Integrand size = 43, antiderivative size = 475 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=-\frac {2 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sin (c+d x)}{693 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)} \]
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Time = 1.58 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {3126, 3110, 3100, 2827, 2716, 2719, 2720} \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {2 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right )}{231 d}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{15 d}+\frac {2 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{693 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{15 d \sqrt {\cos (c+d x)}}+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)} \]
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Rule 2716
Rule 2719
Rule 2720
Rule 2827
Rule 3100
Rule 3110
Rule 3126
Rubi steps \begin{align*} \text {integral}& = \frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2}{11} \int \frac {(a+b \cos (c+d x))^3 \left (\frac {1}{2} (8 A b+11 a B)+\frac {1}{2} (9 a A+11 b B+11 a C) \cos (c+d x)+\frac {1}{2} b (A+11 C) \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx \\ & = \frac {2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {4}{99} \int \frac {(a+b \cos (c+d x))^2 \left (\frac {3}{4} \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right )+\frac {1}{4} \left (146 a A b+77 a^2 B+99 b^2 B+198 a b C\right ) \cos (c+d x)+\frac {1}{4} b (17 A b+11 a B+99 b C) \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx \\ & = \frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {8}{693} \int \frac {(a+b \cos (c+d x)) \left (\frac {1}{8} \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right )+\frac {1}{8} \left (1441 a^2 b B+693 b^3 B+45 a^3 (9 A+11 C)+a b^2 (1381 A+2079 C)\right ) \cos (c+d x)+\frac {1}{8} b \left (242 a b B+9 a^2 (9 A+11 C)+b^2 (167 A+693 C)\right ) \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}-\frac {16 \int \frac {-\frac {15}{16} \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right )-\frac {231}{16} \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \cos (c+d x)-\frac {5}{16} b^2 \left (242 a b B+9 a^2 (9 A+11 C)+b^2 (167 A+693 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{3465} \\ & = \frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sin (c+d x)}{693 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}-\frac {32 \int \frac {-\frac {693}{32} \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right )-\frac {45}{32} \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)} \, dx}{10395} \\ & = \frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sin (c+d x)}{693 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}-\frac {1}{15} \left (-7 a^4 B-54 a^2 b^2 B-15 b^4 B-12 a b^3 (3 A+5 C)-4 a^3 b (7 A+9 C)\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)} \, dx-\frac {1}{231} \left (-220 a^3 b B-308 a b^3 B-77 b^4 (A+3 C)-66 a^2 b^2 (5 A+7 C)-5 a^4 (9 A+11 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sin (c+d x)}{693 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}-\frac {1}{15} \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = -\frac {2 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sin (c+d x)}{693 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)} \\ \end{align*}
Time = 13.16 (sec) , antiderivative size = 381, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {-154 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {2 \left (385 a^3 (4 A b+a B)+77 a \left (36 A b^3+7 a^3 B+54 a b^2 B+4 a^2 b (7 A+9 C)\right ) \cos ^2(c+d x)+15 \left (77 A b^4+220 a^3 b B+308 a b^3 B+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \cos ^3(c+d x)+231 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \cos ^4(c+d x)\right ) \sin (c+d x)+45 \left (a^2 \left (66 A b^2+44 a b B+a^2 (9 A+11 C)\right ) \sin (2 (c+d x))+14 a^4 A \tan (c+d x)\right )}{3 \cos ^{\frac {9}{2}}(c+d x)}}{1155 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1522\) vs. \(2(499)=998\).
Time = 12.78 (sec) , antiderivative size = 1523, normalized size of antiderivative = 3.21
method | result | size |
default | \(\text {Expression too large to display}\) | \(1523\) |
parts | \(\text {Expression too large to display}\) | \(1790\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.18 (sec) , antiderivative size = 588, normalized size of antiderivative = 1.24 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=-\frac {15 \, \sqrt {2} {\left (5 i \, {\left (9 \, A + 11 \, C\right )} a^{4} + 220 i \, B a^{3} b + 66 i \, {\left (5 \, A + 7 \, C\right )} a^{2} b^{2} + 308 i \, B a b^{3} + 77 i \, {\left (A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{6} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-5 i \, {\left (9 \, A + 11 \, C\right )} a^{4} - 220 i \, B a^{3} b - 66 i \, {\left (5 \, A + 7 \, C\right )} a^{2} b^{2} - 308 i \, B a b^{3} - 77 i \, {\left (A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{6} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 \, \sqrt {2} {\left (7 i \, B a^{4} + 4 i \, {\left (7 \, A + 9 \, C\right )} a^{3} b + 54 i \, B a^{2} b^{2} + 12 i \, {\left (3 \, A + 5 \, C\right )} a b^{3} + 15 i \, B b^{4}\right )} \cos \left (d x + c\right )^{6} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 \, \sqrt {2} {\left (-7 i \, B a^{4} - 4 i \, {\left (7 \, A + 9 \, C\right )} a^{3} b - 54 i \, B a^{2} b^{2} - 12 i \, {\left (3 \, A + 5 \, C\right )} a b^{3} - 15 i \, B b^{4}\right )} \cos \left (d x + c\right )^{6} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (231 \, {\left (7 \, B a^{4} + 4 \, {\left (7 \, A + 9 \, C\right )} a^{3} b + 54 \, B a^{2} b^{2} + 12 \, {\left (3 \, A + 5 \, C\right )} a b^{3} + 15 \, B b^{4}\right )} \cos \left (d x + c\right )^{5} + 315 \, A a^{4} + 15 \, {\left (5 \, {\left (9 \, A + 11 \, C\right )} a^{4} + 220 \, B a^{3} b + 66 \, {\left (5 \, A + 7 \, C\right )} a^{2} b^{2} + 308 \, B a b^{3} + 77 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} + 77 \, {\left (7 \, B a^{4} + 4 \, {\left (7 \, A + 9 \, C\right )} a^{3} b + 54 \, B a^{2} b^{2} + 36 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 45 \, {\left ({\left (9 \, A + 11 \, C\right )} a^{4} + 44 \, B a^{3} b + 66 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 385 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{3465 \, d \cos \left (d x + c\right )^{6}} \]
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Timed out. \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {13}{2}}} \,d x } \]
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\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {13}{2}}} \,d x } \]
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Time = 12.68 (sec) , antiderivative size = 1161, normalized size of antiderivative = 2.44 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\text {Too large to display} \]
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