Optimal. Leaf size=894 \[ \frac{C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d \sqrt{\sec (c+d x)}}+\frac{(10 b B-3 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d \sqrt{\sec (c+d x)}}+\frac{\left (-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d \sqrt{\sec (c+d x)}}+\frac{\left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d}+\frac{\left (-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{320 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 a b^2 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left (45 C a^4-30 b (5 B+C) a^3-4 b^2 (660 A+295 B+423 C) a^2-8 b^3 (260 A+355 B+193 C) a-16 b^4 (80 A+45 B+64 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left (-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{128 b^3 d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 4.05813, antiderivative size = 894, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.178, Rules used = {4221, 3049, 3061, 3053, 2809, 2998, 2816, 2994} \[ \frac{C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d \sqrt{\sec (c+d x)}}+\frac{(10 b B-3 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d \sqrt{\sec (c+d x)}}+\frac{\left (-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d \sqrt{\sec (c+d x)}}+\frac{\left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d}+\frac{\left (-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{320 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 a b^2 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left (45 C a^4-30 b (5 B+C) a^3-4 b^2 (660 A+295 B+423 C) a^2-8 b^3 (260 A+355 B+193 C) a-16 b^4 (80 A+45 B+64 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left (-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{128 b^3 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3049
Rule 3061
Rule 3053
Rule 2809
Rule 2998
Rule 2816
Rule 2994
Rubi steps
\begin{align*} \int \frac{(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{C (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 C}{2}+b (5 A+4 C) \cos (c+d x)+\frac{1}{2} (10 b B-3 a C) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{5 b}\\ &=\frac{(10 b B-3 a C) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt{\sec (c+d x)}}+\frac{C (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 b B+a C)+\frac{1}{2} b (40 a A+30 b B+27 a C) \cos (c+d x)+\frac{1}{4} \left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{20 b}\\ &=\frac{\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d \sqrt{\sec (c+d x)}}+\frac{(10 b B-3 a C) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt{\sec (c+d x)}}+\frac{C (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 (80 A b^2+110 a b B+15 a^2 C+64 b^2 C\right )+\frac{1}{4} b \left (310 a b B+32 b^2 (5 A+4 C)+3 a^2 (80 A+49 C)\right ) \cos (c+d x)+\frac{3}{8} \left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{60 b}\\ &=\frac{\left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{320 b d \sqrt{\sec (c+d x)}}+\frac{\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d \sqrt{\sec (c+d x)}}+\frac{(10 b B-3 a C) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt{\sec (c+d x)}}+\frac{C (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 b B+360 b^3 B+15 a^3 C+4 a b^2 (260 A+193 C)\right )+\frac{1}{8} b \left (1610 a^2 b B+360 b^3 B+4 a b^2 (380 A+289 C)+a^3 (960 A+573 C)\right ) \cos (c+d x)+\frac{1}{16} \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\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 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{320 b d \sqrt{\sec (c+d x)}}+\frac{\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d \sqrt{\sec (c+d x)}}+\frac{(10 b B-3 a C) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt{\sec (c+d x)}}+\frac{C (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 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\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 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right )+\frac{1}{8} a b \left (590 a^2 b B+360 b^3 B+15 a^3 C+4 a b^2 (260 A+193 C)\right ) \cos (c+d x)-\frac{15}{16} \left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 (2 A+C)-80 a b^4 (4 A+3 C)\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 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{320 b d \sqrt{\sec (c+d x)}}+\frac{\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d \sqrt{\sec (c+d x)}}+\frac{(10 b B-3 a C) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt{\sec (c+d x)}}+\frac{C (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 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\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 