Optimal. Leaf size=220 \[ -\frac{(A-9 C) \sin (c+d x)}{10 d \sqrt{\sec (c+d x)} \left (a^3 \cos (c+d x)+a^3\right )}+\frac{(A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{6 a^3 d}+\frac{(A-9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac{(A+C) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}+\frac{2 (2 A-3 C) \sin (c+d x)}{15 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.564981, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4221, 3042, 2977, 2978, 2748, 2641, 2639} \[ -\frac{(A-9 C) \sin (c+d x)}{10 d \sqrt{\sec (c+d x)} \left (a^3 \cos (c+d x)+a^3\right )}+\frac{(A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{6 a^3 d}+\frac{(A-9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac{(A+C) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}+\frac{2 (2 A-3 C) \sin (c+d x)}{15 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3042
Rule 2977
Rule 2978
Rule 2748
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sqrt{\sec (c+d x)}} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{\cos (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{3}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{\cos (c+d x)} \left (\frac{1}{2} a (7 A-3 C)-\frac{1}{2} a (A-9 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{3}{2}}(c+d x)}+\frac{2 (2 A-3 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2 \sqrt{\sec (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{a^2 (2 A-3 C)+\frac{1}{2} a^2 (A+21 C) \cos (c+d x)}{\sqrt{\cos (c+d x)} (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{3}{2}}(c+d x)}+\frac{2 (2 A-3 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2 \sqrt{\sec (c+d x)}}-\frac{(A-9 C) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right ) \sqrt{\sec (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{5}{4} a^3 (A+3 C)+\frac{3}{4} a^3 (A-9 C) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx}{15 a^6}\\ &=-\frac{(A+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (2 A-3 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2 \sqrt{\sec (c+d x)}}-\frac{(A-9 C) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right ) \sqrt{\sec (c+d x)}}+\frac{\left ((A-9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{20 a^3}+\frac{\left ((A+3 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{(A-9 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+3 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{3}{2}}(c+d x)}+\frac{2 (2 A-3 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2 \sqrt{\sec (c+d x)}}-\frac{(A-9 C) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right ) \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 6.97093, size = 787, normalized size = 3.58 \[ \frac{\cos ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{\sec (c+d x)} \left (-\frac{2 \sec \left (\frac{c}{2}\right ) \sec ^5\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (A \sin \left (\frac{d x}{2}\right )+C \sin \left (\frac{d x}{2}\right )\right )}{5 d}+\frac{8 \sec \left (\frac{c}{2}\right ) \sec ^3\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (A \sin \left (\frac{d x}{2}\right )+6 C \sin \left (\frac{d x}{2}\right )\right )}{15 d}-\frac{2 (A+C) \tan \left (\frac{c}{2}\right ) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{5 d}+\frac{8 (A+6 C) \tan \left (\frac{c}{2}\right ) \sec ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{15 d}+\frac{4 \sec \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}+\frac{d x}{2}\right ) \left (A \sin \left (\frac{d x}{2}\right )-9 C \sin \left (\frac{d x}{2}\right )\right )}{3 d}-\frac{2 (A-9 C) \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \cos (d x)}{5 d}+\frac{4 (A-9 C) \tan \left (\frac{c}{2}\right )}{3 d}\right )}{(a \cos (c+d x)+a)^3}-\frac{\sqrt{2} A \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) 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 ) \cos ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{15 d (a \cos (c+d x)+a)^3}+\frac{2 A \sin (c) \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \sqrt{\cos (c+d x)} \cos ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d (a \cos (c+d x)+a)^3}+\frac{3 \sqrt{2} C \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) 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 ) \cos ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{5 d (a \cos (c+d x)+a)^3}+\frac{2 C \sin (c) \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \sqrt{\cos (c+d x)} \cos ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.468, size = 451, normalized size = 2.1 \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 )}^{3} \sqrt{\sec \left (d x + c\right )}}\,{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^{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 )} \sqrt{\sec \left (d x + c\right )}}, 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} \sqrt{\sec \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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