Integrand size = 35, antiderivative size = 275 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\frac {32 a^2 (168 A+187 B) \sin (c+d x)}{3465 d \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {16 a^2 (168 A+187 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{3465 d \sqrt {a+a \sec (c+d x)}}+\frac {4 a^2 (168 A+187 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (168 A+187 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (12 A+11 B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a A \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{11 d} \]
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Time = 0.89 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3034, 4102, 4100, 3890, 3889} \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\frac {2 a^2 (12 A+11 B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{99 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^2 (168 A+187 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{693 d \sqrt {a \sec (c+d x)+a}}+\frac {4 a^2 (168 A+187 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{1155 d \sqrt {a \sec (c+d x)+a}}+\frac {16 a^2 (168 A+187 B) \sin (c+d x) \sqrt {\cos (c+d x)}}{3465 d \sqrt {a \sec (c+d x)+a}}+\frac {32 a^2 (168 A+187 B) \sin (c+d x)}{3465 d \sqrt {\cos (c+d x)} \sqrt {a \sec (c+d x)+a}}+\frac {2 a A \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{11 d} \]
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Rule 3034
Rule 3889
Rule 3890
Rule 4100
Rule 4102
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx \\ & = \frac {2 a A \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{11 d}+\frac {1}{11} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \sec (c+d x)} \left (\frac {1}{2} a (12 A+11 B)+\frac {1}{2} a (8 A+11 B) \sec (c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx \\ & = \frac {2 a^2 (12 A+11 B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a A \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{11 d}+\frac {1}{99} \left (a (168 A+187 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sec ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 a^2 (168 A+187 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (12 A+11 B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a A \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{11 d}+\frac {1}{231} \left (2 a (168 A+187 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sec ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {4 a^2 (168 A+187 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (168 A+187 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (12 A+11 B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a A \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{11 d}+\frac {\left (8 a (168 A+187 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{1155} \\ & = \frac {16 a^2 (168 A+187 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{3465 d \sqrt {a+a \sec (c+d x)}}+\frac {4 a^2 (168 A+187 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (168 A+187 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (12 A+11 B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a A \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{11 d}+\frac {\left (16 a (168 A+187 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{3465} \\ & = \frac {32 a^2 (168 A+187 B) \sin (c+d x)}{3465 d \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {16 a^2 (168 A+187 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{3465 d \sqrt {a+a \sec (c+d x)}}+\frac {4 a^2 (168 A+187 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (168 A+187 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (12 A+11 B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a A \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{11 d} \\ \end{align*}
Time = 1.85 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.48 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\frac {2 a \sqrt {\cos (c+d x)} \left (2688 A+2992 B+8 (168 A+187 B) \cos (c+d x)+6 (168 A+187 B) \cos ^2(c+d x)+(840 A+935 B) \cos ^3(c+d x)+35 (21 A+11 B) \cos ^4(c+d x)+315 A \cos ^5(c+d x)\right ) \sqrt {a (1+\sec (c+d x))} \sin (c+d x)}{3465 d (1+\cos (c+d x))} \]
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Time = 5.38 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.50
method | result | size |
default | \(-\frac {2 a \left (\left (315 \cos \left (d x +c \right )^{5}+735 \cos \left (d x +c \right )^{4}+840 \cos \left (d x +c \right )^{3}+1008 \cos \left (d x +c \right )^{2}+1344 \cos \left (d x +c \right )+2688\right ) A +\left (385 \cos \left (d x +c \right )^{4}+935 \cos \left (d x +c \right )^{3}+1122 \cos \left (d x +c \right )^{2}+1496 \cos \left (d x +c \right )+2992\right ) B \right ) \sqrt {\cos \left (d x +c \right )}\, \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}{3465 d}\) | \(137\) |
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Time = 0.28 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.52 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\frac {2 \, {\left (315 \, A a \cos \left (d x + c\right )^{5} + 35 \, {\left (21 \, A + 11 \, B\right )} a \cos \left (d x + c\right )^{4} + 5 \, {\left (168 \, A + 187 \, B\right )} a \cos \left (d x + c\right )^{3} + 6 \, {\left (168 \, A + 187 \, B\right )} a \cos \left (d x + c\right )^{2} + 8 \, {\left (168 \, A + 187 \, B\right )} a \cos \left (d x + c\right ) + 16 \, {\left (168 \, A + 187 \, B\right )} a\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{3465 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
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Timed out. \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 703 vs. \(2 (239) = 478\).
Time = 0.47 (sec) , antiderivative size = 703, normalized size of antiderivative = 2.56 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\text {Too large to display} \]
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\[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{\frac {11}{2}} \,d x } \]
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Timed out. \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\int {\cos \left (c+d\,x\right )}^{11/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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