Optimal. Leaf size=343 \[ \frac {4 a^3 (210 A+253 B+264 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{1155 d}+\frac {4 a^3 (105 A+121 B+143 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{231 d}+\frac {4 a^3 (15 A+17 B+21 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{693 d}+\frac {4 a^3 (105 A+121 B+143 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}-\frac {4 a^3 (15 A+17 B+21 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 (6 A+11 B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{99 a d}+\frac {2 A \sin (c+d x) \sec ^{\frac {11}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d} \]
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Rubi [A] time = 0.84, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.209, Rules used = {4221, 3043, 2975, 2968, 3021, 2748, 2636, 2641, 2639} \[ \frac {4 a^3 (210 A+253 B+264 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{1155 d}+\frac {4 a^3 (105 A+121 B+143 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{231 d}+\frac {4 a^3 (15 A+17 B+21 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{693 d}+\frac {4 a^3 (105 A+121 B+143 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}-\frac {4 a^3 (15 A+17 B+21 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 (6 A+11 B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{99 a d}+\frac {2 A \sin (c+d x) \sec ^{\frac {11}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 2641
Rule 2748
Rule 2968
Rule 2975
Rule 3021
Rule 3043
Rule 4221
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 {13}{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 {13}{2}}(c+d x)} \, dx\\ &=\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 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+11 B)+\frac {1}{2} a (3 A+11 C) \cos (c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx}{11 a}\\ &=\frac {2 (6 A+11 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 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 (105 A+143 B+99 C)+\frac {3}{4} a^2 (15 A+11 B+33 C) \cos (c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx}{99 a}\\ &=\frac {2 (105 A+143 B+99 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 (6 A+11 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 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 (210 A+253 B+264 C)+\frac {15}{4} a^3 (21 A+22 B+33 C) \cos (c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx}{693 a}\\ &=\frac {2 (105 A+143 B+99 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 (6 A+11 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {3}{4} a^4 (210 A+253 B+264 C)+\left (\frac {15}{4} a^4 (21 A+22 B+33 C)+\frac {3}{4} a^4 (210 A+253 B+264 C)\right ) \cos (c+d x)+\frac {15}{4} a^4 (21 A+22 B+33 C) \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x)} \, dx}{693 a}\\ &=\frac {4 a^3 (210 A+253 B+264 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d}+\frac {2 (105 A+143 B+99 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 (6 A+11 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {\left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {45}{8} a^4 (105 A+121 B+143 C)+\frac {231}{8} a^4 (15 A+17 B+21 C) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{3465 a}\\ &=\frac {4 a^3 (210 A+253 B+264 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d}+\frac {2 (105 A+143 B+99 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 (6 A+11 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {1}{15} \left (2 a^3 (15 A+17 B+21 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)} \, dx+\frac {1}{77} \left (2 a^3 (105 A+121 B+143 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {4 a^3 (15 A+17 B+21 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^3 (105 A+121 B+143 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {4 a^3 (210 A+253 B+264 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d}+\frac {2 (105 A+143 B+99 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 (6 A+11 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}-\frac {1}{15} \left (2 a^3 (15 A+17 B+21 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{231} \left (2 a^3 (105 A+121 B+143 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 (15 A+17 B+21 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 (105 A+121 B+143 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {4 a^3 (15 A+17 B+21 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^3 (105 A+121 B+143 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {4 a^3 (210 A+253 B+264 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d}+\frac {2 (105 A+143 B+99 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 (6 A+11 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}\\ \end {align*}
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Mathematica [A] time = 3.19, size = 242, normalized size = 0.71 \[ \frac {a^3 \sec ^{\frac {11}{2}}(c+d x) \left (480 (105 A+121 B+143 C) \cos ^{\frac {11}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-7392 (15 A+17 B+21 C) \cos ^{\frac {11}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 \sin (c+d x) (154 (375 A+377 B+396 C) \cos (c+d x)+60 (336 A+341 B+319 C) \cos (2 (c+d x))+21945 A \cos (3 (c+d x))+3150 A \cos (4 (c+d x))+3465 A \cos (5 (c+d x))+19530 A+24871 B \cos (3 (c+d x))+3630 B \cos (4 (c+d x))+3927 B \cos (5 (c+d x))+16830 B+28413 C \cos (3 (c+d x))+4290 C \cos (4 (c+d x))+4851 C \cos (5 (c+d x))+14850 C)\right )}{27720 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C a^{3} \cos \left (d x + c\right )^{5} + {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + {\left (A + 3 \, B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + {\left (3 \, A + 3 \, B + C\right )} a^{3} \cos \left (d x + c\right )^{2} + {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) + A a^{3}\right )} \sec \left (d x + c\right )^{\frac {13}{2}}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {13}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 15.05, size = 1424, normalized size = 4.15 \[ \text {result too large to display} \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {13}{2}}\,{d x} \]
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 \[ \int {\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{13/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
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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