Optimal. Leaf size=228 \[ \frac{(3 A-13 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left (a^3 \sec (c+d x)+a^3\right )}+\frac{(3 A-13 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{6 a^3 d}-\frac{(9 A-49 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{(3 A-8 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}+\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \sec (c+d x)+a)^3} \]
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Rubi [A] time = 0.576693, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2960, 4020, 3787, 3771, 2639, 2641} \[ \frac{(3 A-13 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left (a^3 \sec (c+d x)+a^3\right )}+\frac{(3 A-13 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{6 a^3 d}-\frac{(9 A-49 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{(3 A-8 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}+\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \sec (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2960
Rule 4020
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x)} \, dx &=\int \frac{B+A \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3} \, dx\\ &=\frac{(A-B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{\int \frac{-\frac{1}{2} a (A-11 B)+\frac{5}{2} a (A-B) \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=\frac{(A-B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{(3 A-8 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac{\int \frac{-\frac{1}{2} a^2 (6 A-41 B)+\frac{3}{2} a^2 (3 A-8 B) \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))} \, dx}{15 a^4}\\ &=\frac{(A-B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{(3 A-8 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac{(3 A-13 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{6 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac{\int \frac{-\frac{3}{4} a^3 (9 A-49 B)+\frac{5}{4} a^3 (3 A-13 B) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{15 a^6}\\ &=\frac{(A-B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{(3 A-8 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac{(3 A-13 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{6 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{(9 A-49 B) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{20 a^3}+\frac{(3 A-13 B) \int \sqrt{\sec (c+d x)} \, dx}{12 a^3}\\ &=\frac{(A-B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{(3 A-8 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac{(3 A-13 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{6 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{\left ((9 A-49 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{20 a^3}+\frac{\left ((3 A-13 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{12 a^3}\\ &=-\frac{(9 A-49 B) \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{(3 A-13 B) \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-B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{(3 A-8 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac{(3 A-13 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{6 d \left (a^3+a^3 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 7.10992, size = 817, normalized size = 3.58 \[ \frac{3 \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 )}{5 d (\cos (c+d x) a+a)^3}-\frac{49 \sqrt{2} B 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 )}{d (\cos (c+d x) a+a)^3}-\frac{26 B \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 )-B \sin \left (\frac{d x}{2}\right )\right ) \sec ^5\left (\frac{c}{2}+\frac{d x}{2}\right )}{5 d}-\frac{2 (A-B) \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 (12 A \sin \left (\frac{d x}{2}\right )-17 B \sin \left (\frac{d x}{2}\right )\right ) \sec ^3\left (\frac{c}{2}+\frac{d x}{2}\right )}{15 d}+\frac{4 (12 A-17 B) \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 (9 A \sin \left (\frac{d x}{2}\right )-23 B \sin \left (\frac{d x}{2}\right )\right ) \sec \left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d}-\frac{2 (-9 A+39 B+10 B \cos (2 c)) \cos (d x) \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right )}{5 d}+\frac{16 B \cos (c) \sin (d x)}{d}-\frac{4 (9 A-23 B) \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 = 3.719, size = 451, normalized size = 2. \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{B \cos \left (d x + c\right ) + 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{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{B \cos \left (d x + c\right ) + A}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3} \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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