Integrand size = 43, antiderivative size = 342 \[ \int (a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=-\frac {2 \left (18 a b B+3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (10 a A b+5 a^2 B+7 b^2 B+14 a b C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (18 a b B+3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 \left (10 a A b+5 a^2 B+7 b^2 B+14 a b C\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 \left (4 A b^2+18 a b B+a^2 (7 A+9 C)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 a (4 A b+9 a B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+b \cos (c+d x))^2 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d} \]
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Time = 0.97 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {4306, 3126, 3110, 3100, 2827, 2716, 2720, 2719} \[ \int (a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\frac {2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (a^2 (7 A+9 C)+18 a b B+4 A b^2\right )}{45 d}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 B+10 a A b+14 a b C+7 b^2 B\right )}{21 d}+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right )}{15 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (5 a^2 B+10 a A b+14 a b C+7 b^2 B\right )}{21 d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right )}{15 d}+\frac {2 a (9 a B+4 A b) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{63 d}+\frac {2 A \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d} \]
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Rule 2716
Rule 2719
Rule 2720
Rule 2827
Rule 3100
Rule 3110
Rule 3126
Rule 4306
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^2 \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+b \cos (c+d x))^2 \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)) \left (\frac {1}{2} (4 A b+9 a B)+\frac {1}{2} (7 a A+9 b B+9 a C) \cos (c+d x)+\frac {3}{2} b (A+3 C) \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx \\ & = \frac {2 a (4 A b+9 a B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+b \cos (c+d x))^2 \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 {-\frac {7}{4} \left (4 A b^2+18 a b B+a^2 (7 A+9 C)\right )-\frac {9}{4} \left (10 a A b+5 a^2 B+7 b^2 B+14 a b C\right ) \cos (c+d x)-\frac {21}{4} b^2 (A+3 C) \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 \left (4 A b^2+18 a b B+a^2 (7 A+9 C)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 a (4 A b+9 a B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+b \cos (c+d x))^2 \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 {-\frac {45}{8} \left (10 a A b+5 a^2 B+7 b^2 B+14 a b C\right )-\frac {21}{8} \left (18 a b B+3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {2 \left (4 A b^2+18 a b B+a^2 (7 A+9 C)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 a (4 A b+9 a B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+b \cos (c+d x))^2 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{7} \left (\left (10 a A b+5 a^2 B+7 b^2 B+14 a b C\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\cos ^{\frac {5}{2}}(c+d x)} \, dx+\frac {1}{15} \left (\left (18 a b B+3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \frac {2 \left (18 a b B+3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 \left (10 a A b+5 a^2 B+7 b^2 B+14 a b C\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 \left (4 A b^2+18 a b B+a^2 (7 A+9 C)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 a (4 A b+9 a B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+b \cos (c+d x))^2 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{21} \left (\left (10 a A b+5 a^2 B+7 b^2 B+14 a b 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 (18 a b B+3 b^2 (3 A+5 C)+a^2 (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 (18 a b B+3 b^2 (3 A+5 C)+a^2 (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 {2 \left (10 a A b+5 a^2 B+7 b^2 B+14 a b C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (18 a b B+3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 \left (10 a A b+5 a^2 B+7 b^2 B+14 a b C\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 \left (4 A b^2+18 a b B+a^2 (7 A+9 C)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 a (4 A b+9 a B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+b \cos (c+d x))^2 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d} \\ \end{align*}
Time = 8.03 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.04 \[ \int (a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\frac {\frac {2 \left (-49 a^2 A-63 A b^2-126 a b B-63 a^2 C-105 b^2 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}+2 \left (50 a A b+25 a^2 B+35 b^2 B+70 a b C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{105 d}+\frac {\sqrt {\sec (c+d x)} \left (\frac {2}{15} \left (7 a^2 A+9 A b^2+18 a b B+9 a^2 C+15 b^2 C\right ) \sin (c+d x)+\frac {2}{7} \sec ^3(c+d x) \left (2 a A b \sin (c+d x)+a^2 B \sin (c+d x)\right )+\frac {2}{45} \sec ^2(c+d x) \left (7 a^2 A \sin (c+d x)+9 A b^2 \sin (c+d x)+18 a b B \sin (c+d x)+9 a^2 C \sin (c+d x)\right )+\frac {2}{21} \sec (c+d x) \left (10 a A b \sin (c+d x)+5 a^2 B \sin (c+d x)+7 b^2 B \sin (c+d x)+14 a b C \sin (c+d x)\right )+\frac {2}{9} a^2 A \sec ^3(c+d x) \tan (c+d x)\right )}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1168\) vs. \(2(362)=724\).
Time = 1463.60 (sec) , antiderivative size = 1169, normalized size of antiderivative = 3.42
method | result | size |
default | \(\text {Expression too large to display}\) | \(1169\) |
parts | \(\text {Expression too large to display}\) | \(1422\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.17 \[ \int (a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=-\frac {15 \, \sqrt {2} {\left (5 i \, B a^{2} + 2 i \, {\left (5 \, A + 7 \, C\right )} a b + 7 i \, B b^{2}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-5 i \, B a^{2} - 2 i \, {\left (5 \, A + 7 \, C\right )} a b - 7 i \, B b^{2}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (i \, {\left (7 \, A + 9 \, C\right )} a^{2} + 18 i \, B a b + 3 i \, {\left (3 \, A + 5 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-i \, {\left (7 \, A + 9 \, C\right )} a^{2} - 18 i \, B a b - 3 i \, {\left (3 \, A + 5 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (21 \, {\left ({\left (7 \, A + 9 \, C\right )} a^{2} + 18 \, B a b + 3 \, {\left (3 \, A + 5 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{4} + 15 \, {\left (5 \, B a^{2} + 2 \, {\left (5 \, A + 7 \, C\right )} a b + 7 \, B b^{2}\right )} \cos \left (d x + c\right )^{3} + 35 \, A a^{2} + 7 \, {\left ({\left (7 \, A + 9 \, C\right )} a^{2} + 18 \, B a b + 9 \, A b^{2}\right )} \cos \left (d x + c\right )^{2} + 45 \, {\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, d \cos \left (d x + c\right )^{4}} \]
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Timed out. \[ \int (a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\text {Timed out} \]
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\[ \int (a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \]
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\[ \int (a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \]
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Timed out. \[ \int (a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int {\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^2\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
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