Optimal. Leaf size=237 \[ \frac{2 a^2 (33 A+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^2 (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{8 a^2 (3 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{4 a^2 (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{8 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2}{7 d} \]
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Rubi [A] time = 0.562733, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.257, Rules used = {4221, 3044, 2975, 2968, 3021, 2748, 2636, 2639, 2641} \[ \frac{2 a^2 (33 A+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^2 (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{8 a^2 (3 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{4 a^2 (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{8 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2}{7 d} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3044
Rule 2975
Rule 2968
Rule 3021
Rule 2748
Rule 2636
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac{9}{2}}(c+d x) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 A (a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^2 \left (2 a A+\frac{1}{2} a (A+7 C) \cos (c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx}{7 a}\\ &=\frac{8 A \left (a^2+a^2 \cos (c+d x)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 A (a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x)) \left (\frac{1}{4} a^2 (33 A+35 C)+\frac{1}{4} a^2 (9 A+35 C) \cos (c+d x)\right )}{\cos ^{\frac{5}{2}}(c+d x)} \, dx}{35 a}\\ &=\frac{8 A \left (a^2+a^2 \cos (c+d x)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 A (a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{4} a^3 (33 A+35 C)+\left (\frac{1}{4} a^3 (9 A+35 C)+\frac{1}{4} a^3 (33 A+35 C)\right ) \cos (c+d x)+\frac{1}{4} a^3 (9 A+35 C) \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx}{35 a}\\ &=\frac{2 a^2 (33 A+35 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{8 A \left (a^2+a^2 \cos (c+d x)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 A (a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{21}{4} a^3 (3 A+5 C)+\frac{5}{2} a^3 (3 A+7 C) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx}{105 a}\\ &=\frac{2 a^2 (33 A+35 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{8 A \left (a^2+a^2 \cos (c+d x)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 A (a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{1}{5} \left (2 a^2 (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx+\frac{1}{21} \left (4 a^2 (3 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{8 a^2 (3 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{4 a^2 (3 A+5 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 a^2 (33 A+35 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{8 A \left (a^2+a^2 \cos (c+d x)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 A (a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}-\frac{1}{5} \left (2 a^2 (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{4 a^2 (3 A+5 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 d}+\frac{8 a^2 (3 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{4 a^2 (3 A+5 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 a^2 (33 A+35 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{8 A \left (a^2+a^2 \cos (c+d x)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 A (a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [C] time = 4.20427, size = 399, normalized size = 1.68 \[ \frac{a^2 \csc (c) e^{-i d x} (\cos (c+d x)+1)^2 \sec ^4\left (\frac{1}{2} (c+d x)\right ) \left (7 \sqrt{2} \left (-1+e^{2 i c}\right ) (3 A+5 C) e^{2 i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right )-\frac{\left (-1+e^{2 i c}\right ) e^{-i (c-d x)} \sqrt{\sec (c+d x)} \left (6 A \left (7 e^{i (c+d x)}-20 e^{2 i (c+d x)}+63 e^{3 i (c+d x)}+20 e^{4 i (c+d x)}+77 e^{5 i (c+d x)}+10 e^{6 i (c+d x)}+21 e^{7 i (c+d x)}-10\right )+35 C \left (6 e^{i (c+d x)}+e^{2 i (c+d x)}+6 e^{3 i (c+d x)}-1\right ) \left (1+e^{2 i (c+d x)}\right )^2\right )}{2 \left (1+e^{2 i (c+d x)}\right )^3}+20 (3 A+7 C) \sin (c) e^{i d x} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{210 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.711, size = 918, normalized size = 3.9 \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 (a \cos \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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C a^{2} \cos \left (d x + c\right )^{4} + 2 \, C a^{2} \cos \left (d x + c\right )^{3} +{\left (A + C\right )} a^{2} \cos \left (d x + c\right )^{2} + 2 \, A a^{2} \cos \left (d x + c\right ) + A a^{2}\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{\left (C \cos \left (d x + c\right )^{2} + A\right )}{\left (a \cos \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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