Optimal. Leaf size=199 \[ \frac{4 a^3 (4 A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 (A-B) \sin (c+d x) \sqrt{\sec (c+d x)} \left (a^3 \sec (c+d x)+a^3\right )}{3 d}+\frac{20 a^3 (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{4 a^3 (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{3 d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 0.491089, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {2960, 4017, 4018, 3997, 3787, 3771, 2639, 2641} \[ \frac{4 a^3 (4 A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 (A-B) \sin (c+d x) \sqrt{\sec (c+d x)} \left (a^3 \sec (c+d x)+a^3\right )}{3 d}+\frac{20 a^3 (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{4 a^3 (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{3 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2960
Rule 4017
Rule 4018
Rule 3997
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx &=\int \frac{(a+a \sec (c+d x))^3 (B+A \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a B (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2}{3} \int \frac{(a+a \sec (c+d x))^2 \left (\frac{1}{2} a (3 A+7 B)+\frac{3}{2} a (A-B) \sec (c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 a B (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 (A-B) \sqrt{\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{3 d}+\frac{4}{9} \int \frac{(a+a \sec (c+d x)) \left (\frac{3}{2} a^2 (A+4 B)+\frac{3}{2} a^2 (4 A+B) \sec (c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{4 a^3 (4 A+B) \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d}+\frac{2 a B (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 (A-B) \sqrt{\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{3 d}+\frac{8}{9} \int \frac{-\frac{9}{4} a^3 (A-B)+\frac{15}{4} a^3 (A+B) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{4 a^3 (4 A+B) \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d}+\frac{2 a B (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 (A-B) \sqrt{\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{3 d}-\left (2 a^3 (A-B)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{3} \left (10 a^3 (A+B)\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{4 a^3 (4 A+B) \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d}+\frac{2 a B (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 (A-B) \sqrt{\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{3 d}-\left (2 a^3 (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{3} \left (10 a^3 (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{4 a^3 (A-B) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{d}+\frac{20 a^3 (A+B) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{3 d}+\frac{4 a^3 (4 A+B) \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d}+\frac{2 a B (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 (A-B) \sqrt{\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [C] time = 1.89163, size = 202, normalized size = 1.02 \[ \frac{a^3 e^{-i d x} \sec ^{\frac{3}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left (4 i (A-B) \left (1+e^{2 i (c+d x)}\right )^{3/2} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right )+40 (A+B) \cos ^{\frac{3}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+4 A \sin (c+d x)+18 A \sin (2 (c+d x))-12 i A \cos (2 (c+d x))-12 i A+B \sin (c+d x)+6 B \sin (2 (c+d x))+B \sin (3 (c+d x))+12 i B \cos (2 (c+d x))+12 i B\right )}{6 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.417, size = 654, normalized size = 3.3 \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{\left (B \cos \left (d x + c\right ) + A\right )}{\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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B a^{3} \cos \left (d x + c\right )^{4} +{\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 3 \,{\left (A + B\right )} a^{3} \cos \left (d x + c\right )^{2} +{\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) + A 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{\left (B \cos \left (d x + c\right ) + A\right )}{\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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