Integrand size = 26, antiderivative size = 73 \[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {8 i a^2 \sec ^9(c+d x)}{99 d (a+i a \tan (c+d x))^{9/2}}+\frac {2 i a \sec ^9(c+d x)}{11 d (a+i a \tan (c+d x))^{7/2}} \]
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Time = 0.16 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3575, 3574} \[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {8 i a^2 \sec ^9(c+d x)}{99 d (a+i a \tan (c+d x))^{9/2}}+\frac {2 i a \sec ^9(c+d x)}{11 d (a+i a \tan (c+d x))^{7/2}} \]
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Rule 3574
Rule 3575
Rubi steps \begin{align*} \text {integral}& = \frac {2 i a \sec ^9(c+d x)}{11 d (a+i a \tan (c+d x))^{7/2}}+\frac {1}{11} (4 a) \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx \\ & = \frac {8 i a^2 \sec ^9(c+d x)}{99 d (a+i a \tan (c+d x))^{9/2}}+\frac {2 i a \sec ^9(c+d x)}{11 d (a+i a \tan (c+d x))^{7/2}} \\ \end{align*}
Time = 1.61 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.10 \[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {2 \sec ^7(c+d x) (\cos (2 (c+d x))-i \sin (2 (c+d x))) (-13 i+9 \tan (c+d x))}{99 a^2 d (-i+\tan (c+d x))^2 \sqrt {a+i a \tan (c+d x)}} \]
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Timed out.
\[\int \frac {\sec ^{9}\left (d x +c \right )}{\left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (57) = 114\).
Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.60 \[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=-\frac {64 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-11 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 2 i\right )}}{99 \, {\left (a^{3} d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} d\right )}} \]
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\[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\int \frac {\sec ^{9}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}}}\, dx \]
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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. \(2 (57) = 114\).
Time = 0.43 (sec) , antiderivative size = 626, normalized size of antiderivative = 8.58 \[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=-\frac {2 \, {\left (-13 i \, \sqrt {a} - \frac {34 \, \sqrt {a} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {46 i \, \sqrt {a} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {174 \, \sqrt {a} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {54 i \, \sqrt {a} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {394 \, \sqrt {a} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {22 i \, \sqrt {a} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {550 \, \sqrt {a} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {550 \, \sqrt {a} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {22 i \, \sqrt {a} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {394 \, \sqrt {a} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac {54 i \, \sqrt {a} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {174 \, \sqrt {a} \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} + \frac {46 i \, \sqrt {a} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} - \frac {34 \, \sqrt {a} \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} + \frac {13 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 {5}{2}} {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}^{\frac {5}{2}}}{99 \, {\left (a^{3} - \frac {8 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {28 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {56 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {70 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {56 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {28 \, a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {8 \, a^{3} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {a^{3} \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 {5}{2}}} \]
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\[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{9}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 7.06 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.25 \[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {64\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,11{}\mathrm {i}+2{}\mathrm {i}\right )\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}}{99\,a^3\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^5} \]
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