Optimal. Leaf size=667 \[ -\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (-a^2 b^2 (3 A-61 C)+15 a^3 b B-35 a^4 C-33 a b^3 B+b^4 (21 A-8 C)\right )}{12 b^3 d \left (a^2-b^2\right )^2}-\frac{\sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (a^2 b^2 (A+13 C)+3 a^3 b B-7 a^4 C-9 a b^3 B+5 A b^4\right )}{4 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (-a^2 b^2 (3 A-61 C)+15 a^3 b B-35 a^4 C-33 a b^3 B+b^4 (21 A-8 C)\right )}{12 b^3 d \left (a^2-b^2\right )^2}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left (-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+15 a^4 b B-35 a^5 C+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{4 b^4 d \left (a^2-b^2\right )^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+15 a^4 b B-35 a^5 C+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{4 b^4 d \left (a^2-b^2\right )^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+38 a^3 b^3 B-15 a^5 b B+35 a^6 C-35 a b^5 B+15 A b^6\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 b^4 d (a-b)^2 (a+b)^3} \]
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Rubi [A] time = 2.24793, antiderivative size = 667, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.209, Rules used = {4098, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ -\frac{\sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (a^2 b^2 (A+13 C)+3 a^3 b B-7 a^4 C-9 a b^3 B+5 A b^4\right )}{4 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (-a^2 b^2 (3 A-61 C)+15 a^3 b B-35 a^4 C-33 a b^3 B+b^4 (21 A-8 C)\right )}{12 b^3 d \left (a^2-b^2\right )^2}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left (-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+15 a^4 b B-35 a^5 C+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{4 b^4 d \left (a^2-b^2\right )^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-a^2 b^2 (3 A-61 C)+15 a^3 b B-35 a^4 C-33 a b^3 B+b^4 (21 A-8 C)\right )}{12 b^3 d \left (a^2-b^2\right )^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+15 a^4 b B-35 a^5 C+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{4 b^4 d \left (a^2-b^2\right )^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+38 a^3 b^3 B-15 a^5 b B+35 a^6 C-35 a b^5 B+15 A b^6\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 b^4 d (a-b)^2 (a+b)^3} \]
Antiderivative was successfully verified.
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Rule 4098
Rule 4102
Rule 4106
Rule 3849
Rule 2805
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{\sec ^{\frac{7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx &=-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac{\int \frac{\sec ^{\frac{5}{2}}(c+d x) \left (\frac{5}{2} \left (A b^2-a (b B-a C)\right )+2 b (b B-a (A+C)) \sec (c+d x)-\frac{1}{2} \left (3 A b^2-3 a b B+7 a^2 C-4 b^2 C\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac{\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left (\frac{3}{4} \left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right )+b \left (a^2 b B+2 b^3 B+a^3 C-a b^2 (3 A+4 C)\right ) \sec (c+d x)-\frac{1}{4} \left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac{\int \frac{\sqrt{\sec (c+d x)} \left (-\frac{1}{8} a \left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right )+\frac{1}{2} b \left (3 a^3 b B-12 a b^3 B-7 a^4 C+2 b^4 (3 A+C)+a^2 b^2 (3 A+14 C)\right ) \sec (c+d x)+\frac{3}{8} \left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{3 b^3 \left (a^2-b^2\right )^2}\\ &=\frac{\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac{2 \int \frac{-\frac{3}{16} a \left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right )-\frac{1}{4} b \left (15 a^4 b B-30 a^2 b^3 B+6 b^5 B-a^3 b^2 (3 A-64 C)+4 a b^4 (3 A-5 C)-35 a^5 C\right ) \sec (c+d x)-\frac{1}{16} \left (45 a^5 b B-99 a^3 b^3 B+72 a b^5 B-a^4 b^2 (9 A-223 C)+a^2 b^4 (15 A-128 C)-105 a^6 C-8 b^6 (3 A+C)\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{3 b^4 \left (a^2-b^2\right )^2}\\ &=\frac{\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac{2 \int \frac{-\frac{3}{16} a^2 \left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right )-\left (-\frac{3}{16} a b \left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right )+\frac{1}{4} a b \left (15 a^4 b B-30 a^2 b^3 B+6 b^5 B-a^3 b^2 (3 A-64 C)+4 a b^4 (3 A-5 C)-35 a^5 C\right )\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{3 a^2 b^4 \left (a^2-b^2\right )^2}+\frac{\left (15 A b^6-15 a^5 b B+38 a^3 b^3 B-35 a b^5 B+a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+35 a^6 C\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{8 b^4 \left (a^2-b^2\right )^2}\\ &=\frac{\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac{\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \int \sqrt{\sec (c+d x)} \, dx}{24 b^3 \left (a^2-b^2\right )^2}-\frac{\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{8 b^4 \left (a^2-b^2\right )^2}+\frac{\left (\left (15 A b^6-15 a^5 b B+38 a^3 b^3 B-35 a b^5 B+a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+35 a^6 C\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{8 b^4 \left (a^2-b^2\right )^2}\\ &=\frac{\left (15 A b^6-15 a^5 b B+38 a^3 b^3 B-35 a b^5 B+a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+35 a^6 C\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{4 (a-b)^2 b^4 (a+b)^3 d}+\frac{\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac{\left (\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{24 b^3 \left (a^2-b^2\right )^2}-\frac{\left (\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{8 b^4 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{4 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{12 b^3 \left (a^2-b^2\right )^2 d}+\frac{\left (15 A b^6-15 a^5 b B+38 a^3 b^3 B-35 a b^5 B+a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+35 a^6 C\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{4 (a-b)^2 b^4 (a+b)^3 d}+\frac{\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 7.78109, size = 1161, normalized size = 1.74 \[ \frac{\sec (c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (-\frac{2 \left (-48 B b^6-96 a A b^5+160 a C b^5+240 a^2 B b^4+24 a^3 A b^3-512 a^3 C b^3-120 a^4 B b^2+280 a^5 C b\right ) \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac{2 \left (315 C a^6-135 b B a^5+27 A b^2 a^4-641 b^2 C a^4+285 b^3 B a^3-57 A b^4 a^2+328 b^4 C a^2-168 b^5 B a+48 A b^6+16 b^6 C\right ) \left (\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right ),-1\right )+\Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right ) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}-\frac{2 \left (105 C a^6-45 b B a^5+9 A b^2 a^4-195 b^2 C a^4+87 b^3 B a^3-27 A b^4 a^2+72 b^4 C a^2-24 b^5 B a\right ) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left (\Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2-2 b \sec ^2(c+d x) a+2 b a+2 b E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a+(a-2 b) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right ),-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 b^2 \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right ) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right ) \sqrt{\sec (c+d x)} \left (2-\sec ^2(c+d x)\right )}\right ) (b+a \cos (c+d x))^3}{24 (a-b)^2 b^4 (a+b)^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (-\frac{\left (35 C a^5-15 b B a^4+3 A b^2 a^3-65 b^2 C a^3+29 b^3 B a^2-9 A b^4 a+24 b^4 C a-8 b^5 B\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^2}+\frac{-C \sin (c+d x) a^3+b B \sin (c+d x) a^2-A b^2 \sin (c+d x) a}{b^2 \left (b^2-a^2\right ) (b+a \cos (c+d x))^2}+\frac{9 C \sin (c+d x) a^5-5 b B \sin (c+d x) a^4+A b^2 \sin (c+d x) a^3-15 b^2 C \sin (c+d x) a^3+11 b^3 B \sin (c+d x) a^2-7 A b^4 \sin (c+d x) a}{2 b^3 \left (b^2-a^2\right )^2 (b+a \cos (c+d x))}+\frac{4 C \tan (c+d x)}{3 b^3}\right ) (b+a \cos (c+d x))^3}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 20.032, size = 2185, normalized size = 3.3 \begin{align*} \text{result 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 )} \sec \left (d x + c\right )^{\frac{7}{2}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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