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right )+\frac{1}{8} a b \left (590 a^2 b B+360 b^3 B+15 a^3 C+4 a b^2 (260 A+193 C)\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 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 (2 A+C)-80 a b^4 (4 A+3 C)\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 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 (2 A+C)-80 a b^4 (4 A+3 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left (\frac{a+b}{b};\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}}}{128 b^3 d \sqrt{\sec (c+d x)}}+\frac{\left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{320 b d \sqrt{\sec (c+d x)}}+\frac{\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d \sqrt{\sec (c+d x)}}+\frac{(10 b B-3 a C) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt{\sec (c+d x)}}+\frac{C (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 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\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 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 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}{3840 b^2}-\frac{\left (a \left (45 a^4 C-30 a^3 b (5 B+C)-16 b^4 (80 A+45 B+64 C)-8 a b^3 (260 A+355 B+193 C)-4 a^2 b^2 (660 A+295 B+423 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}{3840 b^2}\\ &=-\frac{(a-b) \sqrt{a+b} \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \sqrt{\cos (c+d x)} \csc (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}}}{1920 a b^2 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left (45 a^4 C-30 a^3 b (5 B+C)-16 b^4 (80 A+45 B+64 C)-8 a b^3 (260 A+355 B+193 C)-4 a^2 b^2 (660 A+295 B+423 C)\right ) \sqrt{\cos (c+d x)} \csc (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}}}{1920 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 (2 A+C)-80 a b^4 (4 A+3 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left (\frac{a+b}{b};\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}}}{128 b^3 d \sqrt{\sec (c+d x)}}+\frac{\left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{320 b d \sqrt{\sec (c+d x)}}+\frac{\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d \sqrt{\sec (c+d x)}}+\frac{(10 b B-3 a C) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d \sqrt{\sec (c+d x)}}+\frac{C (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 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{1920 b^2 d}\\ \end{align*}
Mathematica [C] time = 20.6159, size = 667, normalized size = 0.75 \[ \frac{\sqrt{a+b \cos (c+d x)} \left (\frac{2 \sin (c+d x) \left (590 a^2 b B+15 a^3 C+16 a b^2 (65 A+64 C)+480 b^3 B\right )}{b}+2 \tan (c+d x) \left (93 a^2 C+170 a b B+80 A b^2+88 b^2 C\right )+\frac{\tan \left (\frac{1}{2} (c+d x)\right ) \left (12 a^2 b^2 (220 A+141 C)+150 a^3 b B-45 a^4 C+2840 a b^3 B+256 b^4 (5 A+4 C)\right )}{b^2}+2 \sin (3 (c+d x)) \sec (c+d x) \left (93 a^2 C+170 a b B+80 A b^2+100 b^2 C\right )-\frac{i \sqrt{\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a+b \cos (c+d x))}{a+b}} \left (-2 (a-b) \left (-4 a^2 b^2 (180 A+185 B+129 C)-30 a^3 b (5 B-C)+45 a^4 C-8 a b^3 (220 A+45 B+161 C)-720 b^4 B\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{a-b}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{a+b}{a-b}\right )+(a-b) \left (-12 a^2 b^2 (220 A+141 C)-150 a^3 b B+45 a^4 C-2840 a b^3 B-256 b^4 (5 A+4 C)\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{a-b}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{a+b}{a-b}\right )+30 \left (40 a^3 b^2 (2 A+C)+240 a^2 b^3 B-10 a^4 b B+3 a^5 C+80 a b^4 (4 A+3 C)+96 b^5 B\right ) \Pi \left (\frac{a+b}{a-b};i \sinh ^{-1}\left (\sqrt{\frac{a-b}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{a+b}{a-b}\right )\right )}{b^2 \sqrt{\frac{a-b}{a+b}} \sqrt{\cos (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right )} (a+b \cos (c+d x))}+6 b (21 a C+10 b B) \sin (4 (c+d x)) \sec (c+d x)+24 b^2 C \sin (5 (c+d x)) \sec (c+d x)\right )}{1920 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.753, size = 7064, normalized size = 7.9 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{\sqrt{\sec \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C b^{2} \cos \left (d x + c\right )^{4} +{\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{3} + A a^{2} +{\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )\right )} \sqrt{b \cos \left (d x + c\right ) + a}}{\sqrt{\sec \left (d x + c\right )}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{\sqrt{\sec \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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