Integrand size = 26, antiderivative size = 268 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {11 i a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{64 \sqrt {2} d}+\frac {11 i a^4 \cos (c+d x)}{96 d \sqrt {a+i a \tan (c+d x)}}-\frac {11 i a^3 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{64 d}-\frac {11 i a^3 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{120 d}-\frac {11 i a^2 \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{140 d}-\frac {11 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{126 d}-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2}}{9 d} \]
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Time = 0.52 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3578, 3583, 3571, 3570, 212} \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {11 i a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{64 \sqrt {2} d}+\frac {11 i a^4 \cos (c+d x)}{96 d \sqrt {a+i a \tan (c+d x)}}-\frac {11 i a^3 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{120 d}-\frac {11 i a^3 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{64 d}-\frac {11 i a^2 \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{140 d}-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2}}{9 d}-\frac {11 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{126 d} \]
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Rule 212
Rule 3570
Rule 3571
Rule 3578
Rule 3583
Rubi steps \begin{align*} \text {integral}& = -\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2}}{9 d}+\frac {1}{18} (11 a) \int \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx \\ & = -\frac {11 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{126 d}-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2}}{9 d}+\frac {1}{28} \left (11 a^2\right ) \int \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx \\ & = -\frac {11 i a^2 \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{140 d}-\frac {11 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{126 d}-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2}}{9 d}+\frac {1}{40} \left (11 a^3\right ) \int \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx \\ & = -\frac {11 i a^3 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{120 d}-\frac {11 i a^2 \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{140 d}-\frac {11 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{126 d}-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2}}{9 d}+\frac {1}{48} \left (11 a^4\right ) \int \frac {\cos (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx \\ & = \frac {11 i a^4 \cos (c+d x)}{96 d \sqrt {a+i a \tan (c+d x)}}-\frac {11 i a^3 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{120 d}-\frac {11 i a^2 \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{140 d}-\frac {11 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{126 d}-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2}}{9 d}+\frac {1}{64} \left (11 a^3\right ) \int \cos (c+d x) \sqrt {a+i a \tan (c+d x)} \, dx \\ & = \frac {11 i a^4 \cos (c+d x)}{96 d \sqrt {a+i a \tan (c+d x)}}-\frac {11 i a^3 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{64 d}-\frac {11 i a^3 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{120 d}-\frac {11 i a^2 \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{140 d}-\frac {11 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{126 d}-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2}}{9 d}+\frac {1}{128} \left (11 a^4\right ) \int \frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx \\ & = \frac {11 i a^4 \cos (c+d x)}{96 d \sqrt {a+i a \tan (c+d x)}}-\frac {11 i a^3 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{64 d}-\frac {11 i a^3 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{120 d}-\frac {11 i a^2 \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{140 d}-\frac {11 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{126 d}-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2}}{9 d}+\frac {\left (11 i a^4\right ) \text {Subst}\left (\int \frac {1}{2-a x^2} \, dx,x,\frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}}\right )}{64 d} \\ & = \frac {11 i a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{64 \sqrt {2} d}+\frac {11 i a^4 \cos (c+d x)}{96 d \sqrt {a+i a \tan (c+d x)}}-\frac {11 i a^3 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{64 d}-\frac {11 i a^3 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{120 d}-\frac {11 i a^2 \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{140 d}-\frac {11 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{126 d}-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2}}{9 d} \\ \end{align*}
Time = 3.65 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.70 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {i a^3 e^{-3 i (c+d x)} \sqrt {\frac {a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left (-315+4303 e^{2 i (c+d x)}+7034 e^{4 i (c+d x)}+3754 e^{6 i (c+d x)}+1798 e^{8 i (c+d x)}+530 e^{10 i (c+d x)}+70 e^{12 i (c+d x)}-3465 e^{2 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )\right )}{20160 \sqrt {2} d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1210 vs. \(2 (219 ) = 438\).
Time = 4.77 (sec) , antiderivative size = 1211, normalized size of antiderivative = 4.52
\[\text {Expression too large to display}\]
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Time = 0.29 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.17 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {{\left (3465 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} d e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-\frac {11 \, {\left (-i \, a^{4} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{32 \, d}\right ) - 3465 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} d e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-\frac {11 \, {\left (-i \, a^{4} - \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{32 \, d}\right ) - \sqrt {2} {\left (-70 i \, a^{3} e^{\left (12 i \, d x + 12 i \, c\right )} - 530 i \, a^{3} e^{\left (10 i \, d x + 10 i \, c\right )} - 1798 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - 3754 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 7034 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 4303 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 315 i \, a^{3}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{40320 \, d} \]
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Timed out. \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\text {Timed out} \]
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\[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \cos \left (d x + c\right )^{9} \,d x } \]
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Timed out. \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\int {\cos \left (c+d\,x\right )}^9\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2} \,d x \]
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