Optimal. Leaf size=236 \[ \frac{4 (5 A+14 C) \sin (c+d x)}{15 a^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 (A+3 C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{(A+3 C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1) \sec ^{\frac{5}{2}}(c+d x)}-\frac{5 (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^2 d}+\frac{4 (5 A+14 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a^2 d}-\frac{(A+C) \sin (c+d x)}{3 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.461364, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{4 (5 A+14 C) \sin (c+d x)}{15 a^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 (A+3 C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{(A+3 C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1) \sec ^{\frac{5}{2}}(c+d x)}-\frac{5 (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^2 d}+\frac{4 (5 A+14 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a^2 d}-\frac{(A+C) \sin (c+d x)}{3 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 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))^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+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx\\ &=-\frac{(A+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 C)+\frac{1}{2} a (5 A+11 C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=-\frac{(A+C) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}-\frac{(A+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+3 C)+2 a^2 (5 A+14 C) \cos (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac{(A+C) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}-\frac{(A+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+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 (2 (5 A+14 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{5}{2}}(c+d x) \, dx}{3 a^2}\\ &=-\frac{(A+C) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}-\frac{(A+3 C) \sin (c+d x)}{a^2 d (1+\cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 (5 A+14 C) \sin (c+d x)}{15 a^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 (A+3 C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{\left (5 (A+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 (2 (5 A+14 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 a^2}\\ &=\frac{4 (5 A+14 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+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+C) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}-\frac{(A+3 C) \sin (c+d x)}{a^2 d (1+\cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 (5 A+14 C) \sin (c+d x)}{15 a^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 (A+3 C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 6.86583, size = 813, normalized size = 3.44 \[ -\frac{4 \sqrt{2} A 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)}} \csc \left (\frac{c}{2}\right ) \left (e^{2 i d x} \left (-1+e^{2 i c}\right ) \, _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 ) \sec \left (\frac{c}{2}\right ) \cos ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d (\cos (c+d x) a+a)^2}-\frac{56 \sqrt{2} 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)}} \csc \left (\frac{c}{2}\right ) \left (e^{2 i d x} \left (-1+e^{2 i c}\right ) \, _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 ) \sec \left (\frac{c}{2}\right ) \cos ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{15 d (\cos (c+d x) a+a)^2}-\frac{10 A \sqrt{\cos (c+d x)} \csc \left (\frac{c}{2}\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sec \left (\frac{c}{2}\right ) \sqrt{\sec (c+d x)} \sin (c) \cos ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d (\cos (c+d x) a+a)^2}-\frac{10 C \sqrt{\cos (c+d x)} \csc \left (\frac{c}{2}\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sec \left (\frac{c}{2}\right ) \sqrt{\sec (c+d x)} \sin (c) \cos ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{d (\cos (c+d x) a+a)^2}+\frac{\sqrt{\sec (c+d x)} \left (-\frac{2 \sec \left (\frac{c}{2}\right ) \left (A \sin \left (\frac{d x}{2}\right )+C \sin \left (\frac{d x}{2}\right )\right ) \sec ^3\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d}-\frac{2 (A+C) \tan \left (\frac{c}{2}\right ) \sec ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d}+\frac{4 \sec \left (\frac{c}{2}\right ) \left (7 A \sin \left (\frac{d x}{2}\right )+13 C \sin \left (\frac{d x}{2}\right )\right ) \sec \left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d}-\frac{(20 \cos (2 c) A+60 A+151 C+73 C \cos (2 c)) \cos (d x) \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right )}{10 d}-\frac{8 C \cos (2 d x) \sin (2 c)}{3 d}+\frac{2 C \cos (3 d x) \sin (3 c)}{5 d}+\frac{2 (20 A+73 C) \cos (c) \sin (d x)}{5 d}-\frac{8 C \cos (2 c) \sin (2 d x)}{3 d}+\frac{2 C \cos (3 c) \sin (3 d x)}{5 d}+\frac{4 (7 A+13 C) \tan \left (\frac{c}{2}\right )}{3 d}\right ) \cos ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{(\cos (c+d x) a+a)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.15, size = 451, 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] 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 )}^{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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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^{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} + 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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