Optimal. Leaf size=292 \[ \frac{2 \left (a^2 (7 A+9 C)+4 A b^2\right ) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{45 d}+\frac{2 \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}-\frac{2 \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{4 a b (5 A+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{4 a b (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{8 a A b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d} \]
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Rubi [A] time = 0.629894, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {4221, 3048, 3031, 3021, 2748, 2636, 2641, 2639} \[ \frac{2 \left (a^2 (7 A+9 C)+4 A b^2\right ) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{45 d}+\frac{2 \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}-\frac{2 \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{4 a b (5 A+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{4 a b (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{8 a A b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3048
Rule 3031
Rule 3021
Rule 2748
Rule 2636
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac{11}{2}}(c+d x) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac{11}{2}}(c+d x)} \, dx\\ &=\frac{2 A (a+b \cos (c+d x))^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{1}{9} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x)) \left (2 A b+\frac{1}{2} a (7 A+9 C) \cos (c+d x)+\frac{3}{2} b (A+3 C) \cos ^2(c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{8 a A b \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 A (a+b \cos (c+d x))^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}-\frac{1}{63} \left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{7}{4} \left (4 A b^2+a^2 (7 A+9 C)\right )-\frac{9}{2} a b (5 A+7 C) \cos (c+d x)-\frac{21}{4} b^2 (A+3 C) \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 \left (4 A b^2+a^2 (7 A+9 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{8 a A b \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 A (a+b \cos (c+d x))^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}-\frac{1}{315} \left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{45}{4} a b (5 A+7 C)-\frac{21}{8} \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 \left (4 A b^2+a^2 (7 A+9 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{8 a A b \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 A (a+b \cos (c+d x))^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{1}{7} \left (2 a b (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\cos ^{\frac{5}{2}}(c+d x)} \, dx+\frac{1}{15} \left (\left (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 \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{4 a b (5 A+7 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 \left (4 A b^2+a^2 (7 A+9 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{8 a A b \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 A (a+b \cos (c+d x))^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{1}{21} \left (2 a b (5 A+7 C) \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 (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 (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{4 a b (5 A+7 C) \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 \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{4 a b (5 A+7 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 \left (4 A b^2+a^2 (7 A+9 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{8 a A b \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 A (a+b \cos (c+d x))^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}\\ \end{align*}
Mathematica [A] time = 6.41383, size = 286, normalized size = 0.98 \[ \frac{\sqrt{\sec (c+d x)} \left (\frac{2}{15} \left (7 a^2 A+9 a^2 C+9 A b^2+15 b^2 C\right ) \sin (c+d x)+\frac{2}{45} \sec ^2(c+d x) \left (7 a^2 A \sin (c+d x)+9 a^2 C \sin (c+d x)+9 A b^2 \sin (c+d x)\right )+\frac{2}{9} a^2 A \tan (c+d x) \sec ^3(c+d x)+\frac{4}{21} \sec (c+d x) (5 a A b \sin (c+d x)+7 a b C \sin (c+d x))+\frac{4}{7} a A b \tan (c+d x) \sec ^2(c+d x)\right )}{d}+\frac{\frac{2 \left (-49 a^2 A-63 a^2 C-63 A b^2-105 b^2 C\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}+2 (50 a A b+70 a b C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{105 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 5.346, size = 1179, normalized size = 4. \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac{11}{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 ({\left (C b^{2} \cos \left (d x + c\right )^{4} + 2 \, C a b \cos \left (d x + c\right )^{3} + 2 \, A a b \cos \left (d x + c\right ) + A a^{2} +{\left (C a^{2} + A b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sec \left (d x + c\right )^{\frac{11}{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{\left (C \cos \left (d x + c\right )^{2} + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac{11}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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