Integrand size = 35, antiderivative size = 839 \[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {(a-b) \sqrt {a+b} \left (150 a^3 A b+2840 a A b^3-45 a^4 B+1692 a^2 b^2 B+1024 b^4 B\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{1920 a b^2 d \sqrt {\sec (c+d x)}}-\frac {\sqrt {a+b} \left (45 a^4 B-30 a^3 b (5 A+B)-16 b^4 (45 A+64 B)-8 a b^3 (355 A+193 B)-4 a^2 b^2 (295 A+423 B)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{1920 b^2 d \sqrt {\sec (c+d x)}}+\frac {\sqrt {a+b} \left (10 a^4 A b-240 a^2 A b^3-96 A b^5-3 a^5 B-40 a^3 b^2 B-240 a b^4 B\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{128 b^3 d \sqrt {\sec (c+d x)}}+\frac {\left (50 a^2 A b+120 A b^3-15 a^3 B+172 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{320 b d \sqrt {\sec (c+d x)}}+\frac {\left (50 a A b-15 a^2 B+64 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d \sqrt {\sec (c+d x)}}+\frac {(10 A b-3 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt {\sec (c+d x)}}+\frac {B (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d \sqrt {\sec (c+d x)}}+\frac {\left (150 a^3 A b+2840 a A b^3-45 a^4 B+1692 a^2 b^2 B+1024 b^4 B\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{1920 b^2 d} \]
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Time = 4.58 (sec) , antiderivative size = 839, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {3040, 3069, 3128, 3140, 3132, 2888, 3077, 2895, 3073} \[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d \sqrt {\sec (c+d x)}}+\frac {(10 A b-3 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d \sqrt {\sec (c+d x)}}+\frac {\left (-15 B a^2+50 A b a+64 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d \sqrt {\sec (c+d x)}}+\frac {\left (-45 B a^4+150 A b a^3+1692 b^2 B a^2+2840 A b^3 a+1024 b^4 B\right ) \sqrt {\sec (c+d x)} \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{1920 b^2 d}+\frac {\left (-15 B a^3+50 A b a^2+172 b^2 B a+120 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{320 b d \sqrt {\sec (c+d x)}}-\frac {(a-b) \sqrt {a+b} \left (-45 B a^4+150 A b a^3+1692 b^2 B a^2+2840 A b^3 a+1024 b^4 B\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}}}{1920 a b^2 d \sqrt {\sec (c+d x)}}-\frac {\sqrt {a+b} \left (45 B a^4-30 b (5 A+B) a^3-4 b^2 (295 A+423 B) a^2-8 b^3 (355 A+193 B) a-16 b^4 (45 A+64 B)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}}}{1920 b^2 d \sqrt {\sec (c+d x)}}+\frac {\sqrt {a+b} \left (-3 B a^5+10 A b a^4-40 b^2 B a^3-240 A b^3 a^2-240 b^4 B a-96 A b^5\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}}}{128 b^3 d \sqrt {\sec (c+d x)}} \]
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Rule 2888
Rule 2895
Rule 3040
Rule 3069
Rule 3073
Rule 3077
Rule 3128
Rule 3132
Rule 3140
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx \\ & = \frac {B (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d \sqrt {\sec (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^{5/2} \left (\frac {a B}{2}+4 b B \cos (c+d x)+\frac {1}{2} (10 A b-3 a B) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{5 b} \\ & = \frac {(10 A b-3 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt {\sec (c+d x)}}+\frac {B (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d \sqrt {\sec (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^{3/2} \left (\frac {5}{4} a (2 A b+a B)+\frac {3}{2} b (10 A b+9 a B) \cos (c+d x)+\frac {1}{4} \left (50 a A b-15 a^2 B+64 b^2 B\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{20 b} \\ & = \frac {\left (50 a A b-15 a^2 B+64 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d \sqrt {\sec (c+d x)}}+\frac {(10 A b-3 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt {\sec (c+d x)}}+\frac {B (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d \sqrt {\sec (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{8} a \left (110 a A b+15 a^2 B+64 b^2 B\right )+\frac {1}{4} b \left (310 a A b+147 a^2 B+128 b^2 B\right ) \cos (c+d x)+\frac {3}{8} \left (50 a^2 A b+120 A b^3-15 a^3 B+172 a b^2 B\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{60 b} \\ & = \frac {\left (50 a^2 A b+120 A b^3-15 a^3 B+172 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{320 b d \sqrt {\sec (c+d x)}}+\frac {\left (50 a A b-15 a^2 B+64 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d \sqrt {\sec (c+d x)}}+\frac {(10 A b-3 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt {\sec (c+d x)}}+\frac {B (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d \sqrt {\sec (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{16} a \left (590 a^2 A b+360 A b^3+15 a^3 B+772 a b^2 B\right )+\frac {1}{8} b \left (1610 a^2 A b+360 A b^3+573 a^3 B+1156 a b^2 B\right ) \cos (c+d x)+\frac {1}{16} \left (150 a^3 A b+2840 a A b^3-45 a^4 B+1692 a^2 b^2 B+1024 b^4 B\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{120 b} \\ & = \frac {\left (50 a^2 A b+120 A b^3-15 a^3 B+172 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{320 b d \sqrt {\sec (c+d x)}}+\frac {\left (50 a A b-15 a^2 B+64 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d \sqrt {\sec (c+d x)}}+\frac {(10 A b-3 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt {\sec (c+d x)}}+\frac {B (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d \sqrt {\sec (c+d x)}}+\frac {\left (150 a^3 A b+2840 a A b^3-45 a^4 B+1692 a^2 b^2 B+1024 b^4 B\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{1920 b^2 d}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{16} a \left (150 a^3 A b+2840 a A b^3-45 a^4 B+1692 a^2 b^2 B+1024 b^4 B\right )+\frac {1}{8} a b \left (590 a^2 A b+360 A b^3+15 a^3 B+772 a b^2 B\right ) \cos (c+d x)-\frac {15}{16} \left (10 a^4 A b-240 a^2 A b^3-96 A b^5-3 a^5 B-40 a^3 b^2 B-240 a b^4 B\right ) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{240 b^2} \\ & = \frac {\left (50 a^2 A b+120 A b^3-15 a^3 B+172 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{320 b d \sqrt {\sec (c+d x)}}+\frac {\left (50 a A b-15 a^2 B+64 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d \sqrt {\sec (c+d x)}}+\frac {(10 A b-3 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt {\sec (c+d x)}}+\frac {B (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d \sqrt {\sec (c+d x)}}+\frac {\left (150 a^3 A b+2840 a A b^3-45 a^4 B+1692 a^2 b^2 B+1024 b^4 B\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{1920 b^2 d}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{16} a \left (150 a^3 A b+2840 a A b^3-45 a^4 B+1692 a^2 b^2 B+1024 b^4 B\right )+\frac {1}{8} a b \left (590 a^2 A b+360 A b^3+15 a^3 B+772 a b^2 B\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{240 b^2}-\frac {\left (\left (10 a^4 A b-240 a^2 A b^3-96 A b^5-3 a^5 B-40 a^3 b^2 B-240 a b^4 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}} \, dx}{256 b^2} \\ & = \frac {\sqrt {a+b} \left (10 a^4 A b-240 a^2 A b^3-96 A b^5-3 a^5 B-40 a^3 b^2 B-240 a b^4 B\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\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}}}{128 b^3 d \sqrt {\sec (c+d x)}}+\frac {\left (50 a^2 A b+120 A b^3-15 a^3 B+172 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{320 b d \sqrt {\sec (c+d x)}}+\frac {\left (50 a A b-15 a^2 B+64 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d \sqrt {\sec (c+d x)}}+\frac {(10 A b-3 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt {\sec (c+d x)}}+\frac {B (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d \sqrt {\sec (c+d x)}}+\frac {\left (150 a^3 A b+2840 a A