Optimal. Leaf size=272 \[ \frac{(3 A-13 B+33 C) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}-\frac{(9 A-49 B+119 C) \sin (c+d x)}{30 d \sec ^{\frac{3}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}+\frac{(3 A-13 B+33 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{6 a^3 d}-\frac{(9 A-49 B+119 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac{(A-B+C) \sin (c+d x)}{5 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^3}+\frac{(B-2 C) \sin (c+d x)}{3 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.685166, antiderivative size = 272, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {4221, 3041, 2977, 2748, 2639, 2635, 2641} \[ \frac{(3 A-13 B+33 C) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}-\frac{(9 A-49 B+119 C) \sin (c+d x)}{30 d \sec ^{\frac{3}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}+\frac{(3 A-13 B+33 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{6 a^3 d}-\frac{(9 A-49 B+119 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac{(A-B+C) \sin (c+d x)}{5 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^3}+\frac{(B-2 C) \sin (c+d x)}{3 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3041
Rule 2977
Rule 2748
Rule 2639
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx\\ &=-\frac{(A-B+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{5}{2}}(c+d x) \left (\frac{1}{2} a (3 A+7 B-7 C)+\frac{1}{2} a (3 A-3 B+13 C) \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{(A-B+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x)}+\frac{(B-2 C) \sin (c+d x)}{3 a d (a+a \cos (c+d x))^2 \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{3}{2}}(c+d x) \left (\frac{25}{2} a^2 (B-2 C)+\frac{3}{2} a^2 (3 A-8 B+23 C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{15 a^4}\\ &=-\frac{(A-B+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x)}+\frac{(B-2 C) \sin (c+d x)}{3 a d (a+a \cos (c+d x))^2 \sec ^{\frac{5}{2}}(c+d x)}-\frac{(9 A-49 B+119 C) \sin (c+d x)}{30 d \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \left (-\frac{3}{4} a^3 (9 A-49 B+119 C)+\frac{15}{4} a^3 (3 A-13 B+33 C) \cos (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac{(A-B+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x)}+\frac{(B-2 C) \sin (c+d x)}{3 a d (a+a \cos (c+d x))^2 \sec ^{\frac{5}{2}}(c+d x)}-\frac{(9 A-49 B+119 C) \sin (c+d x)}{30 d \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left ((3 A-13 B+33 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx}{4 a^3}-\frac{\left ((9 A-49 B+119 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{20 a^3}\\ &=-\frac{(9 A-49 B+119 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{(A-B+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x)}+\frac{(B-2 C) \sin (c+d x)}{3 a d (a+a \cos (c+d x))^2 \sec ^{\frac{5}{2}}(c+d x)}-\frac{(9 A-49 B+119 C) \sin (c+d x)}{30 d \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac{3}{2}}(c+d x)}+\frac{(3 A-13 B+33 C) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}+\frac{\left ((3 A-13 B+33 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{12 a^3}\\ &=-\frac{(9 A-49 B+119 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{10 a^3 d}+\frac{(3 A-13 B+33 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{6 a^3 d}-\frac{(A-B+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x)}+\frac{(B-2 C) \sin (c+d x)}{3 a d (a+a \cos (c+d x))^2 \sec ^{\frac{5}{2}}(c+d x)}-\frac{(9 A-49 B+119 C) \sin (c+d x)}{30 d \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac{3}{2}}(c+d x)}+\frac{(3 A-13 B+33 C) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 2.90274, size = 206, normalized size = 0.76 \[ -\frac{2 \cos ^6\left (\frac{1}{2} (c+d x)\right ) \sqrt{\sec (c+d x)} \left (-10 (3 A-13 B+33 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+6 (9 A-49 B+119 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\frac{1}{8} \left (\sin \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{3}{2} (c+d x)\right )\right ) \sec ^5\left (\frac{1}{2} (c+d x)\right ) ((72 A-292 B+782 C) \cos (c+d x)+3 (9 A-29 B+79 C) \cos (2 (c+d x))+57 A-217 B+10 C \cos (3 (c+d x))+567 C)\right )}{15 a^3 d (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.493, size = 638, normalized size = 2.4 \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(-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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (a^{3} \cos \left (d x + c\right )^{3} + 3 \, a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}\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 \frac{C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3} \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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