Integrand size = 26, antiderivative size = 110 \[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {64 i a^3 \sec ^9(c+d x)}{1287 d (a+i a \tan (c+d x))^{9/2}}+\frac {16 i a^2 \sec ^9(c+d x)}{143 d (a+i a \tan (c+d x))^{7/2}}+\frac {2 i a \sec ^9(c+d x)}{13 d (a+i a \tan (c+d x))^{5/2}} \]
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Time = 0.23 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3575, 3574} \[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {64 i a^3 \sec ^9(c+d x)}{1287 d (a+i a \tan (c+d x))^{9/2}}+\frac {16 i a^2 \sec ^9(c+d x)}{143 d (a+i a \tan (c+d x))^{7/2}}+\frac {2 i a \sec ^9(c+d x)}{13 d (a+i a \tan (c+d x))^{5/2}} \]
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Rule 3574
Rule 3575
Rubi steps \begin{align*} \text {integral}& = \frac {2 i a \sec ^9(c+d x)}{13 d (a+i a \tan (c+d x))^{5/2}}+\frac {1}{13} (8 a) \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx \\ & = \frac {16 i a^2 \sec ^9(c+d x)}{143 d (a+i a \tan (c+d x))^{7/2}}+\frac {2 i a \sec ^9(c+d x)}{13 d (a+i a \tan (c+d x))^{5/2}}+\frac {1}{143} \left (32 a^2\right ) \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx \\ & = \frac {64 i a^3 \sec ^9(c+d x)}{1287 d (a+i a \tan (c+d x))^{9/2}}+\frac {16 i a^2 \sec ^9(c+d x)}{143 d (a+i a \tan (c+d x))^{7/2}}+\frac {2 i a \sec ^9(c+d x)}{13 d (a+i a \tan (c+d x))^{5/2}} \\ \end{align*}
Time = 1.50 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.84 \[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {2 \sec ^8(c+d x) (52+151 \cos (2 (c+d x))+135 i \sin (2 (c+d x))) (\cos (3 (c+d x))-i \sin (3 (c+d x)))}{1287 a d (-i+\tan (c+d x)) \sqrt {a+i a \tan (c+d x)}} \]
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Timed out.
\[\int \frac {\sec ^{9}\left (d x +c \right )}{\left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}d x\]
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none
Time = 0.28 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.30 \[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {128 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-143 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 52 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 8 i\right )}}{1287 \, {\left (a^{2} d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, a^{2} d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, a^{2} d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}} \]
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\[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {\sec ^{9}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 626 vs. \(2 (86) = 172\).
Time = 0.43 (sec) , antiderivative size = 626, normalized size of antiderivative = 5.69 \[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {2 \, {\left (-203 i \, \sqrt {a} - \frac {678 \, \sqrt {a} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2 i \, \sqrt {a} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1802 \, \sqrt {a} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {26 i \, \sqrt {a} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {3614 \, \sqrt {a} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {858 i \, \sqrt {a} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {6578 \, \sqrt {a} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {6578 \, \sqrt {a} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {858 i \, \sqrt {a} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {3614 \, \sqrt {a} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac {26 i \, \sqrt {a} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {1802 \, \sqrt {a} \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} + \frac {2 i \, \sqrt {a} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} - \frac {678 \, \sqrt {a} \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} + \frac {203 i \, \sqrt {a} \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}}\right )} {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}} {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}^{\frac {3}{2}}}{1287 \, {\left (a^{2} - \frac {8 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {28 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {56 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {70 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {56 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {28 \, a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {8 \, a^{2} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {a^{2} \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}}\right )} d {\left (-\frac {2 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )}^{\frac {3}{2}}} \]
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\[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{9}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 9.55 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.95 \[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {128\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,52{}\mathrm {i}+{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,143{}\mathrm {i}+8{}\mathrm {i}\right )}{1287\,a^2\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^6} \]
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