Optimal. Leaf size=230 \[ \frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (5 a A+7 a C+7 b B)}{21 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} (3 a B+3 A b+5 b C)}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (5 a A+7 a C+7 b B)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (3 a B+3 A b+5 b C)}{5 d}+\frac{2 (a B+A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d} \]
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Rubi [A] time = 0.376289, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {4221, 3031, 3021, 2748, 2636, 2641, 2639} \[ \frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (5 a A+7 a C+7 b B)}{21 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} (3 a B+3 A b+5 b C)}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (5 a A+7 a C+7 b B)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (3 a B+3 A b+5 b C)}{5 d}+\frac{2 (a B+A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3031
Rule 3021
Rule 2748
Rule 2636
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int (a+b \cos (c+d x)) \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)) \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 \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{-\frac{7}{2} (A b+a B)-\frac{1}{2} (5 a A+7 b B+7 a C) \cos (c+d x)-\frac{7}{2} b C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 (A b+a B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2 a A \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} (5 a A+7 b B+7 a C)-\frac{7}{4} (3 A b+3 a B+5 b C) \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 (A b+a B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2 a A \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{1}{7} \left ((5 a A+7 b B+7 a 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}{5} \left ((3 A b+3 a B+5 b C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 (3 A b+3 a B+5 b C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 (5 a A+7 b B+7 a C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 (A b+a B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2 a A \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{1}{21} \left ((5 a A+7 b B+7 a C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx-\frac{1}{5} \left ((3 A b+3 a B+5 b C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 (3 A b+3 a B+5 b 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{2 (5 a A+7 b B+7 a 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 (3 A b+3 a B+5 b C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 (5 a A+7 b B+7 a C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 (A b+a B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2 a A \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 3.17863, size = 191, normalized size = 0.83 \[ \frac{\sec ^{\frac{7}{2}}(c+d x) \left (40 \cos ^{\frac{7}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (5 a A+7 a C+7 b B)-168 \cos ^{\frac{7}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (3 a B+3 A b+5 b C)+2 \sin (c+d x) (21 \cos (c+d x) (13 a B+13 A b+15 b C)+10 \cos (2 (c+d x)) (5 a A+7 a C+7 b B)+110 a A+63 a B \cos (3 (c+d x))+70 a C+63 A b \cos (3 (c+d x))+70 b B+105 b C \cos (3 (c+d x)))\right )}{420 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 4.913, size = 851, normalized size = 3.7 \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] 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 )} \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 \cos \left (d x + c\right )^{3} +{\left (C a + B b\right )} \cos \left (d x + c\right )^{2} + A a +{\left (B 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 )} \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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