Integrand size = 45, antiderivative size = 284 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {16 a^2 (336 A+374 B+429 C) \sin (c+d x)}{3465 d \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {8 a^2 (336 A+374 B+429 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{3465 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (336 A+374 B+429 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (84 A+110 B+99 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (3 A+11 B) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{99 d}+\frac {2 A \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d} \]
[Out]
Time = 1.07 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4350, 4171, 4102, 4100, 3890, 3889} \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 a^2 (84 A+110 B+99 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{693 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^2 (336 A+374 B+429 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{1155 d \sqrt {a \sec (c+d x)+a}}+\frac {8 a^2 (336 A+374 B+429 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3465 d \sqrt {a \sec (c+d x)+a}}+\frac {16 a^2 (336 A+374 B+429 C) \sin (c+d x)}{3465 d \sqrt {\cos (c+d x)} \sqrt {a \sec (c+d x)+a}}+\frac {2 a (3 A+11 B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{99 d}+\frac {2 A \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d} \]
[In]
[Out]
Rule 3889
Rule 3890
Rule 4100
Rule 4102
Rule 4171
Rule 4350
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} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx \\ & = \frac {2 A \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \sec (c+d x))^{3/2} \left (\frac {1}{2} a (3 A+11 B)+\frac {1}{2} a (6 A+11 C) \sec (c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx}{11 a} \\ & = \frac {2 a (3 A+11 B) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{99 d}+\frac {2 A \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \sec (c+d x)} \left (\frac {1}{4} a^2 (84 A+110 B+99 C)+\frac {3}{4} a^2 (24 A+22 B+33 C) \sec (c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx}{99 a} \\ & = \frac {2 a^2 (84 A+110 B+99 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (3 A+11 B) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{99 d}+\frac {2 A \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac {1}{231} \left (a (336 A+374 B+429 C) \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 {2 a^2 (336 A+374 B+429 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (84 A+110 B+99 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (3 A+11 B) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{99 d}+\frac {2 A \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac {\left (4 a (336 A+374 B+429 C) \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 {8 a^2 (336 A+374 B+429 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{3465 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (336 A+374 B+429 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (84 A+110 B+99 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (3 A+11 B) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{99 d}+\frac {2 A \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac {\left (8 a (336 A+374 B+429 C) \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 {16 a^2 (336 A+374 B+429 C) \sin (c+d x)}{3465 d \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {8 a^2 (336 A+374 B+429 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{3465 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (336 A+374 B+429 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (84 A+110 B+99 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (3 A+11 B) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{99 d}+\frac {2 A \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d} \\ \end{align*}
Time = 6.54 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.56 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a \sqrt {\cos (c+d x)} (55482 A+59158 B+65208 C+(34734 A+44 (799 B+759 C)) \cos (c+d x)+8 (1743 A+1507 B+1287 C) \cos (2 (c+d x))+4935 A \cos (3 (c+d x))+3740 B \cos (3 (c+d x))+1980 C \cos (3 (c+d x))+1470 A \cos (4 (c+d x))+770 B \cos (4 (c+d x))+315 A \cos (5 (c+d x))) \sqrt {a (1+\sec (c+d x))} \tan \left (\frac {1}{2} (c+d x)\right )}{27720 d} \]
[In]
[Out]
Time = 0.75 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.60
method | result | size |
default | \(-\frac {2 a \left (\left (315 \cos \left (d x +c \right )^{5}+735 \cos \left (d x +c \right )^{4}+840 \cos \left (d x +c \right )^{3}+1008 \cos \left (d x +c \right )^{2}+1344 \cos \left (d x +c \right )+2688\right ) A +\left (385 \cos \left (d x +c \right )^{4}+935 \cos \left (d x +c \right )^{3}+1122 \cos \left (d x +c \right )^{2}+1496 \cos \left (d x +c \right )+2992\right ) B +\left (495 \cos \left (d x +c \right )^{3}+1287 \cos \left (d x +c \right )^{2}+1716 \cos \left (d x +c \right )+3432\right ) C \right ) \sqrt {\cos \left (d x +c \right )}\, \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}{3465 d}\) | \(169\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.54 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (315 \, A a \cos \left (d x + c\right )^{5} + 35 \, {\left (21 \, A + 11 \, B\right )} a \cos \left (d x + c\right )^{4} + 5 \, {\left (168 \, A + 187 \, B + 99 \, C\right )} a \cos \left (d x + c\right )^{3} + 3 \, {\left (336 \, A + 374 \, B + 429 \, C\right )} a \cos \left (d x + c\right )^{2} + 4 \, {\left (336 \, A + 374 \, B + 429 \, C\right )} a \cos \left (d x + c\right ) + 8 \, {\left (336 \, A + 374 \, B + 429 \, C\right )} a\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{3465 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
[In]
[Out]
Timed out. \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 862 vs. \(2 (248) = 496\).
Time = 0.51 (sec) , antiderivative size = 862, normalized size of antiderivative = 3.04 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{\frac {11}{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^{11/2}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]
[In]
[Out]