Optimal. Leaf size=290 \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (5 a^2 B+10 a A b+14 a b C+7 b^2 B\right )}{21 d}+\frac{2 \sin (c+d x) \left (a^2 (7 A+9 C)+18 a b B+4 A b^2\right )}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (5 a^2 B+10 a A b+14 a b C+7 b^2 B\right )}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right )}{15 d}+\frac{2 a (9 a B+4 A b) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac{7}{2}}(c+d x)} \]
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Rubi [A] time = 0.552037, antiderivative size = 290, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.186, Rules used = {4094, 4074, 4047, 3769, 3771, 2641, 4045, 2639} \[ \frac{2 \sin (c+d x) \left (a^2 (7 A+9 C)+18 a b B+4 A b^2\right )}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (5 a^2 B+10 a A b+14 a b C+7 b^2 B\right )}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (5 a^2 B+10 a A b+14 a b C+7 b^2 B\right )}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right )}{15 d}+\frac{2 a (9 a B+4 A b) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4094
Rule 4074
Rule 4047
Rule 3769
Rule 3771
Rule 2641
Rule 4045
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac{9}{2}}(c+d x)} \, dx &=\frac{2 A (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2}{9} \int \frac{(a+b \sec (c+d x)) \left (\frac{1}{2} (4 A b+9 a B)+\frac{1}{2} (7 a A+9 b B+9 a C) \sec (c+d x)+\frac{3}{2} b (A+3 C) \sec ^2(c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 a (4 A b+9 a B) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}-\frac{4}{63} \int \frac{-\frac{7}{4} \left (4 A b^2+18 a b B+a^2 (7 A+9 C)\right )-\frac{9}{4} \left (10 a A b+5 a^2 B+7 b^2 B+14 a b C\right ) \sec (c+d x)-\frac{21}{4} b^2 (A+3 C) \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a (4 A b+9 a B) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}-\frac{4}{63} \int \frac{-\frac{7}{4} \left (4 A b^2+18 a b B+a^2 (7 A+9 C)\right )-\frac{21}{4} b^2 (A+3 C) \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx-\frac{1}{7} \left (-10 a A b-5 a^2 B-7 b^2 B-14 a b C\right ) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a (4 A b+9 a B) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (4 A b^2+18 a b B+a^2 (7 A+9 C)\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (10 a A b+5 a^2 B+7 b^2 B+14 a b C\right ) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}-\frac{1}{21} \left (-10 a A b-5 a^2 B-7 b^2 B-14 a b C\right ) \int \sqrt{\sec (c+d x)} \, dx-\frac{1}{15} \left (-18 a b B-3 b^2 (3 A+5 C)-a^2 (7 A+9 C)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 a (4 A b+9 a B) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (4 A b^2+18 a b B+a^2 (7 A+9 C)\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (10 a A b+5 a^2 B+7 b^2 B+14 a b C\right ) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}-\frac{1}{21} \left (\left (-10 a A b-5 a^2 B-7 b^2 B-14 a b C\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx-\frac{1}{15} \left (\left (-18 a b B-3 b^2 (3 A+5 C)-a^2 (7 A+9 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 \left (18 a b B+3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 \left (10 a A b+5 a^2 B+7 b^2 B+14 a b C\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{21 d}+\frac{2 a (4 A b+9 a B) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (4 A b^2+18 a b B+a^2 (7 A+9 C)\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (10 a A b+5 a^2 B+7 b^2 B+14 a b C\right ) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 3.65549, size = 286, normalized size = 0.99 \[ \frac{(a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (120 \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (5 a^2 B+2 a b (5 A+7 C)+7 b^2 B\right )+\sin (2 (c+d x)) \left (7 \cos (c+d x) \left (a^2 (43 A+36 C)+72 a b B+36 A b^2\right )+5 \left (7 a^2 A \cos (3 (c+d x))+78 a^2 B+18 a (a B+2 A b) \cos (2 (c+d x))+156 a A b+168 a b C+84 b^2 B\right )\right )+168 \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right )\right )}{630 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+b)^2 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.181, size = 784, normalized size = 2.7 \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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{C b^{2} \sec \left (d x + c\right )^{4} +{\left (2 \, C a b + B b^{2}\right )} \sec \left (d x + c\right )^{3} + A a^{2} +{\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \sec \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, A a b\right )} \sec \left (d x + c\right )}{\sec \left (d x + c\right )^{\frac{9}{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 (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 )}^{2}}{\sec \left (d x + c\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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