Optimal. Leaf size=237 \[ \frac{2 \left (9 a^2 A b+3 a^3 B+3 a b^2 B+A b^3\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 \left (5 a^3 A-15 a^2 b B-15 a A b^2-3 b^3 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a^2 (5 a A-b B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 b^2 (9 a B+5 A b) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{5 d \sec ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.532947, antiderivative size = 237, 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, 4025, 4074, 4047, 3771, 2641, 4046, 2639} \[ \frac{2 \left (9 a^2 A b+3 a^3 B+3 a b^2 B+A b^3\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 \left (5 a^3 A-15 a^2 b B-15 a A b^2-3 b^3 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a^2 (5 a A-b B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 b^2 (9 a B+5 A b) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{5 d \sec ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2960
Rule 4025
Rule 4074
Rule 4047
Rule 3771
Rule 2641
Rule 4046
Rule 2639
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx &=\int \frac{(b+a \sec (c+d x))^3 (B+A \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 b B (b+a \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2}{5} \int \frac{(b+a \sec (c+d x)) \left (-\frac{1}{2} b (5 A b+9 a B)-\frac{1}{2} \left (10 a A b+5 a^2 B+3 b^2 B\right ) \sec (c+d x)-\frac{1}{2} a (5 a A-b B) \sec ^2(c+d x)\right )}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 b^2 (5 A b+9 a B) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 b B (b+a \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4}{15} \int \frac{\frac{3}{4} b \left (15 a A b+14 a^2 B+3 b^2 B\right )+\frac{5}{4} \left (9 a^2 A b+A b^3+3 a^3 B+3 a b^2 B\right ) \sec (c+d x)+\frac{3}{4} a^2 (5 a A-b B) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 b^2 (5 A b+9 a B) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 b B (b+a \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4}{15} \int \frac{\frac{3}{4} b \left (15 a A b+14 a^2 B+3 b^2 B\right )+\frac{3}{4} a^2 (5 a A-b B) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{3} \left (9 a^2 A b+A b^3+3 a^3 B+3 a b^2 B\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{2 b^2 (5 A b+9 a B) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 (5 a A-b B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 b B (b+a \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{1}{5} \left (-5 a^3 A+15 a A b^2+15 a^2 b B+3 b^3 B\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{3} \left (\left (9 a^2 A b+A b^3+3 a^3 B+3 a b^2 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (9 a^2 A b+A b^3+3 a^3 B+3 a b^2 B\right ) \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{2 b^2 (5 A b+9 a B) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 (5 a A-b B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 b B (b+a \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{1}{5} \left (\left (-5 a^3 A+15 a A b^2+15 a^2 b B+3 b^3 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 \left (5 a^3 A-15 a A b^2-15 a^2 b B-3 b^3 B\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 \left (9 a^2 A b+A b^3+3 a^3 B+3 a b^2 B\right ) \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{2 b^2 (5 A b+9 a B) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 (5 a A-b B) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 b B (b+a \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 1.37629, size = 172, normalized size = 0.73 \[ \frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (20 \left (9 a^2 A b+3 a^3 B+3 a b^2 B+A b^3\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+12 \left (-5 a^3 A+15 a^2 b B+15 a A b^2+3 b^3 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\frac{2 \sin (c+d x) \left (3 \left (10 a^3 A+b^3 B \cos (2 (c+d x))+b^3 B\right )+10 b^2 (3 a B+A b) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}}\right )}{30 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 4.28, size = 867, normalized size = 3.7 \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 (b \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 ({\left (B b^{3} \cos \left (d x + c\right )^{4} + A a^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )\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{\left (B \cos \left (d x + c\right ) + A\right )}{\left (b \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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