Optimal. Leaf size=674 \[ \frac{\left (4 a^2 b^2 (1180 A+809 C)+1330 a^3 b B-15 a^4 C+3560 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{2 a}{a+b}\right )}{1920 b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\sin (c+d x) \left (590 a^2 b B+15 a^3 C+4 a b^2 (260 A+193 C)+360 b^3 B\right ) \sqrt{a+b \sec (c+d x)}}{960 b d \cos ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt{a+b \sec (c+d x)}}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\sin (c+d x) \left (12 a^2 b^2 (220 A+141 C)+150 a^3 b B-45 a^4 C+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt{a+b \sec (c+d x)}}{1920 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \left (12 a^2 b^2 (220 A+141 C)+150 a^3 b B-45 a^4 C+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{1920 b^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{\left (-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B+10 a^4 b B-3 a^5 C-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{128 b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{(a C+2 b B) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)} \]
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Rubi [A] time = 2.97369, antiderivative size = 674, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 14, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.311, Rules used = {4265, 4096, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661} \[ \frac{\sin (c+d x) \left (590 a^2 b B+15 a^3 C+4 a b^2 (260 A+193 C)+360 b^3 B\right ) \sqrt{a+b \sec (c+d x)}}{960 b d \cos ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt{a+b \sec (c+d x)}}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\sin (c+d x) \left (12 a^2 b^2 (220 A+141 C)+150 a^3 b B-45 a^4 C+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt{a+b \sec (c+d x)}}{1920 b^2 d \sqrt{\cos (c+d x)}}+\frac{\left (4 a^2 b^2 (1180 A+809 C)+1330 a^3 b B-15 a^4 C+3560 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{1920 b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \left (12 a^2 b^2 (220 A+141 C)+150 a^3 b B-45 a^4 C+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{1920 b^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{\left (-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B+10 a^4 b B-3 a^5 C-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{128 b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{(a C+2 b B) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4265
Rule 4096
Rule 4102
Rule 4108
Rule 3859
Rule 2807
Rule 2805
Rule 4035
Rule 3856
Rule 2655
Rule 2653
Rule 3858
Rule 2663
Rule 2661
Rubi steps
\begin{align*} \int \frac{(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac{3}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{3}{2}}(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{C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{1}{5} \left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac{1}{2} a (10 A+3 C)+(5 A b+5 a B+4 b C) \sec (c+d x)+\frac{5}{2} (2 b B+a C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{(2 b B+a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{1}{20} \left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \left (\frac{1}{4} a (80 a A+30 b B+39 a C)+\frac{1}{2} \left (40 a^2 B+30 b^2 B+a b (80 A+59 C)\right ) \sec (c+d x)+\frac{1}{4} \left (80 A b^2+110 a b B+15 a^2 C+64 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{\left (80 A b^2+110 a b B+15 a^2 C+64 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{(2 b B+a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{1}{60} \left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x) \left (\frac{3}{8} a \left (170 a b B+16 b^2 (5 A+4 C)+a^2 (160 A+93 C)\right )+\frac{1}{4} \left (240 a^3 B+490 a b^2 B+32 b^3 (5 A+4 C)+a^2 (720 A b+501 b C)\right ) \sec (c+d x)+\frac{1}{8} \left (590 a^2 b B+360 b^3 B+15 a^3 C+4 a b^2 (260 A+193 C)\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{\left (80 A b^2+110 a b B+15 a^2 C+64 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\left (590 a^2 b B+360 b^3 B+15 a^3 C+4 a b^2 (260 A+193 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{960 b d \cos ^{\frac{3}{2}}(c+d x)}+\frac{(2 b B+a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{\sec (c+d x)} \left (\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 ) \sec (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 ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{120 b}\\ &=\frac{\left (80 A b^2+110 a b B+15 a^2 C+64 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\left (590 a^2 b B+360 b^3 B+15 a^3 C+4 a b^2 (260 A+193 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{960 b d \cos ^{\frac{3}{2}}(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 \sec (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt{\cos (c+d x)}}+\frac{(2 b B+a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{1}{32} 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}{16} 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 ) \sec (c+d x)-\frac{15}{32} \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 ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx}{120 b^2}\\ &=\frac{\left (80 A b^2+110 a b B+15 a^2 C+64 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\left (590 a^2 b B+360 b^3 B+15 a^3 C+4 a b^2 (260 A+193 