Optimal. Leaf size=253 \[ -\frac {4 a^3 (21 A+5 C) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {2 (11 A+5 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^3 \cos (c+d x)+a^3\right )}{5 d}+\frac {4 a^3 (3 A+5 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}-\frac {4 a^3 (9 A-5 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {4 A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 a d}+\frac {2 A \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{5 d} \]
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Rubi [A] time = 0.68, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {4221, 3044, 2975, 2968, 3023, 2748, 2641, 2639} \[ -\frac {4 a^3 (21 A+5 C) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {2 (11 A+5 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^3 \cos (c+d x)+a^3\right )}{5 d}+\frac {4 a^3 (3 A+5 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}-\frac {4 a^3 (9 A-5 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {4 A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 a d}+\frac {2 A \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{5 d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2748
Rule 2968
Rule 2975
Rule 3023
Rule 3044
Rule 4221
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^3 \left (A+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+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 (3 a A-\frac {1}{2} a (3 A-5 C) \cos (c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{5 a}\\ &=\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 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 {3}{4} a^2 (11 A+5 C)-\frac {3}{4} a^2 (9 A-5 C) \cos (c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx}{15 a}\\ &=\frac {2 (11 A+5 C) \left (a^3+a^3 \cos (c+d x)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 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}{2} a^3 (6 A+5 C)-\frac {3}{4} a^3 (21 A+5 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{15 a}\\ &=\frac {2 (11 A+5 C) \left (a^3+a^3 \cos (c+d x)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 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}{2} a^4 (6 A+5 C)+\left (\frac {3}{2} a^4 (6 A+5 C)-\frac {3}{4} a^4 (21 A+5 C)\right ) \cos (c+d x)-\frac {3}{4} a^4 (21 A+5 C) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx}{15 a}\\ &=-\frac {4 a^3 (21 A+5 C) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {2 (11 A+5 C) \left (a^3+a^3 \cos (c+d x)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 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 C)-\frac {9}{8} a^4 (9 A-5 C) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx}{45 a}\\ &=-\frac {4 a^3 (21 A+5 C) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {2 (11 A+5 C) \left (a^3+a^3 \cos (c+d x)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 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 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 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 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 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+5 C) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {2 (11 A+5 C) \left (a^3+a^3 \cos (c+d x)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 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*}
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Mathematica [C] time = 3.18, size = 279, normalized size = 1.10 \[ \frac {a^3 \csc (c) \sec (c) e^{-i d x} \sqrt {\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (4 \left (-1+e^{4 i c}\right ) (9 A-5 C) e^{-i (c-d x)} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )+\frac {1}{2} \sin (2 c) \sec ^2(c+d x) \left (-36 i (9 A-5 C) \cos (c+d x)+80 (3 A+5 C) \cos ^{\frac {5}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+132 A \sin (c+d x)+60 A \sin (2 (c+d x))+108 A \sin (3 (c+d x))-108 i A \cos (3 (c+d x))+30 C \sin (c+d x)+10 C \sin (2 (c+d x))+30 C \sin (3 (c+d x))+5 C \sin (4 (c+d x))+60 i C \cos (3 (c+d x))\right )\right )}{60 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C a^{3} \cos \left (d x + c\right )^{5} + 3 \, C a^{3} \cos \left (d x + c\right )^{4} + {\left (A + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + {\left (3 \, A + C\right )} a^{3} \cos \left (d x + c\right )^{2} + 3 \, A a^{3} \cos \left (d x + c\right ) + A a^{3}\right )} \sec \left (d x + c\right )^{\frac {7}{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 8.84, size = 939, normalized size = 3.71 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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