Integrand size = 37, antiderivative size = 544 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=-\frac {2 (a-b) \sqrt {a+b} \left (10 A b^4-21 a^4 (7 A+9 C)-3 a^2 b^2 (93 A+161 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{315 a^3 d \sqrt {\sec (c+d x)}}-\frac {2 (a-b) \sqrt {a+b} \left (10 A b^3+21 a^3 (7 A+9 C)+15 a b^2 (11 A+21 C)-6 a^2 b (19 A+28 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{315 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 b \left (5 A b^2+a^2 (163 A+231 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 a d}+\frac {2 \left (15 A b^2+7 a^2 (7 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {10 A b (a+b \cos (c+d x))^{3/2} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d} \]
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Time = 2.11 (sec) , antiderivative size = 544, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {4306, 3127, 3126, 3134, 3077, 2895, 3073} \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\frac {2 \left (7 a^2 (7 A+9 C)+15 A b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}{315 d}+\frac {2 b \left (a^2 (163 A+231 C)+5 A b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}{315 a d}-\frac {2 (a-b) \sqrt {a+b} \left (21 a^3 (7 A+9 C)-6 a^2 b (19 A+28 C)+15 a b^2 (11 A+21 C)+10 A b^3\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{315 a^2 d \sqrt {\sec (c+d x)}}-\frac {2 (a-b) \sqrt {a+b} \left (-21 a^4 (7 A+9 C)-3 a^2 b^2 (93 A+161 C)+10 A b^4\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{315 a^3 d \sqrt {\sec (c+d x)}}+\frac {2 A \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{9 d}+\frac {10 A b \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{63 d} \]
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Rule 2895
Rule 3073
Rule 3077
Rule 3126
Rule 3127
Rule 3134
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))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx \\ & = \frac {2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{9} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^{3/2} \left (\frac {5 A b}{2}+\frac {1}{2} a (7 A+9 C) \cos (c+d x)+\frac {1}{2} b (2 A+9 C) \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx \\ & = \frac {10 A b (a+b \cos (c+d x))^{3/2} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{63} \left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{4} \left (15 A b^2+7 a^2 (7 A+9 C)\right )+\frac {1}{2} a b (44 A+63 C) \cos (c+d x)+\frac {3}{4} b^2 (8 A+21 C) \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 \left (15 A b^2+7 a^2 (7 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {10 A b (a+b \cos (c+d x))^{3/2} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{315} \left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {3}{8} b \left (5 A b^2+a^2 (163 A+231 C)\right )+\frac {1}{8} a \left (21 a^2 (7 A+9 C)+5 b^2 (121 A+189 C)\right ) \cos (c+d x)+\frac {1}{8} b \left (14 a^2 (7 A+9 C)+15 b^2 (10 A+21 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx \\ & = \frac {2 b \left (5 A b^2+a^2 (163 A+231 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 a d}+\frac {2 \left (15 A b^2+7 a^2 (7 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {10 A b (a+b \cos (c+d x))^{3/2} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {\left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {3}{16} \left (10 A b^4-21 a^4 (7 A+9 C)-3 a^2 b^2 (93 A+161 C)\right )+\frac {3}{16} a b \left (5 b^2 (31 A+63 C)+3 a^2 (87 A+119 C)\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{945 a} \\ & = \frac {2 b \left (5 A b^2+a^2 (163 A+231 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 a d}+\frac {2 \left (15 A b^2+7 a^2 (7 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {10 A b (a+b \cos (c+d x))^{3/2} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}-\frac {\left ((a-b) \left (10 A b^3+21 a^3 (7 A+9 C)+15 a b^2 (11 A+21 C)-6 a^2 b (19 A+28 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{315 a}-\frac {\left (\left (10 A b^4-21 a^4 (7 A+9 C)-3 a^2 b^2 (93 A+161 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{315 a} \\ & = -\frac {2 (a-b) \sqrt {a+b} \left (10 A b^4-21 a^4 (7 A+9 C)-3 a^2 b^2 (93 A+161 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\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}}}{315 a^3 d \sqrt {\sec (c+d x)}}-\frac {2 (a-b) \sqrt {a+b} \left (10 A b^3+21 a^3 (7 A+9 C)+15 a b^2 (11 A+21 C)-6 a^2 b (19 A+28 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\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}}}{315 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 b \left (5 A b^2+a^2 (163 A+231 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 a d}+\frac {2 \left (15 A b^2+7 a^2 (7 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {10 A b (a+b \cos (c+d x))^{3/2} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d} \\ \end{align*}
Time = 19.52 (sec) , antiderivative size = 621, normalized size of antiderivative = 1.14 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\frac {2 \sqrt {2} \sqrt {\frac {\cos (c+d x)}{(1+\cos (c+d x))^2}} \sqrt {\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )} \left (\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right )^{3/2} \left (-\left ((a+b) \left (\left (-10 A b^4+21 a^4 (7 A+9 C)+3 a^2 b^2 (93 A+161 C)\right ) E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {-a+b}{a+b}\right )-a \left (-10 A b^3+21 a^3 (7 A+9 C)+15 a b^2 (11 A+21 C)+6 a^2 b (19 A+28 C)\right ) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right )\right ) \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^{3/2} \sqrt {\frac {(a+b \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}} \sec (c+d x)\right )-\left (-10 A b^4+21 a^4 (7 A+9 C)+3 a^2 b^2 (93 A+161 C)\right ) \cos (c+d x) (a+b \cos (c+d x)) \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{315 a^2 d \sqrt {\frac {1}{1+\cos (c+d x)}} \sqrt {a+b \cos (c+d x)} \sec ^2\left (\frac {1}{2} (c+d x)\right )^{3/2}}+\frac {\sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {2 \left (-147 a^4 A-279 a^2 A b^2+10 A b^4-189 a^4 C-483 a^2 b^2 C\right ) \sin (c+d x)}{315 a^2}+\frac {2}{315} \sec ^2(c+d x) \left (49 a^2 A \sin (c+d x)+75 A b^2 \sin (c+d x)+63 a^2 C \sin (c+d x)\right )+\frac {2 \sec (c+d x) \left (163 a^2 A b \sin (c+d x)+5 A b^3 \sin (c+d x)+231 a^2 b C \sin (c+d x)\right )}{315 a}+\frac {38}{63} a A b \sec ^2(c+d x) \tan (c+d x)+\frac {2}{9} a^2 A \sec ^3(c+d x) \tan (c+d x)\right )}{d} \]
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hanged
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\[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \]
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Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\text {Timed out} \]
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\[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \]
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\[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \]
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Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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