Integrand size = 25, antiderivative size = 156 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\frac {64 a^3 \sin (c+d x)}{21 d \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {16 a^2 \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {2 \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{7 d} \]
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Time = 0.42 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {4349, 3897, 3894, 3889} \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\frac {64 a^3 \sin (c+d x)}{21 d \sqrt {\cos (c+d x)} \sqrt {a \sec (c+d x)+a}}+\frac {16 a^2 \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a \sec (c+d x)+a}}{21 d}+\frac {2 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}+\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d} \]
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Rule 3889
Rule 3894
Rule 3897
Rule 4349
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))^{5/2}}{\sec ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {1}{7} \left (5 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \sec (c+d x))^{5/2}}{\sec ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {2 a \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {2 \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {1}{7} \left (8 a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \sec (c+d x))^{3/2}}{\sec ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \frac {16 a^2 \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {2 \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {1}{21} \left (32 a^2 \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 \\ & = \frac {64 a^3 \sin (c+d x)}{21 d \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {16 a^2 \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {2 \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{7 d} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.47 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\frac {a^2 \sqrt {\cos (c+d x)} (208+101 \cos (c+d x)+24 \cos (2 (c+d x))+3 \cos (3 (c+d x))) \sqrt {a (1+\sec (c+d x))} \tan \left (\frac {1}{2} (c+d x)\right )}{42 d} \]
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Time = 0.98 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.48
\[-\frac {2 a^{2} \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \sqrt {\cos \left (d x +c \right )}\, \left (3 \cos \left (d x +c \right )^{4}+9 \cos \left (d x +c \right )^{3}+11 \cos \left (d x +c \right )^{2}+23 \cos \left (d x +c \right )-46\right ) \csc \left (d x +c \right )}{21 d}\]
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Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.59 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\frac {2 \, {\left (3 \, a^{2} \cos \left (d x + c\right )^{3} + 12 \, a^{2} \cos \left (d x + c\right )^{2} + 23 \, a^{2} \cos \left (d x + c\right ) + 46 \, a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{21 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
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Timed out. \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (132) = 264\).
Time = 0.39 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.07 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\frac {\sqrt {2} {\left (315 \, a^{2} \cos \left (\frac {6}{7} \, \arctan \left (\sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ), \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )\right ) \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 77 \, a^{2} \cos \left (\frac {4}{7} \, \arctan \left (\sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ), \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )\right ) \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 21 \, a^{2} \cos \left (\frac {2}{7} \, \arctan \left (\sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ), \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )\right ) \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) - 315 \, a^{2} \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) \sin \left (\frac {6}{7} \, \arctan \left (\sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ), \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )\right ) - 77 \, a^{2} \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) \sin \left (\frac {4}{7} \, \arctan \left (\sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ), \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )\right ) - 21 \, a^{2} \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) \sin \left (\frac {2}{7} \, \arctan \left (\sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ), \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )\right ) + 6 \, a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 21 \, a^{2} \sin \left (\frac {5}{7} \, \arctan \left (\sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ), \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )\right ) + 77 \, a^{2} \sin \left (\frac {3}{7} \, \arctan \left (\sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ), \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )\right ) + 315 \, a^{2} \sin \left (\frac {1}{7} \, \arctan \left (\sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ), \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )\right )\right )} \sqrt {a}}{168 \, d} \]
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\[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{\frac {7}{2}} \,d x } \]
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Timed out. \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\int {\cos \left (c+d\,x\right )}^{7/2}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]
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