Optimal. Leaf size=290 \[ \frac{7 (7 A+33 C) \sin (c+d x)}{30 a^3 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(13 A+63 C) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}-\frac{(13 A+63 C) \sin (c+d x)}{10 d \sec ^{\frac{5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}-\frac{(13 A+63 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{6 a^3 d}+\frac{7 (7 A+33 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac{2 (A+6 C) \sin (c+d x)}{15 a d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.656942, antiderivative size = 290, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4221, 3042, 2977, 2748, 2635, 2641, 2639} \[ \frac{7 (7 A+33 C) \sin (c+d x)}{30 a^3 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(13 A+63 C) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}-\frac{(13 A+63 C) \sin (c+d x)}{10 d \sec ^{\frac{5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}-\frac{(13 A+63 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{6 a^3 d}+\frac{7 (7 A+33 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac{2 (A+6 C) \sin (c+d x)}{15 a d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3042
Rule 2977
Rule 2748
Rule 2635
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx\\ &=-\frac{(A+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac{9}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{7}{2}}(c+d x) \left (\frac{1}{2} a (A-9 C)+\frac{5}{2} a (A+3 C) \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{(A+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac{9}{2}}(c+d x)}-\frac{2 (A+6 C) \sin (c+d x)}{15 a 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 (-7 a^2 (A+6 C)+\frac{5}{2} a^2 (5 A+21 C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{15 a^4}\\ &=-\frac{(A+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac{9}{2}}(c+d x)}-\frac{2 (A+6 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}-\frac{(13 A+63 C) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right ) \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}{4} a^3 (13 A+63 C)+\frac{35}{4} a^3 (7 A+33 C) \cos (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac{(A+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac{9}{2}}(c+d x)}-\frac{2 (A+6 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}-\frac{(13 A+63 C) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (7 (7 A+33 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{5}{2}}(c+d x) \, dx}{12 a^3}-\frac{\left ((13 A+63 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{(A+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac{9}{2}}(c+d x)}-\frac{2 (A+6 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}-\frac{(13 A+63 C) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac{5}{2}}(c+d x)}+\frac{7 (7 A+33 C) \sin (c+d x)}{30 a^3 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(13 A+63 C) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}+\frac{\left (7 (7 A+33 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{20 a^3}-\frac{\left ((13 A+63 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{7 (7 A+33 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+63 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+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac{9}{2}}(c+d x)}-\frac{2 (A+6 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}-\frac{(13 A+63 C) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac{5}{2}}(c+d x)}+\frac{7 (7 A+33 C) \sin (c+d x)}{30 a^3 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(13 A+63 C) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 5.39056, size = 623, normalized size = 2.15 \[ -\frac{\cos ^6\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \sec ^5\left (\frac{1}{2} (c+d x)\right ) \left (2 (806 A+3795 C) \cos \left (\frac{1}{2} (c-d x)\right )+2 (664 A+3135 C) \cos \left (\frac{1}{2} (3 c+d x)\right )+940 A \cos \left (\frac{1}{2} (c+3 d x)\right )+530 A \cos \left (\frac{1}{2} (5 c+3 d x)\right )+234 A \cos \left (\frac{1}{2} (3 c+5 d x)\right )+60 A \cos \left (\frac{1}{2} (7 c+5 d x)\right )+4500 C \cos \left (\frac{1}{2} (c+3 d x)\right )+2430 C \cos \left (\frac{1}{2} (5 c+3 d x)\right )+1110 C \cos \left (\frac{1}{2} (3 c+5 d x)\right )+276 C \cos \left (\frac{1}{2} (7 c+5 d x)\right )+15 C \cos \left (\frac{1}{2} (5 c+7 d x)\right )-15 C \cos \left (\frac{1}{2} (9 c+7 d x)\right )-3 C \cos \left (\frac{1}{2} (7 c+9 d x)\right )+3 C \cos \left (\frac{1}{2} (11 c+9 d x)\right )\right )}{16 \sqrt{\sec (c+d x)}}+98 \sqrt{2} A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left (\left (-1+e^{2 i c}\right ) e^{2 i d x} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right )+260 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+462 \sqrt{2} C \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left (\left (-1+e^{2 i c}\right ) e^{2 i d x} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right )+1260 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{15 a^3 d (\cos (c+d x)+1)^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 1.222, size = 479, normalized size = 1.7 \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} + 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{7}{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} + A}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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