Optimal. Leaf size=270 \[ -\frac{4 a^3 (21 A+20 B+5 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 (33 A+35 B+15 C) \sin (c+d x) \sqrt{\sec (c+d x)} \left (a^3 \cos (c+d x)+a^3\right )}{15 d}+\frac{4 a^3 (3 A+5 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{4 a^3 (9 A+5 B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 (6 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{15 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{5 d} \]
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Rubi [A] time = 0.790965, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.186, Rules used = {4221, 3043, 2975, 2968, 3023, 2748, 2641, 2639} \[ -\frac{4 a^3 (21 A+20 B+5 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 (33 A+35 B+15 C) \sin (c+d x) \sqrt{\sec (c+d x)} \left (a^3 \cos (c+d x)+a^3\right )}{15 d}+\frac{4 a^3 (3 A+5 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{4 a^3 (9 A+5 B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 (6 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{15 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{5 d} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3043
Rule 2975
Rule 2968
Rule 3023
Rule 2748
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 A (a+a \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^3 \left (\frac{1}{2} a (6 A+5 B)-\frac{1}{2} a (3 A-5 C) \cos (c+d x)\right )}{\cos ^{\frac{5}{2}}(c+d x)} \, dx}{5 a}\\ &=\frac{2 (6 A+5 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 a d}+\frac{2 A (a+a \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^2 \left (\frac{1}{4} a^2 (33 A+35 B+15 C)-\frac{3}{4} a^2 (9 A+5 B-5 C) \cos (c+d x)\right )}{\cos ^{\frac{3}{2}}(c+d x)} \, dx}{15 a}\\ &=\frac{2 (33 A+35 B+15 C) \left (a^3+a^3 \cos (c+d x)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 (6 A+5 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 a d}+\frac{2 A (a+a \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x)) \left (\frac{3}{4} a^3 (12 A+15 B+10 C)-\frac{3}{4} a^3 (21 A+20 B+5 C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{15 a}\\ &=\frac{2 (33 A+35 B+15 C) \left (a^3+a^3 \cos (c+d x)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 (6 A+5 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 a d}+\frac{2 A (a+a \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{3}{4} a^4 (12 A+15 B+10 C)+\left (-\frac{3}{4} a^4 (21 A+20 B+5 C)+\frac{3}{4} a^4 (12 A+15 B+10 C)\right ) \cos (c+d x)-\frac{3}{4} a^4 (21 A+20 B+5 C) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx}{15 a}\\ &=-\frac{4 a^3 (21 A+20 B+5 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 (33 A+35 B+15 C) \left (a^3+a^3 \cos (c+d x)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 (6 A+5 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 a d}+\frac{2 A (a+a \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{\left (16 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{15}{8} a^4 (3 A+5 (B+C))-\frac{9}{8} a^4 (9 A+5 B-5 C) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx}{45 a}\\ &=-\frac{4 a^3 (21 A+20 B+5 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 (33 A+35 B+15 C) \left (a^3+a^3 \cos (c+d x)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 (6 A+5 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 a d}+\frac{2 A (a+a \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}-\frac{1}{5} \left (2 a^3 (9 A+5 B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{3} \left (2 a^3 (3 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{4 a^3 (9 A+5 B-5 C) \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{4 a^3 (3 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)}}{3 d}-\frac{4 a^3 (21 A+20 B+5 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 (33 A+35 B+15 C) \left (a^3+a^3 \cos (c+d x)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 (6 A+5 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 a d}+\frac{2 A (a+a \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 2.10876, size = 157, normalized size = 0.58 \[ \frac{a^3 \sec ^{\frac{5}{2}}(c+d x) \left (80 (3 A+5 (B+C)) \cos ^{\frac{5}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-48 (9 A+5 B-5 C) \cos ^{\frac{5}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+2 \sin (c+d x) (5 (12 A+4 B+3 C) \cos (c+d x)+6 (18 A+5 (3 B+C)) \cos (2 (c+d x))+5 (6 (4 A+3 B+C)+C \cos (3 (c+d x))))\right )}{60 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 4.743, size = 1328, normalized size = 4.9 \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 (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 )}^{3} \sec \left (d x + c\right )^{\frac{7}{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 (C a^{3} \cos \left (d x + c\right )^{5} +{\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} +{\left (A + 3 \, B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} +{\left (3 \, A + 3 \, B + C\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{7}{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 (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 )}^{3} \sec \left (d x + c\right )^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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