Optimal. Leaf size=307 \[ \frac {4 a^3 (32 A+41 B+42 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{105 d}+\frac {4 a^3 (17 A+21 B+27 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{315 d}+\frac {4 a^3 (11 A+13 B+21 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {4 a^3 (17 A+21 B+27 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 (2 A+3 B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{21 a d}+\frac {2 A \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) (a \cos (c+d x)+a)^3}{9 d} \]
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Rubi [A] time = 0.82, antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 10, 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, 2639, 2641} \[ \frac {4 a^3 (32 A+41 B+42 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{105 d}+\frac {4 a^3 (17 A+21 B+27 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{315 d}+\frac {4 a^3 (11 A+13 B+21 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {4 a^3 (17 A+21 B+27 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 (2 A+3 B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{21 a d}+\frac {2 A \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) (a \cos (c+d x)+a)^3}{9 d} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 2641
Rule 2748
Rule 2968
Rule 2975
Rule 3021
Rule 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 {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+B \cos (c+d x)+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 (\frac {3}{2} a (2 A+3 B)+\frac {1}{2} a (A+9 C) \cos (c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx}{9 a}\\ &=\frac {2 (2 A+3 B) \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+99 B+63 C)+\frac {1}{4} a^2 (13 A+9 B+63 C) \cos (c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx}{63 a}\\ &=\frac {2 (73 A+99 B+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 {2 (2 A+3 B) \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}{4} a^3 (32 A+41 B+42 C)+\frac {3}{4} a^3 (23 A+24 B+63 C) \cos (c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{315 a}\\ &=\frac {2 (73 A+99 B+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 {2 (2 A+3 B) \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}{4} a^4 (32 A+41 B+42 C)+\left (\frac {9}{4} a^4 (32 A+41 B+42 C)+\frac {3}{4} a^4 (23 A+24 B+63 C)\right ) \cos (c+d x)+\frac {3}{4} a^4 (23 A+24 B+63 C) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{315 a}\\ &=\frac {4 a^3 (32 A+41 B+42 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 (73 A+99 B+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 {2 (2 A+3 B) \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+21 B+27 C)+\frac {45}{8} a^4 (11 A+13 B+21 C) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)} \, dx}{945 a}\\ &=\frac {4 a^3 (32 A+41 B+42 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 (73 A+99 B+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 {2 (2 A+3 B) \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+13 B+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+21 B+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+13 B+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+21 B+27 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^3 (32 A+41 B+42 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 (73 A+99 B+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 {2 (2 A+3 B) \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+21 B+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+21 B+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+13 B+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+21 B+27 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^3 (32 A+41 B+42 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 (73 A+99 B+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 {2 (2 A+3 B) \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 [A] time = 2.36, size = 209, normalized size = 0.68 \[ \frac {a^3 \sec ^{\frac {9}{2}}(c+d x) \left (240 (11 A+13 B+21 C) \cos ^{\frac {9}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-336 (17 A+21 B+27 C) \cos ^{\frac {9}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 \sin (c+d x) (45 (34 A+30 B+21 C) \cos (c+d x)+14 (136 A+153 B+171 C) \cos (2 (c+d x))+330 A \cos (3 (c+d x))+357 A \cos (4 (c+d x))+1687 A+390 B \cos (3 (c+d x))+441 B \cos (4 (c+d x))+1701 B+315 C \cos (3 (c+d x))+567 C \cos (4 (c+d x))+1827 C)\right )}{1260 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C a^{3} \cos \left (d x + c\right )^{5} + {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + {\left (A + 3 \, B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + {\left (3 \, A + 3 \, B + C\right )} a^{3} \cos \left (d x + c\right )^{2} + {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) + A a^{3}\right )} \sec \left (d x + c\right )^{\frac {11}{2}}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {11}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 12.92, size = 1262, normalized size = 4.11 \[ \text {result too large to display} \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {11}{2}}\,{d x} \]
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 \[ \int {\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
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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