Integrand size = 26, antiderivative size = 342 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {195 i a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{1024 \sqrt {2} d}+\frac {65 i a^4 \cos (c+d x)}{512 d \sqrt {a+i a \tan (c+d x)}}+\frac {39 i a^4 \cos ^3(c+d x)}{448 d \sqrt {a+i a \tan (c+d x)}}-\frac {195 i a^3 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{1024 d}-\frac {13 i a^3 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{128 d}-\frac {13 i a^3 \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{168 d}-\frac {65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac {5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d}-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d} \]
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Time = 0.68 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3578, 3583, 3571, 3570, 212} \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {195 i a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{1024 \sqrt {2} d}+\frac {39 i a^4 \cos ^3(c+d x)}{448 d \sqrt {a+i a \tan (c+d x)}}+\frac {65 i a^4 \cos (c+d x)}{512 d \sqrt {a+i a \tan (c+d x)}}-\frac {13 i a^3 \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{168 d}-\frac {13 i a^3 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{128 d}-\frac {195 i a^3 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{1024 d}-\frac {65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}-\frac {5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d} \]
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Rule 212
Rule 3570
Rule 3571
Rule 3578
Rule 3583
Rubi steps \begin{align*} \text {integral}& = -\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}+\frac {1}{22} (15 a) \int \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx \\ & = -\frac {5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d}-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}+\frac {1}{132} \left (65 a^2\right ) \int \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx \\ & = -\frac {65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac {5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d}-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}+\frac {1}{168} \left (65 a^3\right ) \int \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx \\ & = -\frac {13 i a^3 \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{168 d}-\frac {65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac {5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d}-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}+\frac {1}{112} \left (39 a^4\right ) \int \frac {\cos ^3(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx \\ & = \frac {39 i a^4 \cos ^3(c+d x)}{448 d \sqrt {a+i a \tan (c+d x)}}-\frac {13 i a^3 \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{168 d}-\frac {65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac {5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d}-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}+\frac {1}{128} \left (39 a^3\right ) \int \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx \\ & = \frac {39 i a^4 \cos ^3(c+d x)}{448 d \sqrt {a+i a \tan (c+d x)}}-\frac {13 i a^3 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{128 d}-\frac {13 i a^3 \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{168 d}-\frac {65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac {5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d}-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}+\frac {1}{256} \left (65 a^4\right ) \int \frac {\cos (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx \\ & = \frac {65 i a^4 \cos (c+d x)}{512 d \sqrt {a+i a \tan (c+d x)}}+\frac {39 i a^4 \cos ^3(c+d x)}{448 d \sqrt {a+i a \tan (c+d x)}}-\frac {13 i a^3 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{128 d}-\frac {13 i a^3 \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{168 d}-\frac {65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac {5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d}-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}+\frac {\left (195 a^3\right ) \int \cos (c+d x) \sqrt {a+i a \tan (c+d x)} \, dx}{1024} \\ & = \frac {65 i a^4 \cos (c+d x)}{512 d \sqrt {a+i a \tan (c+d x)}}+\frac {39 i a^4 \cos ^3(c+d x)}{448 d \sqrt {a+i a \tan (c+d x)}}-\frac {195 i a^3 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{1024 d}-\frac {13 i a^3 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{128 d}-\frac {13 i a^3 \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{168 d}-\frac {65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac {5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d}-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}+\frac {\left (195 a^4\right ) \int \frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx}{2048} \\ & = \frac {65 i a^4 \cos (c+d x)}{512 d \sqrt {a+i a \tan (c+d x)}}+\frac {39 i a^4 \cos ^3(c+d x)}{448 d \sqrt {a+i a \tan (c+d x)}}-\frac {195 i a^3 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{1024 d}-\frac {13 i a^3 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{128 d}-\frac {13 i a^3 \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{168 d}-\frac {65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac {5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d}-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d}+\frac {\left (195 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 )}{1024 d} \\ & = \frac {195 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 )}{1024 \sqrt {2} d}+\frac {65 i a^4 \cos (c+d x)}{512 d \sqrt {a+i a \tan (c+d x)}}+\frac {39 i a^4 \cos ^3(c+d x)}{448 d \sqrt {a+i a \tan (c+d x)}}-\frac {195 i a^3 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{1024 d}-\frac {13 i a^3 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{128 d}-\frac {13 i a^3 \cos ^5(c+d x) \sqrt {a+i a \tan (c+d x)}}{168 d}-\frac {65 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^{3/2}}{924 d}-\frac {5 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^{5/2}}{66 d}-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2}}{11 d} \\ \end{align*}
Time = 6.02 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.57 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {i a^3 e^{-5 i (c+d x)} \left (-462-7161 e^{2 i (c+d x)}+47413 e^{4 i (c+d x)}+78800 e^{6 i (c+d x)}+38512 e^{8 i (c+d x)}+19552 e^{10 i (c+d x)}+7184 e^{12 i (c+d x)}+1624 e^{14 i (c+d x)}+168 e^{16 i (c+d x)}-45045 e^{4 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )\right ) \sqrt {a+i a \tan (c+d x)}}{473088 d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1237 vs. \(2 (281 ) = 562\).
Time = 6.02 (sec) , antiderivative size = 1238, normalized size of antiderivative = 3.62
\[\text {Expression too large to display}\]
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Time = 0.31 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.00 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {{\left (45045 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} d e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-\frac {195 \, {\left (-i \, a^{4} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{512 \, d}\right ) - 45045 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} d e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-\frac {195 \, {\left (-i \, a^{4} - \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{512 \, d}\right ) - \sqrt {2} {\left (-168 i \, a^{3} e^{\left (16 i \, d x + 16 i \, c\right )} - 1624 i \, a^{3} e^{\left (14 i \, d x + 14 i \, c\right )} - 7184 i \, a^{3} e^{\left (12 i \, d x + 12 i \, c\right )} - 19552 i \, a^{3} e^{\left (10 i \, d x + 10 i \, c\right )} - 38512 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - 78800 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 47413 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 7161 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 462 i \, a^{3}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{473088 \, d} \]
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Timed out. \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\text {Timed out} \]
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\[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \cos \left (d x + c\right )^{11} \,d x } \]
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Timed out. \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\int {\cos \left (c+d\,x\right )}^{11}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2} \,d x \]
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