Optimal. Leaf size=151 \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (a (A+3 C)+3 b B)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (a B+A b-b C)}{d}+\frac{2 (a B+A b) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d} \]
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Rubi [A] time = 0.320446, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.146, Rules used = {4221, 3031, 3021, 2748, 2641, 2639} \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (a (A+3 C)+3 b B)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (a B+A b-b C)}{d}+\frac{2 (a B+A b) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3031
Rule 3021
Rule 2748
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{5}{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{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a A \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}-\frac{1}{3} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{3}{2} (A b+a B)-\frac{1}{2} (3 b B+a (A+3 C)) \cos (c+d x)-\frac{3}{2} b C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 (A b+a B) \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{2 a A \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}-\frac{1}{3} \left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{4} (-3 b B-a (A+3 C))+\frac{3}{4} (A b+a B-b C) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 (A b+a B) \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{2 a A \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}-\left ((A b+a B-b C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx-\frac{1}{3} \left ((-3 b B-a (A+3 C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 (A b+a B-b C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{d}+\frac{2 (3 b B+a (A+3 C)) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 (A b+a B) \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{2 a A \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 1.06144, size = 112, normalized size = 0.74 \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (a (A+3 C)+3 b B)-3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (a B+A b-b C)+\frac{\sin (c+d x) (3 (a B+A b) \cos (c+d x)+a A)}{\cos ^{\frac{3}{2}}(c+d x)}\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.967, size = 666, normalized size = 4.4 \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} + 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{5}{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{5}{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{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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