Optimal. Leaf size=177 \[ \frac{2 \left (a^2 A+6 a b B+3 A b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 \left (a^2 B+2 a A b-b^2 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a (3 a B+5 A b) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+b)}{3 d} \]
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Rubi [A] time = 0.351616, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {2960, 4026, 4047, 3771, 2641, 4046, 2639} \[ \frac{2 \left (a^2 A+6 a b B+3 A b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 \left (a^2 B+2 a A b-b^2 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a (3 a B+5 A b) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+b)}{3 d} \]
Antiderivative was successfully verified.
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Rule 2960
Rule 4026
Rule 4047
Rule 3771
Rule 2641
Rule 4046
Rule 2639
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx &=\int \frac{(b+a \sec (c+d x))^2 (B+A \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 a A \sqrt{\sec (c+d x)} (b+a \sec (c+d x)) \sin (c+d x)}{3 d}+\frac{2}{3} \int \frac{-\frac{1}{2} b (a A-3 b B)+\frac{1}{2} \left (a^2 A+3 A b^2+6 a b B\right ) \sec (c+d x)+\frac{1}{2} a (5 A b+3 a B) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 a A \sqrt{\sec (c+d x)} (b+a \sec (c+d x)) \sin (c+d x)}{3 d}+\frac{2}{3} \int \frac{-\frac{1}{2} b (a A-3 b B)+\frac{1}{2} a (5 A b+3 a B) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{3} \left (a^2 A+3 A b^2+6 a b B\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{2 a (5 A b+3 a B) \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d}+\frac{2 a A \sqrt{\sec (c+d x)} (b+a \sec (c+d x)) \sin (c+d x)}{3 d}+\left (-2 a A b-a^2 B+b^2 B\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{3} \left (\left (a^2 A+3 A b^2+6 a b B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (a^2 A+3 A b^2+6 a b B\right ) \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 (5 A b+3 a B) \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d}+\frac{2 a A \sqrt{\sec (c+d x)} (b+a \sec (c+d x)) \sin (c+d x)}{3 d}+\left (\left (-2 a A b-a^2 B+b^2 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 \left (2 a A b+a^2 B-b^2 B\right ) \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 \left (a^2 A+3 A b^2+6 a b B\right ) \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 (5 A b+3 a B) \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d}+\frac{2 a A \sqrt{\sec (c+d x)} (b+a \sec (c+d x)) \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 1.11888, size = 125, normalized size = 0.71 \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (\left (a^2 A+6 a b B+3 A b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-3 \left (a^2 B+2 a A b-b^2 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\frac{a \sin (c+d x) (3 (a B+2 A b) \cos (c+d x)+a A)}{\cos ^{\frac{3}{2}}(c+d x)}\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 7.943, size = 677, normalized size = 3.8 \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 (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{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 (B b^{2} \cos \left (d x + c\right )^{3} + A a^{2} +{\left (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{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 (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{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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