b^3-45 a^4 B+1692 a^2 b^2 B+1024 b^4 B\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{1920 b^2 d}-\frac {\left (a \left (150 a^3 A b+2840 a A b^3-45 a^4 B+1692 a^2 b^2 B+1024 b^4 B\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}{3840 b^2}-\frac {\left (a \left (45 a^4 B-30 a^3 b (5 A+B)-16 b^4 (45 A+64 B)-8 a b^3 (355 A+193 B)-4 a^2 b^2 (295 A+423 B)\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}{3840 b^2} \\ & = -\frac {(a-b) \sqrt {a+b} \left (150 a^3 A b+2840 a A b^3-45 a^4 B+1692 a^2 b^2 B+1024 b^4 B\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}}}{1920 a b^2 d \sqrt {\sec (c+d x)}}-\frac {\sqrt {a+b} \left (45 a^4 B-30 a^3 b (5 A+B)-16 b^4 (45 A+64 B)-8 a b^3 (355 A+193 B)-4 a^2 b^2 (295 A+423 B)\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}}}{1920 b^2 d \sqrt {\sec (c+d x)}}+\frac {\sqrt {a+b} \left (10 a^4 A b-240 a^2 A b^3-96 A b^5-3 a^5 B-40 a^3 b^2 B-240 a b^4 B\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\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}}}{128 b^3 d \sqrt {\sec (c+d x)}}+\frac {\left (50 a^2 A b+120 A b^3-15 a^3 B+172 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{320 b d \sqrt {\sec (c+d x)}}+\frac {\left (50 a A b-15 a^2 B+64 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d \sqrt {\sec (c+d x)}}+\frac {(10 A b-3 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt {\sec (c+d x)}}+\frac {B (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d \sqrt {\sec (c+d x)}}+\frac {\left (150 a^3 A b+2840 a A b^3-45 a^4 B+1692 a^2 b^2 B+1024 b^4 B\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{1920 b^2 d} \\ \end{align*}
Time = 13.17 (sec) , antiderivative size = 703, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{960} \left (170 a A b+93 a^2 B+88 b^2 B\right ) \sin (c+d x)+\frac {\left (590 a^2 A b+480 A b^3+15 a^3 B+1024 a b^2 B\right ) \sin (2 (c+d x))}{1920 b}+\frac {1}{960} \left (170 a A b+93 a^2 B+100 b^2 B\right ) \sin (3 (c+d x))+\frac {1}{320} b (10 A b+21 a B) \sin (4 (c+d x))+\frac {1}{80} b^2 B \sin (5 (c+d x))\right )}{d}-\frac {-b (a+b) \left (150 a^3 A b+2840 a A b^3-45 a^4 B+1692 a^2 b^2 B+1024 b^4 B\right ) E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {-a+b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(a+b \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+a (a+b) \left (45 a^4 B-30 a^3 b (5 A+3 B)+60 a^2 b^2 (5 A+11 B)+16 b^4 (45 A+64 B)+8 a b^3 (265 A+129 B)\right ) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(a+b \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+15 \left (10 a^4 A b-240 a^2 A b^3-96 A b^5-3 a^5 B-40 a^3 b^2 B-240 a b^4 B\right ) \left ((a-b) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right )+2 b \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right )\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(a+b \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}-b \left (150 a^3 A b+2840 a A b^3-45 a^4 B+1692 a^2 b^2 B+1024 b^4 B\right ) (a+b \cos (c+d x)) \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^{3/2} \sec (c+d x) \tan \left (\frac {1}{2} (c+d x)\right )}{1920 b^3 d \sqrt {a+b \cos (c+d x)} \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^{3/2} \sec ^{\frac {3}{2}}(c+d x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(6969\) vs. \(2(767)=1534\).
Time = 23.12 (sec) , antiderivative size = 6970, normalized size of antiderivative = 8.31
method | result | size |
parts | \(\text {Expression too large to display}\) | \(6970\) |
default | \(\text {Expression too large to display}\) | \(7057\) |
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Timed out. \[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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