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{960 b d \cos ^{\frac{3}{2}}(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 \sec (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt{\cos (c+d x)}}+\frac{(2 b B+a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{1}{32} 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}{16} 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 ) \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx}{120 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{\sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{256 b^2}\\ &=\frac{\left (80 A b^2+110 a b B+15 a^2 C+64 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\left (590 a^2 b B+360 b^3 B+15 a^3 C+4 a b^2 (260 A+193 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{960 b d \cos ^{\frac{3}{2}}(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 \sec (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt{\cos (c+d x)}}+\frac{(2 b B+a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\left (\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{\sqrt{a+b \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx}{3840 b^2}+\frac{\left (\left (1330 a^3 b B+3560 a b^3 B-15 a^4 C+256 b^4 (5 A+4 C)+4 a^2 b^2 (1180 A+809 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+b \sec (c+d x)}} \, dx}{3840 b}-\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{b+a \cos (c+d x)}\right ) \int \frac{\sec (c+d x)}{\sqrt{b+a \cos (c+d x)}} \, dx}{256 b^2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}\\ &=\frac{\left (80 A b^2+110 a b B+15 a^2 C+64 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\left (590 a^2 b B+360 b^3 B+15 a^3 C+4 a b^2 (260 A+193 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{960 b d \cos ^{\frac{3}{2}}(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 \sec (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt{\cos (c+d x)}}+\frac{(2 b B+a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\left (\left (1330 a^3 b B+3560 a b^3 B-15 a^4 C+256 b^4 (5 A+4 C)+4 a^2 b^2 (1180 A+809 C)\right ) \sqrt{b+a \cos (c+d x)}\right ) \int \frac{1}{\sqrt{b+a \cos (c+d x)}} \, dx}{3840 b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\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{\frac{b+a \cos (c+d x)}{a+b}}\right ) \int \frac{\sec (c+d x)}{\sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}}} \, dx}{256 b^2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left (\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{a+b \sec (c+d x)}\right ) \int \sqrt{b+a \cos (c+d x)} \, dx}{3840 b^2 \sqrt{b+a \cos (c+d x)}}\\ &=-\frac{\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{\frac{b+a \cos (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{128 b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left (80 A b^2+110 a b B+15 a^2 C+64 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\left (590 a^2 b B+360 b^3 B+15 a^3 C+4 a b^2 (260 A+193 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{960 b d \cos ^{\frac{3}{2}}(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 \sec (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt{\cos (c+d x)}}+\frac{(2 b B+a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\left (\left (1330 a^3 b B+3560 a b^3 B-15 a^4 C+256 b^4 (5 A+4 C)+4 a^2 b^2 (1180 A+809 C)\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}}} \, dx}{3840 b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left (\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{a+b \sec (c+d x)}\right ) \int \sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}} \, dx}{3840 b^2 \sqrt{\frac{b+a \cos (c+d x)}{a+b}}}\\ &=\frac{\left (1330 a^3 b B+3560 a b^3 B-15 a^4 C+256 b^4 (5 A+4 C)+4 a^2 b^2 (1180 A+809 C)\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{1920 b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\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{\frac{b+a \cos (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{128 b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \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{\cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right ) \sqrt{a+b \sec (c+d x)}}{1920 b^2 d \sqrt{\frac{b+a \cos (c+d x)}{a+b}}}+\frac{\left (80 A b^2+110 a b B+15 a^2 C+64 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\left (590 a^2 b B+360 b^3 B+15 a^3 C+4 a b^2 (260 A+193 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{960 b d \cos ^{\frac{3}{2}}(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 \sec (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt{\cos (c+d x)}}+\frac{(2 b B+a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}\\ \end{align*}
Mathematica [C] time = 37.5182, size = 211844, normalized size = 314.31 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 1.381, size = 5292, 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(-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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Fricas [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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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 \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 )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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