Optimal. Leaf size=291 \[ \frac{2 a^2 (19 A+27 B+21 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d}+\frac{4 a^2 (5 A+6 B+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{4 a^2 (8 A+9 B+12 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^2 (5 A+6 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{4 a^2 (8 A+9 B+12 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 (4 A+9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d} \]
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Rubi [A] time = 0.651897, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.209, Rules used = {4221, 3043, 2975, 2968, 3021, 2748, 2636, 2641, 2639} \[ \frac{2 a^2 (19 A+27 B+21 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d}+\frac{4 a^2 (5 A+6 B+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{4 a^2 (8 A+9 B+12 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^2 (5 A+6 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{4 a^2 (8 A+9 B+12 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 (4 A+9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3043
Rule 2975
Rule 2968
Rule 3021
Rule 2748
Rule 2636
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^2 \left (A+B \cos (c+d x)+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+a \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac{11}{2}}(c+d x)} \, dx\\ &=\frac{2 A (a+a \cos (c+d x))^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 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 (\frac{1}{2} a (4 A+9 B)+\frac{3}{2} a (A+3 C) \cos (c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx}{9 a}\\ &=\frac{2 (4 A+9 B) \left (a^2+a^2 \cos (c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 A (a+a \cos (c+d x))^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 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{3}{4} a^2 (19 A+27 B+21 C)+\frac{3}{4} a^2 (11 A+9 B+21 C) \cos (c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx}{63 a}\\ &=\frac{2 (4 A+9 B) \left (a^2+a^2 \cos (c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 A (a+a \cos (c+d x))^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{3}{4} a^3 (19 A+27 B+21 C)+\left (\frac{3}{4} a^3 (11 A+9 B+21 C)+\frac{3}{4} a^3 (19 A+27 B+21 C)\right ) \cos (c+d x)+\frac{3}{4} a^3 (11 A+9 B+21 C) \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx}{63 a}\\ &=\frac{2 a^2 (19 A+27 B+21 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 (4 A+9 B) \left (a^2+a^2 \cos (c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 A (a+a \cos (c+d x))^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{45}{4} a^3 (5 A+6 B+7 C)+\frac{21}{4} a^3 (8 A+9 B+12 C) \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx}{315 a}\\ &=\frac{2 a^2 (19 A+27 B+21 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 (4 A+9 B) \left (a^2+a^2 \cos (c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 A (a+a \cos (c+d x))^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{1}{7} \left (2 a^2 (5 A+6 B+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 (2 a^2 (8 A+9 B+12 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{4 a^2 (8 A+9 B+12 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{4 a^2 (5 A+6 B+7 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 a^2 (19 A+27 B+21 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 (4 A+9 B) \left (a^2+a^2 \cos (c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 A (a+a \cos (c+d x))^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{1}{21} \left (2 a^2 (5 A+6 B+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 (2 a^2 (8 A+9 B+12 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{4 a^2 (8 A+9 B+12 C) \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^2 (5 A+6 B+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 (8 A+9 B+12 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{4 a^2 (5 A+6 B+7 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 a^2 (19 A+27 B+21 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 (4 A+9 B) \left (a^2+a^2 \cos (c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 A (a+a \cos (c+d x))^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}\\ \end{align*}
Mathematica [A] time = 2.29949, size = 209, normalized size = 0.72 \[ \frac{a^2 \sec ^{\frac{9}{2}}(c+d x) \left (240 (5 A+6 B+7 C) \cos ^{\frac{9}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-336 (8 A+9 B+12 C) \cos ^{\frac{9}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+2 \sin (c+d x) (90 (9 A+8 B+7 C) \cos (c+d x)+14 (64 A+72 B+81 C) \cos (2 (c+d x))+150 A \cos (3 (c+d x))+168 A \cos (4 (c+d x))+868 A+180 B \cos (3 (c+d x))+189 B \cos (4 (c+d x))+819 B+210 C \cos (3 (c+d x))+252 C \cos (4 (c+d x))+882 C)\right )}{1260 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 5.187, size = 1181, normalized size = 4.1 \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} + B \cos \left (d x + c\right ) + A\right )}{\left (a \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 a^{2} \cos \left (d x + c\right )^{4} +{\left (B + 2 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} +{\left (A + 2 \, B + C\right )} a^{2} \cos \left (d x + c\right )^{2} +{\left (2 \, A + B\right )} a^{2} \cos \left (d x + c\right ) + A a^{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} + B \cos \left (d x + c\right ) + A\right )}{\left (a \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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