Integrand size = 35, antiderivative size = 270 \[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=-\frac {16 a^2 (2 A+3 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^2 (5 A+7 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {16 a^2 (2 A+3 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^2 (5 A+7 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a^2 (19 A+21 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {8 A \left (a^2+a^2 \cos (c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+a \cos (c+d x))^2 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d} \]
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Time = 0.62 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {4306, 3123, 3054, 3047, 3100, 2827, 2716, 2720, 2719} \[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\frac {2 a^2 (19 A+21 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{105 d}+\frac {4 a^2 (5 A+7 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {16 a^2 (2 A+3 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^2 (5 A+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}-\frac {16 a^2 (2 A+3 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {8 A \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{63 d}+\frac {2 A \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d} \]
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Rule 2716
Rule 2719
Rule 2720
Rule 2827
Rule 3047
Rule 3054
Rule 3100
Rule 3123
Rule 4306
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx \\ & = \frac {2 A (a+a \cos (c+d x))^2 \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))^2 \left (2 a A+\frac {3}{2} a (A+3 C) \cos (c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx}{9 a} \\ & = \frac {8 A \left (a^2+a^2 \cos (c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+a \cos (c+d x))^2 \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)) \left (\frac {3}{4} a^2 (19 A+21 C)+\frac {3}{4} a^2 (11 A+21 C) \cos (c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx}{63 a} \\ & = \frac {8 A \left (a^2+a^2 \cos (c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+a \cos (c+d x))^2 \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 {\frac {3}{4} a^3 (19 A+21 C)+\left (\frac {3}{4} a^3 (11 A+21 C)+\frac {3}{4} a^3 (19 A+21 C)\right ) \cos (c+d x)+\frac {3}{4} a^3 (11 A+21 C) \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x)} \, dx}{63 a} \\ & = \frac {2 a^2 (19 A+21 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {8 A \left (a^2+a^2 \cos (c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+a \cos (c+d x))^2 \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 {45}{4} a^3 (5 A+7 C)+21 a^3 (2 A+3 C) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{315 a} \\ & = \frac {2 a^2 (19 A+21 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {8 A \left (a^2+a^2 \cos (c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+a \cos (c+d x))^2 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{15} \left (8 a^2 (2 A+3 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}{7} \left (2 a^2 (5 A+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\cos ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {16 a^2 (2 A+3 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^2 (5 A+7 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a^2 (19 A+21 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {8 A \left (a^2+a^2 \cos (c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+a \cos (c+d x))^2 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}-\frac {1}{15} \left (8 a^2 (2 A+3 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (2 a^2 (5 A+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = -\frac {16 a^2 (2 A+3 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^2 (5 A+7 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {16 a^2 (2 A+3 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^2 (5 A+7 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a^2 (19 A+21 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {8 A \left (a^2+a^2 \cos (c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+a \cos (c+d x))^2 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 8.73 (sec) , antiderivative size = 655, normalized size of antiderivative = 2.43 \[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\frac {4 \sqrt {2} A e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} (a+a \cos (c+d x))^2 \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{45 d}+\frac {2 \sqrt {2} C e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} (a+a \cos (c+d x))^2 \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{15 d}+\frac {5 A \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {C \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\sec (c+d x)}}{3 d}+(a+a \cos (c+d x))^2 \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\sec (c+d x)} \left (\frac {4 (2 A+3 C) \cos (d x) \csc (c)}{15 d}+\frac {A \sec (c) \sec ^4(c+d x) \sin (d x)}{18 d}+\frac {\sec (c) \sec ^3(c+d x) (7 A \sin (c)+18 A \sin (d x))}{126 d}+\frac {\sec (c) \sec ^2(c+d x) (90 A \sin (c)+112 A \sin (d x)+63 C \sin (d x))}{630 d}+\frac {\sec (c) \sec (c+d x) (112 A \sin (c)+63 C \sin (c)+150 A \sin (d x)+210 C \sin (d x))}{630 d}+\frac {(5 A+7 C) \tan (c)}{21 d}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1140\) vs. \(2(290)=580\).
Time = 152.41 (sec) , antiderivative size = 1141, normalized size of antiderivative = 4.23
method | result | size |
default | \(\text {Expression too large to display}\) | \(1141\) |
parts | \(\text {Expression too large to display}\) | \(1403\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.03 \[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 84 i \, \sqrt {2} {\left (2 \, A + 3 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 84 i \, \sqrt {2} {\left (2 \, A + 3 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (168 \, {\left (2 \, A + 3 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 30 \, {\left (5 \, A + 7 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 7 \, {\left (16 \, A + 9 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 90 \, A a^{2} \cos \left (d x + c\right ) + 35 \, A a^{2}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{315 \, d \cos \left (d x + c\right )^{4}} \]
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Timed out. \[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\text {Timed out} \]
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\[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \]
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\[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \]
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Timed out. \[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^2 \,d x \]
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