Optimal. Leaf size=218 \[ \frac{(A-13 C) \sin (c+d x)}{6 d \sqrt{\sec (c+d x)} \left (a^3 \cos (c+d x)+a^3\right )}+\frac{(A-13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{6 a^3 d}-\frac{(A-49 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{2 (A-4 C) \sin (c+d x)}{15 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.575059, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {4221, 3042, 2977, 2748, 2641, 2639} \[ \frac{(A-13 C) \sin (c+d x)}{6 d \sqrt{\sec (c+d x)} \left (a^3 \cos (c+d x)+a^3\right )}+\frac{(A-13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{6 a^3 d}-\frac{(A-49 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{2 (A-4 C) \sin (c+d x)}{15 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3042
Rule 2977
Rule 2748
Rule 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{3}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{3}{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{5}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{3}{2}}(c+d x) \left (\frac{5}{2} a (A-C)+\frac{1}{2} a (A+11 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{5}{2}}(c+d x)}+\frac{2 (A-4 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2 \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 (3 a^2 (A-4 C)+\frac{1}{2} a^2 (A+41 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{5}{2}}(c+d x)}+\frac{2 (A-4 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x)}+\frac{(A-13 C) \sin (c+d x)}{6 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-13 C)-\frac{3}{4} a^3 (A-49 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{5}{2}}(c+d x)}+\frac{2 (A-4 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x)}+\frac{(A-13 C) \sin (c+d x)}{6 d \left (a^3+a^3 \cos (c+d x)\right ) \sqrt{\sec (c+d x)}}-\frac{\left ((A-49 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-13 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-49 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-13 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{5}{2}}(c+d x)}+\frac{2 (A-4 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x)}+\frac{(A-13 C) \sin (c+d x)}{6 d \left (a^3+a^3 \cos (c+d x)\right ) \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 6.95723, size = 813, normalized size = 3.73 \[ \frac{\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 ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{15 d (\cos (c+d x) a+a)^3}-\frac{49 \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 ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{15 d (\cos (c+d x) a+a)^3}+\frac{2 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 ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d (\cos (c+d x) a+a)^3}-\frac{26 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 ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d (\cos (c+d x) a+a)^3}+\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 ^5\left (\frac{c}{2}+\frac{d x}{2}\right )}{5 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{4 \sec \left (\frac{c}{2}\right ) \left (7 A \sin \left (\frac{d x}{2}\right )+17 C \sin \left (\frac{d x}{2}\right )\right ) \sec ^3\left (\frac{c}{2}+\frac{d x}{2}\right )}{15 d}-\frac{4 (7 A+17 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 ) \left (A \sin \left (\frac{d x}{2}\right )+23 C \sin \left (\frac{d x}{2}\right )\right ) \sec \left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d}-\frac{2 (-A+39 C+10 C \cos (2 c)) \cos (d x) \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right )}{5 d}+\frac{16 C \cos (c) \sin (d x)}{d}+\frac{4 (A+23 C) \tan \left (\frac{c}{2}\right )}{3 d}\right ) \cos ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{(\cos (c+d x) a+a)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.124, 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} \sec \left (d x + c\right )^{\frac{3}{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^{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{3}{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{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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