Optimal. Leaf size=839 \[ \frac{B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d \sqrt{\sec (c+d x)}}+\frac{(10 A b-3 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d \sqrt{\sec (c+d x)}}+\frac{\left (-15 B a^2+50 A b a+64 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d \sqrt{\sec (c+d x)}}+\frac{\left (-45 B a^4+150 A b a^3+1692 b^2 B a^2+2840 A b^3 a+1024 b^4 B\right ) \sqrt{\sec (c+d x)} \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d}+\frac{\left (-15 B a^3+50 A b a^2+172 b^2 B a+120 A b^3\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{320 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left (-45 B a^4+150 A b a^3+1692 b^2 B a^2+2840 A b^3 a+1024 b^4 B\right ) \sqrt{\cos (c+d x)} \csc (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 a b^2 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left (45 B a^4-30 b (5 A+B) a^3-4 b^2 (295 A+423 B) a^2-8 b^3 (355 A+193 B) a-16 b^4 (45 A+64 B)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left (-3 B a^5+10 A b a^4-40 b^2 B a^3-240 A b^3 a^2-240 b^4 B a-96 A b^5\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{128 b^3 d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 3.6028, antiderivative size = 839, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.257, Rules used = {2961, 2990, 3049, 3061, 3053, 2809, 2998, 2816, 2994} \[ \frac{B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d \sqrt{\sec (c+d x)}}+\frac{(10 A b-3 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d \sqrt{\sec (c+d x)}}+\frac{\left (-15 B a^2+50 A b a+64 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d \sqrt{\sec (c+d x)}}+\frac{\left (-45 B a^4+150 A b a^3+1692 b^2 B a^2+2840 A b^3 a+1024 b^4 B\right ) \sqrt{\sec (c+d x)} \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d}+\frac{\left (-15 B a^3+50 A b a^2+172 b^2 B a+120 A b^3\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{320 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left (-45 B a^4+150 A b a^3+1692 b^2 B a^2+2840 A b^3 a+1024 b^4 B\right ) \sqrt{\cos (c+d x)} \csc (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 a b^2 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left (45 B a^4-30 b (5 A+B) a^3-4 b^2 (295 A+423 B) a^2-8 b^3 (355 A+193 B) a-16 b^4 (45 A+64 B)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left (-3 B a^5+10 A b a^4-40 b^2 B a^3-240 A b^3 a^2-240 b^4 B a-96 A b^5\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{128 b^3 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2961
Rule 2990
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} (A+B \cos (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx &=\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) \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 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 (\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 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) 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 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) \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 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*}
Mathematica [A] time = 15.5365, size = 705, normalized size = 0.84 \[ \frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left (\frac{1}{960} \left (93 a^2 B+170 a A b+88 b^2 B\right ) \sin (c+d x)+\frac{1}{960} \left (93 a^2 B+170 a A b+100 b^2 B\right ) \sin (3 (c+d x))+\frac{\left (590 a^2 A b+15 a^3 B+1024 a b^2 B+480 A b^3\right ) \sin (2 (c+d x))}{1920 b}+\frac{1}{320} b (21 a B+10 A b) \sin (4 (c+d x))+\frac{1}{80} b^2 B \sin (5 (c+d x))\right )}{d}-\frac{-b \left (150 a^3 A b+1692 a^2 b^2 B-45 a^4 B+2840 a A b^3+1024 b^4 B\right ) \tan \left (\frac{1}{2} (c+d x)\right ) \sec (c+d x) \left (\cos (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right )\right )^{3/2} (a+b \cos (c+d x))+a (a+b) \left (60 a^2 b^2 (5 A+11 B)-30 a^3 b (5 A+3 B)+45 a^4 B+8 a b^3 (265 A+129 B)+16 b^4 (45 A+64 B)\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a+b \cos (c+d x))}{a+b}} F\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{b-a}{a+b}\right )-b (a+b) \left (150 a^3 A b+1692 a^2 b^2 B-45 a^4 B+2840 a A b^3+1024 b^4 B\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a+b \cos (c+d x))}{a+b}} E\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{b-a}{a+b}\right )+15 \left (-240 a^2 A b^3+10 a^4 A b-40 a^3 b^2 B-3 a^5 B-240 a b^4 B-96 A b^5\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a+b \cos (c+d x))}{a+b}} \left ((a-b) F\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{b-a}{a+b}\right )-2 b \Pi \left (-1;-\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{b-a}{a+b}\right )\right )}{1920 b^3 d \sec ^{\frac{3}{2}}(c+d x) \left (\cos (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right )\right )^{3/2} \sqrt{a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.967, size = 5172, normalized size = 6.2 \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 (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} \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 (B b^{2} \cos \left (d x + c\right )^{3} + A a^{2} +{\left (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}}{\sec \left (d x + c\right )^{\frac{3}{2}}}, 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 (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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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