Optimal. Leaf size=277 \[ \frac{(49 A-9 B-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{(13 A-3 B-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac{(13 A-3 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{6 a^3 d}-\frac{(49 A-9 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac{(8 A-3 B-2 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.68803, antiderivative size = 277, 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, 2978, 2748, 2636, 2639, 2641} \[ \frac{(49 A-9 B-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{(13 A-3 B-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac{(13 A-3 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{6 a^3 d}-\frac{(49 A-9 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac{(8 A-3 B-2 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3041
Rule 2978
Rule 2748
Rule 2636
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^3} \, dx\\ &=-\frac{(A-B+C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{2} a (11 A-B+C)-\frac{5}{2} a (A-B-C) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{(A-B+C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(8 A-3 B-2 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{2} a^2 (41 A-6 B+C)-\frac{3}{2} a^2 (8 A-3 B-2 C) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))} \, dx}{15 a^4}\\ &=-\frac{(A-B+C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(8 A-3 B-2 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{(13 A-3 B-C) \sqrt{\sec (c+d x)} \sin (c+d x)}{6 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{3}{4} a^3 (49 A-9 B-C)-\frac{5}{4} a^3 (13 A-3 B-C) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx}{15 a^6}\\ &=-\frac{(A-B+C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(8 A-3 B-2 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{(13 A-3 B-C) \sqrt{\sec (c+d x)} \sin (c+d x)}{6 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac{\left ((49 A-9 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx}{20 a^3}-\frac{\left ((13 A-3 B-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{(13 A-3 B-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{(49 A-9 B-C) \sqrt{\sec (c+d x)} \sin (c+d x)}{10 a^3 d}-\frac{(A-B+C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(8 A-3 B-2 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{(13 A-3 B-C) \sqrt{\sec (c+d x)} \sin (c+d x)}{6 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac{\left ((49 A-9 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{20 a^3}\\ &=-\frac{(49 A-9 B-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{(13 A-3 B-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{(49 A-9 B-C) \sqrt{\sec (c+d x)} \sin (c+d x)}{10 a^3 d}-\frac{(A-B+C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(8 A-3 B-2 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{(13 A-3 B-C) \sqrt{\sec (c+d x)} \sin (c+d x)}{6 d \left (a^3+a^3 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 1.94575, size = 215, normalized size = 0.78 \[ -\frac{2 \cos ^6\left (\frac{1}{2} (c+d x)\right ) \sqrt{\sec (c+d x)} \left (10 (13 A-3 B-C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+6 (49 A-9 B-C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )-\frac{1}{8} \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^4\left (\frac{1}{2} (c+d x)\right ) ((1621 A-261 B+11 C) \cos (c+d x)+4 (188 A-33 B-2 C) \cos (2 (c+d x))+147 A \cos (3 (c+d x))+992 A-27 B \cos (3 (c+d x))-132 B-3 C \cos (3 (c+d x))-8 C)\right )}{15 a^3 d (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.602, size = 793, normalized size = 2.9 \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{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac{3}{2}}}{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}}, 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 )} \sec \left (d x + c\right )^{\frac{3}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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