Integrand size = 43, antiderivative size = 291 \[ \int (a+a \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=-\frac {4 a^2 (8 A+9 B+12 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^2 (5 A+6 B+7 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^2 (8 A+9 B+12 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^2 (5 A+6 B+7 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a^2 (19 A+27 B+21 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 (4 A+9 B) \left (a^2+a^2 \cos (c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 A (a+a \cos (c+d x))^2 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d} \]
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Time = 0.73 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.209, Rules used = {4306, 3122, 3054, 3047, 3100, 2827, 2716, 2720, 2719} \[ \int (a+a \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\frac {2 a^2 (19 A+27 B+21 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{105 d}+\frac {4 a^2 (5 A+6 B+7 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {4 a^2 (8 A+9 B+12 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^2 (5 A+6 B+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}-\frac {4 a^2 (8 A+9 B+12 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 (4 A+9 B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{63 d}+\frac {2 A \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d} \]
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Rule 2716
Rule 2719
Rule 2720
Rule 2827
Rule 3047
Rule 3054
Rule 3100
Rule 3122
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+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))^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 (\frac {1}{2} a (4 A+9 B)+\frac {3}{2} a (A+3 C) \cos (c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx}{9 a} \\ & = \frac {2 (4 A+9 B) \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+27 B+21 C)+\frac {3}{4} a^2 (11 A+9 B+21 C) \cos (c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx}{63 a} \\ & = \frac {2 (4 A+9 B) \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+27 B+21 C)+\left (\frac {3}{4} a^3 (11 A+9 B+21 C)+\frac {3}{4} a^3 (19 A+27 B+21 C)\right ) \cos (c+d x)+\frac {3}{4} a^3 (11 A+9 B+21 C) \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x)} \, dx}{63 a} \\ & = \frac {2 a^2 (19 A+27 B+21 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 (4 A+9 B) \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+6 B+7 C)+\frac {21}{4} a^3 (8 A+9 B+12 C) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{315 a} \\ & = \frac {2 a^2 (19 A+27 B+21 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 (4 A+9 B) \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}{7} \left (2 a^2 (5 A+6 B+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 {1}{15} \left (2 a^2 (8 A+9 B+12 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^2 (8 A+9 B+12 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^2 (5 A+6 B+7 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a^2 (19 A+27 B+21 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 (4 A+9 B) \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}{21} \left (2 a^2 (5 A+6 B+7 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^2 (8 A+9 B+12 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = -\frac {4 a^2 (8 A+9 B+12 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+6 B+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 {4 a^2 (8 A+9 B+12 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^2 (5 A+6 B+7 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a^2 (19 A+27 B+21 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 (4 A+9 B) \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*}
Time = 3.73 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.72 \[ \int (a+a \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\frac {a^2 \sec ^{\frac {9}{2}}(c+d x) \left (-336 (8 A+9 B+12 C) \cos ^{\frac {9}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+240 (5 A+6 B+7 C) \cos ^{\frac {9}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 (868 A+819 B+882 C+90 (9 A+8 B+7 C) \cos (c+d x)+14 (64 A+72 B+81 C) \cos (2 (c+d x))+150 A \cos (3 (c+d x))+180 B \cos (3 (c+d x))+210 C \cos (3 (c+d x))+168 A \cos (4 (c+d x))+189 B \cos (4 (c+d x))+252 C \cos (4 (c+d x))) \sin (c+d x)\right )}{1260 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1153\) vs. \(2(311)=622\).
Time = 162.88 (sec) , antiderivative size = 1154, normalized size of antiderivative = 3.97
method | result | size |
default | \(\text {Expression too large to display}\) | \(1154\) |
parts | \(\text {Expression too large to display}\) | \(1425\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.04 \[ \int (a+a \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (5 \, A + 6 \, B + 7 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (5 \, A + 6 \, B + 7 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 i \, \sqrt {2} {\left (8 \, A + 9 \, B + 12 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} {\left (8 \, A + 9 \, B + 12 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (42 \, {\left (8 \, A + 9 \, B + 12 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 30 \, {\left (5 \, A + 6 \, B + 7 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 7 \, {\left (16 \, A + 18 \, B + 9 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 45 \, {\left (2 \, A + B\right )} a^{2} \cos \left (d x + c\right ) + 35 \, A a^{2}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{315 \, d \cos \left (d x + c\right )^{4}} \]
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Timed out. \[ \int (a+a \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\text {Timed out} \]
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\[ \int (a+a \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \]
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\[ \int (a+a \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \]
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Timed out. \[ \int (a+a \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int {\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^2\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
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