+d x)+825 a b^2 B \sin (c+d x)\right )}{3465 a}+\frac {2 \sec ^2(c+d x) \left (675 a^4 A \sin (c+d x)+1025 a^2 A b^2 \sin (c+d x)-20 A b^4 \sin (c+d x)+1793 a^3 b B \sin (c+d x)+55 a b^3 B \sin (c+d x)\right )}{3465 a^2}+\frac {2 \sec (c+d x) \left (3705 a^4 A b \sin (c+d x)+255 a^2 A b^3 \sin (c+d x)+40 A b^5 \sin (c+d x)+1617 a^5 B \sin (c+d x)+3069 a^3 b^2 B \sin (c+d x)-110 a b^4 B \sin (c+d x)\right )}{3465 a^3}+\frac {2}{11} a^2 A \sec ^5(c+d x) \tan (c+d x)\right )}{d} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(5372\) vs.
\(2(572)=1144\).
time = 0.94, size = 5373, normalized size = 8.64
method | result | size |
default | \(\text {Expression too large to display}\) | \(5373\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (A+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 )}^{13/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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