Optimal. Leaf size=211 \[ \frac{2 \left (4 a^2 C+b^2 (7 A+5 C)\right ) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left (7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{4 a b (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{8 a b C \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 0.513227, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4221, 3050, 3033, 3023, 2748, 2641, 2639} \[ \frac{2 \left (4 a^2 C+b^2 (7 A+5 C)\right ) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left (7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{4 a b (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{8 a b C \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3050
Rule 3033
Rule 3023
Rule 2748
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sqrt{\sec (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+C \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 C (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt{\sec (c+d x)}}+\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} a (7 A+C)+\frac{1}{2} b (7 A+5 C) \cos (c+d x)+2 a C \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{8 a b C \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt{\sec (c+d x)}}+\frac{1}{35} \left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{5}{4} a^2 (7 A+C)+\frac{7}{2} a b (5 A+3 C) \cos (c+d x)+\frac{5}{4} \left (4 a^2 C+b^2 (7 A+5 C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{8 a b C \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (4 a^2 C+b^2 (7 A+5 C)\right ) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt{\sec (c+d x)}}+\frac{1}{105} \left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{5}{8} \left (7 a^2 (3 A+C)+b^2 (7 A+5 C)\right )+\frac{21}{4} a b (5 A+3 C) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{8 a b C \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (4 a^2 C+b^2 (7 A+5 C)\right ) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt{\sec (c+d x)}}+\frac{1}{5} \left (2 a b (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{21} \left (\left (7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{4 a b (5 A+3 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 \left (7 a^2 (3 A+C)+b^2 (7 A+5 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{8 a b C \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (4 a^2 C+b^2 (7 A+5 C)\right ) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 1.07578, size = 148, normalized size = 0.7 \[ \frac{\sqrt{\sec (c+d x)} \left (\sin (2 (c+d x)) \left (70 a^2 C+84 a b C \cos (c+d x)+70 A b^2+15 b^2 C \cos (2 (c+d x))+65 b^2 C\right )+20 \left (7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+168 a b (5 A+3 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{210 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.145, size = 532, normalized size = 2.5 \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} + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{2} \sqrt{\sec \left (d x + c\right )}\,{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} + 2 \, C a b \cos \left (d x + c\right )^{3} + 2 \, A a b \cos \left (d x + c\right ) + A a^{2} +{\left (C a^{2} + A b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt{\sec \left (d x + c\right )}, 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} + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{2} \sqrt{\sec \left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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