Optimal. Leaf size=255 \[ \frac{2 a^2 (21 A+27 B+19 C) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (7 A+6 B+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (7 A+6 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{4 a^2 (12 A+9 B+8 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 (9 B+4 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{63 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \sec ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.611305, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.209, Rules used = {4221, 3045, 2976, 2968, 3023, 2748, 2639, 2635, 2641} \[ \frac{2 a^2 (21 A+27 B+19 C) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (7 A+6 B+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (7 A+6 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{4 a^2 (12 A+9 B+8 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 (9 B+4 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{63 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \sec ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3045
Rule 2976
Rule 2968
Rule 3023
Rule 2748
Rule 2639
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+a \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2 \left (\frac{3}{2} a (3 A+C)+\frac{1}{2} a (9 B+4 C) \cos (c+d x)\right ) \, dx}{9 a}\\ &=\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (9 B+4 C) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{63 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} (a+a \cos (c+d x)) \left (\frac{3}{4} a^2 (21 A+9 B+11 C)+\frac{3}{4} a^2 (21 A+27 B+19 C) \cos (c+d x)\right ) \, dx}{63 a}\\ &=\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (9 B+4 C) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{63 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \left (\frac{3}{4} a^3 (21 A+9 B+11 C)+\left (\frac{3}{4} a^3 (21 A+9 B+11 C)+\frac{3}{4} a^3 (21 A+27 B+19 C)\right ) \cos (c+d x)+\frac{3}{4} a^3 (21 A+27 B+19 C) \cos ^2(c+d x)\right ) \, dx}{63 a}\\ &=\frac{2 a^2 (21 A+27 B+19 C) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (9 B+4 C) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{63 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \left (\frac{21}{4} a^3 (12 A+9 B+8 C)+\frac{45}{4} a^3 (7 A+6 B+5 C) \cos (c+d x)\right ) \, dx}{315 a}\\ &=\frac{2 a^2 (21 A+27 B+19 C) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (9 B+4 C) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{63 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{1}{7} \left (2 a^2 (7 A+6 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{15} \left (2 a^2 (12 A+9 B+8 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{4 a^2 (12 A+9 B+8 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^2 (21 A+27 B+19 C) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (9 B+4 C) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{63 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (7 A+6 B+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{1}{21} \left (2 a^2 (7 A+6 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{4 a^2 (12 A+9 B+8 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{4 a^2 (7 A+6 B+5 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^2 (21 A+27 B+19 C) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (9 B+4 C) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{63 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (7 A+6 B+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.936758, size = 151, normalized size = 0.59 \[ \frac{a^2 \sqrt{\sec (c+d x)} \left (2 \sin (2 (c+d x)) (7 (36 A+72 B+79 C) \cos (c+d x)+840 A+90 (B+2 C) \cos (2 (c+d x))+810 B+35 C \cos (3 (c+d x))+780 C)+480 (7 A+6 B+5 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+672 (12 A+9 B+8 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{2520 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.345, size = 514, 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] 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 )}^{2}}{\sqrt{\sec \left (d x + c\right )}}\,{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^{2} \cos \left (d x + c\right )^{4} +{\left (B + 2 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} +{\left (A + 2 \, B + C\right )} a^{2} \cos \left (d x + c\right )^{2} +{\left (2 \, A + B\right )} a^{2} \cos \left (d x + c\right ) + A a^{2}}{\sqrt{\sec \left (d x + c\right )}}, 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 )}^{2}}{\sqrt{\sec \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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