Optimal. Leaf size=217 \[ \frac{2 a (9 A+9 B+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (7 A+5 (B+C)) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 A+5 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 a (9 A+9 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 a (B+C) \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a C \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)} \]
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Rubi [A] time = 0.345041, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {4221, 3033, 3023, 2748, 2635, 2641, 2639} \[ \frac{2 a (9 A+9 B+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (7 A+5 (B+C)) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 A+5 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 a (9 A+9 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 a (B+C) \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a C \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3033
Rule 3023
Rule 2748
Rule 2635
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac{3}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 a C \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1}{9} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{9 a A}{2}+\frac{1}{2} a (9 A+9 B+7 C) \cos (c+d x)+\frac{9}{2} a (B+C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 a C \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a (B+C) \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{1}{63} \left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{9}{4} a (7 A+5 (B+C))+\frac{7}{4} a (9 A+9 B+7 C) \cos (c+d x)\right ) \, dx\\ &=\frac{2 a C \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a (B+C) \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{1}{9} \left (a (9 A+9 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{5}{2}}(c+d x) \, dx+\frac{1}{7} \left (a (7 A+5 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 a C \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a (B+C) \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a (9 A+9 B+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (7 A+5 (B+C)) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{1}{15} \left (a (9 A+9 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{21} \left (a (7 A+5 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 a (9 A+9 B+7 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 a (7 A+5 (B+C)) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{21 d}+\frac{2 a C \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a (B+C) \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a (9 A+9 B+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (7 A+5 (B+C)) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.860774, size = 149, normalized size = 0.69 \[ \frac{a \sqrt{\sec (c+d x)} \left (2 \sin (2 (c+d x)) (7 (36 A+36 B+43 C) \cos (c+d x)+5 (84 A+18 (B+C) \cos (2 (c+d x))+78 B+7 C \cos (3 (c+d x))+78 C))+240 (7 A+5 (B+C)) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+336 (9 A+9 B+7 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{2520 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.378, size = 512, normalized size = 2.4 \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{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}}{\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 a \cos \left (d x + c\right )^{3} +{\left (B + C\right )} a \cos \left (d x + c\right )^{2} +{\left (A + B\right )} a \cos \left (d x + c\right ) + A a}{\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{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}}{\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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