Optimal. Leaf size=365 \[ \frac {4 a b \left (a^2 (101 A+147 C)+32 A b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{315 d}+\frac {2 \left (7 a^2 (7 A+9 C)+48 A b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{315 d}+\frac {8 a b \left (a^2 (5 A+7 C)+7 b^2 (A+3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 \left (21 a^4 (7 A+9 C)+7 a^2 b^2 (155 A+261 C)+192 A b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{315 d}-\frac {2 \left (a^4 (7 A+9 C)+18 a^2 b^2 (3 A+5 C)+15 b^4 (A-C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 A \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))^4}{9 d}+\frac {16 A b \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^3}{63 d} \]
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Rubi [A] time = 1.28, antiderivative size = 365, 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, 3048, 3047, 3031, 3021, 2748, 2641, 2639} \[ \frac {4 a b \left (a^2 (101 A+147 C)+32 A b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{315 d}+\frac {2 \left (7 a^2 b^2 (155 A+261 C)+21 a^4 (7 A+9 C)+192 A b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{315 d}+\frac {2 \left (7 a^2 (7 A+9 C)+48 A b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{315 d}+\frac {8 a b \left (a^2 (5 A+7 C)+7 b^2 (A+3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {2 \left (18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+15 b^4 (A-C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 A \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))^4}{9 d}+\frac {16 A b \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^3}{63 d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2748
Rule 3021
Rule 3031
Rule 3047
Rule 3048
Rule 4221
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^4 \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+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx\\ &=\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{9} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^3 \left (4 A b+\frac {1}{2} a (7 A+9 C) \cos (c+d x)-\frac {1}{2} b (A-9 C) \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\frac {16 A b (a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{63} \left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^2 \left (\frac {1}{4} \left (48 A b^2+7 a^2 (7 A+9 C)\right )+\frac {1}{2} a b (41 A+63 C) \cos (c+d x)-\frac {3}{4} b^2 (5 A-21 C) \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 \left (48 A b^2+7 a^2 (7 A+9 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {16 A b (a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{315} \left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x)) \left (\frac {3}{4} b \left (32 A b^2+a^2 (101 A+147 C)\right )+\frac {1}{8} a \left (21 a^2 (7 A+9 C)+b^2 (479 A+945 C)\right ) \cos (c+d x)-\frac {1}{8} b \left (3 b^2 (41 A-105 C)+7 a^2 (7 A+9 C)\right ) \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {4 a b \left (32 A b^2+a^2 (101 A+147 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {2 \left (48 A b^2+7 a^2 (7 A+9 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {16 A b (a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}-\frac {1}{945} \left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {3}{16} \left (192 A b^4+21 a^4 (7 A+9 C)+7 a^2 b^2 (155 A+261 C)\right )-\frac {45}{4} a b \left (7 b^2 (A+3 C)+a^2 (5 A+7 C)\right ) \cos (c+d x)+\frac {3}{16} b^2 \left (3 b^2 (41 A-105 C)+7 a^2 (7 A+9 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 \left (192 A b^4+21 a^4 (7 A+9 C)+7 a^2 b^2 (155 A+261 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d}+\frac {4 a b \left (32 A b^2+a^2 (101 A+147 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {2 \left (48 A b^2+7 a^2 (7 A+9 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {16 A b (a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}-\frac {1}{945} \left (32 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {45}{8} a b \left (7 b^2 (A+3 C)+a^2 (5 A+7 C)\right )+\frac {63}{32} \left (15 b^4 (A-C)+18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 \left (192 A b^4+21 a^4 (7 A+9 C)+7 a^2 b^2 (155 A+261 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d}+\frac {4 a b \left (32 A b^2+a^2 (101 A+147 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {2 \left (48 A b^2+7 a^2 (7 A+9 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {16 A b (a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{21} \left (4 a b \left (7 b^2 (A+3 C)+a^2 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx-\frac {1}{15} \left (\left (15 b^4 (A-C)+18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {2 \left (15 b^4 (A-C)+18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) \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 {8 a b \left (7 b^2 (A+3 C)+a^2 (5 A+7 C)\right ) \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 {2 \left (192 A b^4+21 a^4 (7 A+9 C)+7 a^2 b^2 (155 A+261 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d}+\frac {4 a b \left (32 A b^2+a^2 (101 A+147 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {2 \left (48 A b^2+7 a^2 (7 A+9 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {16 A b (a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}\\ \end {align*}
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Mathematica [A] time = 6.75, size = 356, normalized size = 0.98 \[ \frac {\sqrt {\sec (c+d x)} \left (\frac {2}{9} a^4 A \tan (c+d x) \sec ^3(c+d x)+\frac {8}{21} \sec (c+d x) \left (5 a^3 A b \sin (c+d x)+7 a^3 b C \sin (c+d x)+7 a A b^3 \sin (c+d x)\right )+\frac {8}{7} a^3 A b \tan (c+d x) \sec ^2(c+d x)+\frac {2}{45} \sec ^2(c+d x) \left (7 a^4 A \sin (c+d x)+9 a^4 C \sin (c+d x)+54 a^2 A b^2 \sin (c+d x)\right )+\frac {2}{15} \left (7 a^4 A+9 a^4 C+54 a^2 A b^2+90 a^2 b^2 C+15 A b^4\right ) \sin (c+d x)\right )}{d}+\frac {2 \left (100 a^3 A b+140 a^3 b C+140 a A b^3+420 a b^3 C\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\frac {2 \left (-49 a^4 A-63 a^4 C-378 a^2 A b^2-630 a^2 b^2 C-105 A b^4+105 b^4 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}}{105 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{4} \cos \left (d x + c\right )^{6} + 4 \, C a b^{3} \cos \left (d x + c\right )^{5} + 4 \, A a^{3} b \cos \left (d x + c\right ) + A a^{4} + {\left (6 \, C a^{2} b^{2} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (C a^{3} b + A a b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{4} + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sec \left (d x + c\right )^{\frac {11}{2}}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {11}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 14.18, size = 1451, normalized size = 3.98 \[ \text {result too large to display} \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {11}{2}}\,{d x} \]
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 \[ \int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^4 \,d x \]
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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