Optimal. Leaf size=257 \[ -\frac{4 a^3 (35 A-41 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}-\frac{2 (35 A-11 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{35 d \sqrt{\sec (c+d x)}}-\frac{2 (7 A-C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 a d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (35 A+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{4 a^3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^3}{d} \]
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Rubi [A] time = 0.679621, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {4221, 3044, 2976, 2968, 3023, 2748, 2641, 2639} \[ -\frac{4 a^3 (35 A-41 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}-\frac{2 (35 A-11 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{35 d \sqrt{\sec (c+d x)}}-\frac{2 (7 A-C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 a d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (35 A+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{4 a^3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^3}{d} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3044
Rule 2976
Rule 2968
Rule 3023
Rule 2748
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac{3}{2}}(c+d x) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 A (a+a \cos (c+d x))^3 \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^3 \left (3 a A-\frac{1}{2} a (7 A-C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{a}\\ &=-\frac{2 (7 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \cos (c+d x))^3 \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^2 \left (\frac{1}{4} a^2 (35 A+C)-\frac{1}{4} a^2 (35 A-11 C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{7 a}\\ &=-\frac{2 (7 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d \sqrt{\sec (c+d x)}}-\frac{2 (35 A-11 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{35 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \cos (c+d x))^3 \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x)) \left (\frac{1}{2} a^3 (35 A+4 C)-\frac{1}{4} a^3 (35 A-41 C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{35 a}\\ &=-\frac{2 (7 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d \sqrt{\sec (c+d x)}}-\frac{2 (35 A-11 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{35 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \cos (c+d x))^3 \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{2} a^4 (35 A+4 C)+\left (-\frac{1}{4} a^4 (35 A-41 C)+\frac{1}{2} a^4 (35 A+4 C)\right ) \cos (c+d x)-\frac{1}{4} a^4 (35 A-41 C) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx}{35 a}\\ &=-\frac{4 a^3 (35 A-41 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}-\frac{2 (7 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d \sqrt{\sec (c+d x)}}-\frac{2 (35 A-11 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{35 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \cos (c+d x))^3 \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{\left (16 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{5}{8} a^4 (35 A+13 C)+\frac{21}{8} a^4 (5 A+7 C) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx}{105 a}\\ &=-\frac{4 a^3 (35 A-41 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}-\frac{2 (7 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d \sqrt{\sec (c+d x)}}-\frac{2 (35 A-11 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{35 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \cos (c+d x))^3 \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{1}{5} \left (2 a^3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{21} \left (2 a^3 (35 A+13 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 (5 A+7 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{4 a^3 (35 A+13 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{4 a^3 (35 A-41 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}-\frac{2 (7 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d \sqrt{\sec (c+d x)}}-\frac{2 (35 A-11 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{35 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \cos (c+d x))^3 \sqrt{\sec (c+d x)} \sin (c+d x)}{d}\\ \end{align*}
Mathematica [C] time = 1.77697, size = 218, normalized size = 0.85 \[ \frac{a^3 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (-112 i (5 A+7 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right )+80 (35 A+13 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+840 A \sin (c+d x)+140 A \sin (2 (c+d x))+1680 i A \cos (c+d x)+126 C \sin (c+d x)+550 C \sin (2 (c+d x))+126 C \sin (3 (c+d x))+15 C \sin (4 (c+d x))+2352 i C \cos (c+d x)\right )}{420 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.214, size = 569, normalized size = 2.2 \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 (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac{3}{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 a^{3} \cos \left (d x + c\right )^{5} + 3 \, C a^{3} \cos \left (d x + c\right )^{4} +{\left (A + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} +{\left (3 \, A + C\right )} a^{3} \cos \left (d x + c\right )^{2} + 3 \, A a^{3} \cos \left (d x + c\right ) + A a^{3}\right )} \sec \left (d x + c\right )^{\frac{3}{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} + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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