Optimal. Leaf size=307 \[ \frac{4 a^3 (264 A+253 B+210 C) \sin (c+d x)}{1155 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (143 A+121 B+105 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 (99 A+143 B+105 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{693 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (143 A+121 B+105 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{4 a^3 (21 A+17 B+15 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 (11 B+6 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{99 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \sec ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.826813, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.209, Rules used = {4221, 3045, 2976, 2968, 3023, 2748, 2639, 2635, 2641} \[ \frac{4 a^3 (264 A+253 B+210 C) \sin (c+d x)}{1155 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (143 A+121 B+105 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 (99 A+143 B+105 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{693 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (143 A+121 B+105 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{4 a^3 (21 A+17 B+15 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 (11 B+6 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{99 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \sec ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3045
Rule 2976
Rule 2968
Rule 3023
Rule 2748
Rule 2639
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+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 \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3 \left (\frac{1}{2} a (11 A+3 C)+\frac{1}{2} a (11 B+6 C) \cos (c+d x)\right ) \, dx}{11 a}\\ &=\frac{2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (11 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2 \left (\frac{3}{4} a^2 (33 A+11 B+15 C)+\frac{1}{4} a^2 (99 A+143 B+105 C) \cos (c+d x)\right ) \, dx}{99 a}\\ &=\frac{2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (11 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (99 A+143 B+105 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{693 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} (a+a \cos (c+d x)) \left (\frac{15}{4} a^3 (33 A+22 B+21 C)+\frac{3}{4} a^3 (264 A+253 B+210 C) \cos (c+d x)\right ) \, dx}{693 a}\\ &=\frac{2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (11 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (99 A+143 B+105 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{693 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \left (\frac{15}{4} a^4 (33 A+22 B+21 C)+\left (\frac{15}{4} a^4 (33 A+22 B+21 C)+\frac{3}{4} a^4 (264 A+253 B+210 C)\right ) \cos (c+d x)+\frac{3}{4} a^4 (264 A+253 B+210 C) \cos ^2(c+d x)\right ) \, dx}{693 a}\\ &=\frac{4 a^3 (264 A+253 B+210 C) \sin (c+d x)}{1155 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (11 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (99 A+143 B+105 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{693 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (16 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \left (\frac{231}{8} a^4 (21 A+17 B+15 C)+\frac{45}{8} a^4 (143 A+121 B+105 C) \cos (c+d x)\right ) \, dx}{3465 a}\\ &=\frac{4 a^3 (264 A+253 B+210 C) \sin (c+d x)}{1155 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (11 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (99 A+143 B+105 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{693 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{1}{15} \left (2 a^3 (21 A+17 B+15 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{77} \left (2 a^3 (143 A+121 B+105 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{4 a^3 (21 A+17 B+15 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^3 (264 A+253 B+210 C) \sin (c+d x)}{1155 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (11 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (99 A+143 B+105 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{693 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (143 A+121 B+105 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{1}{231} \left (2 a^3 (143 A+121 B+105 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{4 a^3 (21 A+17 B+15 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^3 (143 A+121 B+105 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{231 d}+\frac{4 a^3 (264 A+253 B+210 C) \sin (c+d x)}{1155 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (11 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (99 A+143 B+105 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{693 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (143 A+121 B+105 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 1.56686, size = 174, normalized size = 0.57 \[ \frac{a^3 \sqrt{\sec (c+d x)} \left (2 \sin (2 (c+d x)) (154 (108 A+151 B+165 C) \cos (c+d x)+5 (36 (11 A+33 B+49 C) \cos (2 (c+d x))+7260 A+154 (B+3 C) \cos (3 (c+d x))+6996 B+63 C \cos (4 (c+d x))+6741 C))+960 (143 A+121 B+105 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+14784 (21 A+17 B+15 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{55440 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.523, size = 545, normalized size = 1.8 \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 \frac{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{3}}{\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 (\frac{C a^{3} \cos \left (d x + c\right )^{5} +{\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} +{\left (A + 3 \, B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} +{\left (3 \, A + 3 \, B + C\right )} a^{3} \cos \left (d x + c\right )^{2} +{\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) + A a^{3}}{\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 \frac{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{3}}{\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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