Optimal. Leaf size=254 \[ \frac{(20 A-35 B+56 C) \sin (c+d x)}{15 a^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 (A-2 B+3 C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{(A-2 B+3 C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1) \sec ^{\frac{5}{2}}(c+d x)}-\frac{5 (A-2 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^2 d}+\frac{(20 A-35 B+56 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a^2 d}-\frac{(A-B+C) \sin (c+d x)}{3 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.52053, antiderivative size = 254, 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, 2635, 2641, 2639} \[ \frac{(20 A-35 B+56 C) \sin (c+d x)}{15 a^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 (A-2 B+3 C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{(A-2 B+3 C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1) \sec ^{\frac{5}{2}}(c+d x)}-\frac{5 (A-2 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^2 d}+\frac{(20 A-35 B+56 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a^2 d}-\frac{(A-B+C) \sin (c+d x)}{3 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3041
Rule 2977
Rule 2748
Rule 2635
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2 \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))^2} \, dx\\ &=-\frac{(A-B+C) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2 \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 (A-7 B+7 C)+\frac{1}{2} a (5 A-5 B+11 C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=-\frac{(A-B+C) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}-\frac{(A-2 B+3 C) \sin (c+d x)}{a^2 d (1+\cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) \left (-\frac{15}{2} a^2 (A-2 B+3 C)+\frac{1}{2} a^2 (20 A-35 B+56 C) \cos (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac{(A-B+C) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}-\frac{(A-2 B+3 C) \sin (c+d x)}{a^2 d (1+\cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}-\frac{\left (5 (A-2 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx}{2 a^2}+\frac{\left ((20 A-35 B+56 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{5}{2}}(c+d x) \, dx}{6 a^2}\\ &=-\frac{(A-B+C) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}-\frac{(A-2 B+3 C) \sin (c+d x)}{a^2 d (1+\cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}+\frac{(20 A-35 B+56 C) \sin (c+d x)}{15 a^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 (A-2 B+3 C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{\left (5 (A-2 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{6 a^2}+\frac{\left ((20 A-35 B+56 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{10 a^2}\\ &=\frac{(20 A-35 B+56 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 a^2 d}-\frac{5 (A-2 B+3 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{3 a^2 d}-\frac{(A-B+C) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}-\frac{(A-2 B+3 C) \sin (c+d x)}{a^2 d (1+\cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}+\frac{(20 A-35 B+56 C) \sin (c+d x)}{15 a^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 (A-2 B+3 C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 3.64196, size = 200, normalized size = 0.79 \[ \frac{2 \cos ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{\sec (c+d x)} \left (-50 (A-2 B+3 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+6 (20 A-35 B+56 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\frac{1}{4} \left (\sin \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{3}{2} (c+d x)\right )\right ) \sec ^3\left (\frac{1}{2} (c+d x)\right ) ((60 A-130 B+179 C) \cos (c+d x)+50 A+(8 C-10 B) \cos (2 (c+d x))-110 B-3 C \cos (3 (c+d x))+158 C)\right )}{15 a^2 d (\cos (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.254, size = 491, normalized size = 1.9 \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^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}\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 )}^{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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