Optimal. Leaf size=286 \[ \frac {8 a^3 (16 A+21 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{105 d}+\frac {4 a^3 (17 A+27 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 (73 A+63 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{315 d}+\frac {4 a^3 (11 A+21 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {4 a^3 (17 A+27 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 A \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{21 a d}+\frac {2 A \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) (a \cos (c+d x)+a)^3}{9 d} \]
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Rubi [A] time = 0.73, antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {4221, 3044, 2975, 2968, 3021, 2748, 2636, 2639, 2641} \[ \frac {8 a^3 (16 A+21 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{105 d}+\frac {4 a^3 (17 A+27 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 (73 A+63 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{315 d}+\frac {4 a^3 (11 A+21 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {4 a^3 (17 A+27 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 A \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{21 a d}+\frac {2 A \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) (a \cos (c+d x)+a)^3}{9 d} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 2641
Rule 2748
Rule 2968
Rule 2975
Rule 3021
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 {11}{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 {11}{2}}(c+d x)} \, dx\\ &=\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 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 (A+9 C) \cos (c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx}{9 a}\\ &=\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{21 a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 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 (73 A+63 C)+\frac {1}{4} a^2 (13 A+63 C) \cos (c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx}{63 a}\\ &=\frac {2 (73 A+63 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{21 a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 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 {9}{2} a^3 (16 A+21 C)+\frac {3}{4} a^3 (23 A+63 C) \cos (c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{315 a}\\ &=\frac {2 (73 A+63 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{21 a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {9}{2} a^4 (16 A+21 C)+\left (\frac {9}{2} a^4 (16 A+21 C)+\frac {3}{4} a^4 (23 A+63 C)\right ) \cos (c+d x)+\frac {3}{4} a^4 (23 A+63 C) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{315 a}\\ &=\frac {8 a^3 (16 A+21 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 (73 A+63 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{21 a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {\left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {63}{8} a^4 (17 A+27 C)+\frac {45}{8} a^4 (11 A+21 C) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)} \, dx}{945 a}\\ &=\frac {8 a^3 (16 A+21 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 (73 A+63 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{21 a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{21} \left (2 a^3 (11 A+21 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{15} \left (2 a^3 (17 A+27 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {4 a^3 (11 A+21 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 {4 a^3 (17 A+27 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {8 a^3 (16 A+21 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 (73 A+63 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{21 a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}-\frac {1}{15} \left (2 a^3 (17 A+27 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {4 a^3 (17 A+27 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^3 (11 A+21 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 {4 a^3 (17 A+27 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {8 a^3 (16 A+21 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 (73 A+63 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{21 a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}\\ \end {align*}
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Mathematica [C] time = 6.91, size = 655, normalized size = 2.29 \[ \sqrt {\sec (c+d x)} \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \cos (c+d x)+a)^3 \left (\frac {(17 A+27 C) \csc (c) \cos (d x)}{30 d}+\frac {\sec (c) \sec ^2(c+d x) (135 A \sin (c)+238 A \sin (d x)+63 C \sin (d x))}{1260 d}+\frac {\sec (c) \sec (c+d x) (238 A \sin (c)+330 A \sin (d x)+63 C \sin (c)+315 C \sin (d x))}{1260 d}+\frac {(22 A+21 C) \tan (c)}{84 d}+\frac {A \sec (c) \sin (d x) \sec ^4(c+d x)}{36 d}+\frac {\sec (c) \sec ^3(c+d x) (7 A \sin (c)+27 A \sin (d x))}{252 d}\right )+\frac {17 A \csc (c) e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \left (\left (-1+e^{2 i c}\right ) e^{2 i d x} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt {1+e^{2 i (c+d x)}}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \cos (c+d x)+a)^3}{90 \sqrt {2} d}+\frac {11 A \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (a \cos (c+d x)+a)^3}{42 d}+\frac {3 C \csc (c) e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \left (\left (-1+e^{2 i c}\right ) e^{2 i d x} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt {1+e^{2 i (c+d x)}}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \cos (c+d x)+a)^3}{10 \sqrt {2} d}+\frac {C \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (a \cos (c+d x)+a)^3}{2 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.17, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C a^{3} \cos \left (d x + c\right )^{5} + 3 \, C a^{3} \cos \left (d x + c\right )^{4} + {\left (A + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + {\left (3 \, A + C\right )} a^{3} \cos \left (d x + c\right )^{2} + 3 \, A a^{3} \cos \left (d x + c\right ) + A a^{3}\right )} \sec \left (d x + c\right )^{\frac {11}{2}}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {11}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 11.12, size = 1246, normalized size = 4.36 \[ \text {result too large to display} \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {11}{2}}\,{d x} \]
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 \[ \int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3 \,d x \]
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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