Optimal. Leaf size=288 \[ \frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (a^2 (5 A+7 C)+14 a b B+4 A b^2\right )}{21 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left (3 a^2 B+6 a A b+10 a b C+5 b^2 B\right )}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^2 (5 A+7 C)+14 a b B+7 b^2 (A+3 C)\right )}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^2 B+6 a A b+10 a b C+5 b^2 B\right )}{5 d}+\frac{2 a (7 a B+4 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}{7 d} \]
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Rubi [A] time = 0.698136, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.186, Rules used = {4221, 3047, 3031, 3021, 2748, 2636, 2639, 2641} \[ \frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (a^2 (5 A+7 C)+14 a b B+4 A b^2\right )}{21 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left (3 a^2 B+6 a A b+10 a b C+5 b^2 B\right )}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^2 (5 A+7 C)+14 a b B+7 b^2 (A+3 C)\right )}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^2 B+6 a A b+10 a b C+5 b^2 B\right )}{5 d}+\frac{2 a (7 a B+4 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}{7 d} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3047
Rule 3031
Rule 3021
Rule 2748
Rule 2636
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+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+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 A (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{1}{7} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x)) \left (\frac{1}{2} (4 A b+7 a B)+\frac{1}{2} (5 a A+7 b B+7 a C) \cos (c+d x)+\frac{1}{2} b (A+7 C) \cos ^2(c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 a (4 A b+7 a B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 A (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}-\frac{1}{35} \left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{5}{4} \left (4 A b^2+14 a b B+a^2 (5 A+7 C)\right )-\frac{7}{4} \left (6 a A b+3 a^2 B+5 b^2 B+10 a b C\right ) \cos (c+d x)-\frac{5}{4} b^2 (A+7 C) \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 \left (4 A b^2+14 a b B+a^2 (5 A+7 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 a (4 A b+7 a B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 A (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}-\frac{1}{105} \left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{21}{8} \left (6 a A b+3 a^2 B+5 b^2 B+10 a b C\right )-\frac{5}{8} \left (14 a b B+7 b^2 (A+3 C)+a^2 (5 A+7 C)\right ) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 \left (4 A b^2+14 a b B+a^2 (5 A+7 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 a (4 A b+7 a B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 A (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{1}{5} \left (\left (6 a A b+3 a^2 B+5 b^2 B+10 a b 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{1}{21} \left (\left (14 a b B+7 b^2 (A+3 C)+a^2 (5 A+7 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (14 a b B+7 b^2 (A+3 C)+a^2 (5 A+7 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 \left (6 a A b+3 a^2 B+5 b^2 B+10 a b C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 \left (4 A b^2+14 a b B+a^2 (5 A+7 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 a (4 A b+7 a B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 A (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}-\frac{1}{5} \left (\left (6 a A b+3 a^2 B+5 b^2 B+10 a b C\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 \left (6 a A b+3 a^2 B+5 b^2 B+10 a b C\right ) \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{2 \left (14 a b B+7 b^2 (A+3 C)+a^2 (5 A+7 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 \left (6 a A b+3 a^2 B+5 b^2 B+10 a b C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 \left (4 A b^2+14 a b B+a^2 (5 A+7 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 a (4 A b+7 a B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 A (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 4.36254, size = 221, normalized size = 0.77 \[ \frac{2 \sqrt{\sec (c+d x)} \left (21 \sin (c+d x) \left (3 a^2 B+2 a b (3 A+5 C)+5 b^2 B\right )+5 \tan (c+d x) \left (a^2 (5 A+7 C)+14 a b B+7 A b^2\right )+5 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^2 (5 A+7 C)+14 a b B+7 b^2 (A+3 C)\right )-21 \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^2 B+2 a b (3 A+5 C)+5 b^2 B\right )+15 a^2 A \tan (c+d x) \sec ^2(c+d x)+21 a (a B+2 A b) \tan (c+d x) \sec (c+d x)\right )}{105 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 5.326, size = 947, 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] 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 (b \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 b^{2} \cos \left (d x + c\right )^{4} +{\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{3} + A a^{2} +{\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )\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} + B \cos \left (d x + c\right ) + A\right )}{\left (b \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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