Integrand size = 43, antiderivative size = 774 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {(a-b) \sqrt {a+b} \left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{1920 a^2 b d}-\frac {\sqrt {a+b} \left (45 A b^4-30 a b^3 (A+5 B)-16 a^4 (64 A+45 B+80 C)-8 a^3 b (193 A+355 B+260 C)-4 a^2 b^2 (423 A+295 B+660 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{1920 a^2 d}-\frac {\sqrt {a+b} \left (3 A b^5+96 a^5 B+240 a^3 b^2 B-10 a b^4 B+40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{128 a^3 d}-\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{1920 a^2 d}+\frac {\left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{960 a d}+\frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{240 d}+\frac {(A b+2 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d} \]
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Time = 3.44 (sec) , antiderivative size = 774, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {4179, 4189, 4143, 4006, 3869, 3917, 4089} \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\sin (c+d x) \cos ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{240 d}+\frac {\sin (c+d x) \cos (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{960 a d}-\frac {\sqrt {a+b} \cot (c+d x) \left (-16 a^4 (64 A+45 B+80 C)-8 a^3 b (193 A+355 B+260 C)-4 a^2 b^2 (423 A+295 B+660 C)-30 a b^3 (A+5 B)+45 A b^4\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{1920 a^2 d}-\frac {(a-b) \sqrt {a+b} \cot (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{1920 a^2 b d}-\frac {\sin (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{1920 a^2 d}-\frac {\sqrt {a+b} \cot (c+d x) \left (96 a^5 B+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+40 a^2 b^3 (A+2 C)-10 a b^4 B+3 A b^5\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{128 a^3 d}+\frac {(2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{8 d}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d} \]
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Rule 3869
Rule 3917
Rule 4006
Rule 4089
Rule 4143
Rule 4179
Rule 4189
Rubi steps \begin{align*} \text {integral}& = \frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac {1}{5} \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac {5}{2} (A b+2 a B)+(4 a A+5 b B+5 a C) \sec (c+d x)+\frac {1}{2} b (3 A+10 C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {(A b+2 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac {1}{20} \int \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (\frac {1}{4} \left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right )+\frac {1}{2} \left (30 a^2 B+40 b^2 B+a b (59 A+80 C)\right ) \sec (c+d x)+\frac {1}{4} b (39 A b+30 a B+80 b C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{240 d}+\frac {(A b+2 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac {1}{60} \int \frac {\cos ^2(c+d x) \left (\frac {1}{8} \left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right )+\frac {1}{4} \left (490 a^2 b B+240 b^3 B+32 a^3 (4 A+5 C)+3 a b^2 (167 A+240 C)\right ) \sec (c+d x)+\frac {3}{8} b \left (170 a b B+16 a^2 (4 A+5 C)+b^2 (93 A+160 C)\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {\left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{960 a d}+\frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{240 d}+\frac {(A b+2 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}-\frac {\int \frac {\cos (c+d x) \left (\frac {1}{16} \left (-1024 a^4 A-1692 a^2 A b^2+45 A b^4-2840 a^3 b B-150 a b^3 B-1280 a^4 C-2640 a^2 b^2 C\right )-\frac {1}{8} a \left (360 a^3 B+1610 a b^2 B+3 b^3 (191 A+320 C)+4 a^2 b (289 A+380 C)\right ) \sec (c+d x)-\frac {1}{16} b \left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx}{120 a} \\ & = -\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{1920 a^2 d}+\frac {\left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{960 a d}+\frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{240 d}+\frac {(A b+2 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac {\int \frac {\frac {15}{32} \left (3 A b^5+96 a^5 B+240 a^3 b^2 B-10 a b^4 B+40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)\right )+\frac {1}{16} a b \left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \sec (c+d x)+\frac {1}{32} b \left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sec ^2(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{120 a^2} \\ & = -\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{1920 a^2 d}+\frac {\left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{960 a d}+\frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{240 d}+\frac {(A b+2 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac {\int \frac {\frac {15}{32} \left (3 A b^5+96 a^5 B+240 a^3 b^2 B-10 a b^4 B+40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)\right )+\left (-\frac {1}{32} b \left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right )+\frac {1}{16} a b \left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{120 a^2}+\frac {\left (b \left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right )\right ) \int \frac {\sec (c+d x) (1+\sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx}{3840 a^2} \\ & = -\frac {(a-b) \sqrt {a+b} \left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{1920 a^2 b d}-\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{1920 a^2 d}+\frac {\left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{960 a d}+\frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{240 d}+\frac {(A b+2 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac {\left (3 A b^5+96 a^5 B+240 a^3 b^2 B-10 a b^4 B+40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}} \, dx}{256 a^2}-\frac {\left (b \left (45 A b^4-30 a b^3 (A+5 B)-16 a^4 (64 A+45 B+80 C)-8 a^3 b (193 A+355 B+260 C)-4 a^2 b^2 (423 A+295 B+660 C)\right )\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{3840 a^2} \\ & = -\frac {(a-b) \sqrt {a+b} \left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{1920 a^2 b d}-\frac {\sqrt {a+b} \left (45 A b^4-30 a b^3 (A+5 B)-16 a^4 (64 A+45 B+80 C)-8 a^3 b (193 A+355 B+260 C)-4 a^2 b^2 (423 A+295 B+660 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{1920 a^2 d}-\frac {\sqrt {a+b} \left (3 A b^5+96 a^5 B+240 a^3 b^2 B-10 a b^4 B+40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{128 a^3 d}-\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{1920 a^2 d}+\frac {\left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{960 a d}+\frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{240 d}+\frac {(A b+2 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d} \\ \end{align*}
Time = 28.29 (sec) , antiderivative size = 711, normalized size of antiderivative = 0.92 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {1}{480} \left (88 a^2 A+93 A b^2+170 a b B+80 a^2 C\right ) \sin (c+d x)+\frac {\left (1024 a^2 A b+15 A b^3+480 a^3 B+590 a b^2 B+1040 a^2 b C\right ) \sin (2 (c+d x))}{960 a}+\frac {1}{480} \left (100 a^2 A+93 A b^2+170 a b B+80 a^2 C\right ) \sin (3 (c+d x))+\frac {1}{160} a (21 A b+10 a B) \sin (4 (c+d x))+\frac {1}{40} a^2 A \sin (5 (c+d x))\right )}{d (b+a \cos (c+d x))^2 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}-\frac {\cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-\frac {\left ((a+b) \left (-45 A b^4+2840 a^3 b B+150 a b^3 B+256 a^4 (4 A+5 C)+12 a^2 b^2 (141 A+220 C)\right ) E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right )-2 a \left (-15 A b^4+720 a^4 B-4 a^2 b^2 (193 A-805 B+260 C)+8 a^3 b (289 A-45 B+380 C)+2 a b^3 (573 A-295 B+960 C)\right ) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right )+30 \left (3 A b^5+96 a^5 B+240 a^3 b^2 B-10 a b^4 B+40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)\right ) \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right )\right ) \sqrt {\frac {(b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}}{(b+a \cos (c+d x)) \sqrt {\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )}}-\left (-45 A b^4+2840 a^3 b B+150 a b^3 B+256 a^4 (4 A+5 C)+12 a^2 b^2 (141 A+220 C)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{960 a^2 d (b+a \cos (c+d x))^2 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(9714\) vs. \(2(721)=1442\).
Time = 1264.89 (sec) , antiderivative size = 9715, normalized size of antiderivative = 12.55
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\[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{5} \,d x } \]
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Timed out. \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{5} \,d x } \]
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\[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{5} \,d x } \]
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Timed out. \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^5\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]
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