Integrand size = 23, antiderivative size = 233 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=-\frac {11 a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{64 \sqrt {2} d}-\frac {11 a^5 \cos (c+d x)}{64 d (a+a \sin (c+d x))^{3/2}}+\frac {11 a^4 \sec (c+d x)}{48 d \sqrt {a+a \sin (c+d x)}}+\frac {11 a^3 \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{120 d}+\frac {11 a^2 \sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{140 d}+\frac {11 a \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{126 d}+\frac {\sec ^9(c+d x) (a+a \sin (c+d x))^{7/2}}{9 d} \]
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Time = 0.28 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2754, 2766, 2729, 2728, 212} \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=-\frac {11 a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{64 \sqrt {2} d}-\frac {11 a^5 \cos (c+d x)}{64 d (a \sin (c+d x)+a)^{3/2}}+\frac {11 a^4 \sec (c+d x)}{48 d \sqrt {a \sin (c+d x)+a}}+\frac {11 a^3 \sec ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{120 d}+\frac {11 a^2 \sec ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{140 d}+\frac {\sec ^9(c+d x) (a \sin (c+d x)+a)^{7/2}}{9 d}+\frac {11 a \sec ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{126 d} \]
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Rule 212
Rule 2728
Rule 2729
Rule 2754
Rule 2766
Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^9(c+d x) (a+a \sin (c+d x))^{7/2}}{9 d}+\frac {1}{18} (11 a) \int \sec ^8(c+d x) (a+a \sin (c+d x))^{5/2} \, dx \\ & = \frac {11 a \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{126 d}+\frac {\sec ^9(c+d x) (a+a \sin (c+d x))^{7/2}}{9 d}+\frac {1}{28} \left (11 a^2\right ) \int \sec ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx \\ & = \frac {11 a^2 \sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{140 d}+\frac {11 a \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{126 d}+\frac {\sec ^9(c+d x) (a+a \sin (c+d x))^{7/2}}{9 d}+\frac {1}{40} \left (11 a^3\right ) \int \sec ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = \frac {11 a^3 \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{120 d}+\frac {11 a^2 \sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{140 d}+\frac {11 a \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{126 d}+\frac {\sec ^9(c+d x) (a+a \sin (c+d x))^{7/2}}{9 d}+\frac {1}{48} \left (11 a^4\right ) \int \frac {\sec ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = \frac {11 a^4 \sec (c+d x)}{48 d \sqrt {a+a \sin (c+d x)}}+\frac {11 a^3 \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{120 d}+\frac {11 a^2 \sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{140 d}+\frac {11 a \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{126 d}+\frac {\sec ^9(c+d x) (a+a \sin (c+d x))^{7/2}}{9 d}+\frac {1}{32} \left (11 a^5\right ) \int \frac {1}{(a+a \sin (c+d x))^{3/2}} \, dx \\ & = -\frac {11 a^5 \cos (c+d x)}{64 d (a+a \sin (c+d x))^{3/2}}+\frac {11 a^4 \sec (c+d x)}{48 d \sqrt {a+a \sin (c+d x)}}+\frac {11 a^3 \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{120 d}+\frac {11 a^2 \sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{140 d}+\frac {11 a \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{126 d}+\frac {\sec ^9(c+d x) (a+a \sin (c+d x))^{7/2}}{9 d}+\frac {1}{128} \left (11 a^4\right ) \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = -\frac {11 a^5 \cos (c+d x)}{64 d (a+a \sin (c+d x))^{3/2}}+\frac {11 a^4 \sec (c+d x)}{48 d \sqrt {a+a \sin (c+d x)}}+\frac {11 a^3 \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{120 d}+\frac {11 a^2 \sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{140 d}+\frac {11 a \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{126 d}+\frac {\sec ^9(c+d x) (a+a \sin (c+d x))^{7/2}}{9 d}-\frac {\left (11 a^4\right ) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{64 d} \\ & = -\frac {11 a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{64 \sqrt {2} d}-\frac {11 a^5 \cos (c+d x)}{64 d (a+a \sin (c+d x))^{3/2}}+\frac {11 a^4 \sec (c+d x)}{48 d \sqrt {a+a \sin (c+d x)}}+\frac {11 a^3 \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{120 d}+\frac {11 a^2 \sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{140 d}+\frac {11 a \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{126 d}+\frac {\sec ^9(c+d x) (a+a \sin (c+d x))^{7/2}}{9 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.80 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.27 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {a^3 \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},2,-\frac {7}{2},\frac {1}{2} (1-\sin (c+d x))\right ) \sec ^9(c+d x) (1+\sin (c+d x))^4 \sqrt {a (1+\sin (c+d x))}}{18 d} \]
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Time = 0.55 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.89
\[-\frac {-6930 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right ) a^{\frac {11}{2}}+25410 \left (\cos ^{4}\left (d x +c \right )\right ) a^{\frac {11}{2}}+42504 \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) a^{\frac {11}{2}}+3465 \left (a -a \sin \left (d x +c \right )\right )^{\frac {9}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \sin \left (d x +c \right ) a -50424 \left (\cos ^{2}\left (d x +c \right )\right ) a^{\frac {11}{2}}+3465 \left (a -a \sin \left (d x +c \right )\right )^{\frac {9}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a -12320 \sin \left (d x +c \right ) a^{\frac {11}{2}}+7840 a^{\frac {11}{2}}}{40320 a^{\frac {3}{2}} \left (\sin \left (d x +c \right )-1\right )^{4} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\]
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Time = 0.31 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.48 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {3465 \, {\left (3 \, \sqrt {2} a^{3} \cos \left (d x + c\right )^{5} - 4 \, \sqrt {2} a^{3} \cos \left (d x + c\right )^{3} - {\left (\sqrt {2} a^{3} \cos \left (d x + c\right )^{5} - 4 \, \sqrt {2} a^{3} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a \sin \left (d x + c\right ) + a} {\left (\sqrt {2} \cos \left (d x + c\right ) - \sqrt {2} \sin \left (d x + c\right ) + \sqrt {2}\right )} \sqrt {a} + 3 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) + 4 \, {\left (12705 \, a^{3} \cos \left (d x + c\right )^{4} - 25212 \, a^{3} \cos \left (d x + c\right )^{2} + 3920 \, a^{3} - 77 \, {\left (45 \, a^{3} \cos \left (d x + c\right )^{4} - 276 \, a^{3} \cos \left (d x + c\right )^{2} + 80 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{80640 \, {\left (3 \, d \cos \left (d x + c\right )^{5} - 4 \, d \cos \left (d x + c\right )^{3} - {\left (d \cos \left (d x + c\right )^{5} - 4 \, d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\text {Timed out} \]
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Time = 0.33 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.75 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=-\frac {\sqrt {2} a^{\frac {7}{2}} {\left (\frac {630 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + \frac {4 \, {\left (1575 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 420 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 189 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 90 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35\right )}}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}} - 3465 \, \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) + 3465 \, \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{80640 \, d} \]
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Timed out. \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{7/2}}{{\cos \left (c+d\,x\right )}^{10}} \,d x \]
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