Integrand size = 30, antiderivative size = 122 \[ \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{8/3} \, dx=\frac {54 i a^3 (d \sec (e+f x))^{2/3}}{7 f \sqrt [3]{a+i a \tan (e+f x)}}+\frac {9 i a^2 (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3}}{7 f}+\frac {3 i a (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{5/3}}{7 f} \]
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Time = 0.29 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3575, 3574} \[ \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{8/3} \, dx=\frac {54 i a^3 (d \sec (e+f x))^{2/3}}{7 f \sqrt [3]{a+i a \tan (e+f x)}}+\frac {9 i a^2 (a+i a \tan (e+f x))^{2/3} (d \sec (e+f x))^{2/3}}{7 f}+\frac {3 i a (a+i a \tan (e+f x))^{5/3} (d \sec (e+f x))^{2/3}}{7 f} \]
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Rule 3574
Rule 3575
Rubi steps \begin{align*} \text {integral}& = \frac {3 i a (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{5/3}}{7 f}+\frac {1}{7} (12 a) \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{5/3} \, dx \\ & = \frac {9 i a^2 (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3}}{7 f}+\frac {3 i a (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{5/3}}{7 f}+\frac {1}{7} \left (18 a^2\right ) \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3} \, dx \\ & = \frac {54 i a^3 (d \sec (e+f x))^{2/3}}{7 f \sqrt [3]{a+i a \tan (e+f x)}}+\frac {9 i a^2 (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3}}{7 f}+\frac {3 i a (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{5/3}}{7 f} \\ \end{align*}
Time = 1.50 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.82 \[ \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{8/3} \, dx=\frac {3 a^2 (d \sec (e+f x))^{5/3} (i \cos (e-f x)+\sin (e-f x)) (21+23 \cos (2 (e+f x))+5 i \sin (2 (e+f x))) (a+i a \tan (e+f x))^{2/3}}{14 d f (\cos (f x)+i \sin (f x))^2} \]
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\[\int \left (d \sec \left (f x +e \right )\right )^{\frac {2}{3}} \left (a +i a \tan \left (f x +e \right )\right )^{\frac {8}{3}}d x\]
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none
Time = 0.24 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.88 \[ \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{8/3} \, dx=-\frac {6 \cdot 2^{\frac {1}{3}} {\left (-14 i \, a^{2} e^{\left (4 i \, f x + 4 i \, e\right )} - 21 i \, a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 9 i \, a^{2}\right )} \left (\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac {2}{3}} \left (\frac {d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac {2}{3}} e^{\left (2 i \, f x + 2 i \, e\right )}}{7 \, {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + f e^{\left (2 i \, f x + 2 i \, e\right )}\right )}} \]
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Timed out. \[ \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{8/3} \, dx=\text {Timed out} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 402 vs. \(2 (92) = 184\).
Time = 0.89 (sec) , antiderivative size = 402, normalized size of antiderivative = 3.30 \[ \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{8/3} \, dx=\frac {6 \, {\left (7 \, {\left (-i \cdot 2^{\frac {1}{3}} a^{2} \cos \left (\frac {4}{3} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) - 2^{\frac {1}{3}} a^{2} \sin \left (\frac {4}{3} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right )\right )} \sqrt {\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1} a^{\frac {2}{3}} d^{\frac {2}{3}} + 2 \, {\left (i \cdot 2^{\frac {1}{3}} a^{2} \cos \left (\frac {7}{3} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) + 2^{\frac {1}{3}} a^{2} \sin \left (\frac {7}{3} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) + 7 \, {\left (i \cdot 2^{\frac {1}{3}} a^{2} \cos \left (2 \, f x + 2 \, e\right )^{2} + i \cdot 2^{\frac {1}{3}} a^{2} \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 i \cdot 2^{\frac {1}{3}} a^{2} \cos \left (2 \, f x + 2 \, e\right ) + i \cdot 2^{\frac {1}{3}} a^{2}\right )} \cos \left (\frac {1}{3} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) + 7 \, {\left (2^{\frac {1}{3}} a^{2} \cos \left (2 \, f x + 2 \, e\right )^{2} + 2^{\frac {1}{3}} a^{2} \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \cdot 2^{\frac {1}{3}} a^{2} \cos \left (2 \, f x + 2 \, e\right ) + 2^{\frac {1}{3}} a^{2}\right )} \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right )\right )} a^{\frac {2}{3}} d^{\frac {2}{3}}\right )}}{7 \, {\left (\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )}^{\frac {7}{6}} f} \]
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\[ \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{8/3} \, dx=\int { \left (d \sec \left (f x + e\right )\right )^{\frac {2}{3}} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {8}{3}} \,d x } \]
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Time = 6.36 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00 \[ \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{8/3} \, dx=\frac {3\,a^2\,{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{2/3}\,{\left (\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}\right )}^{2/3}\,\left (\cos \left (2\,e+2\,f\,x\right )\,44{}\mathrm {i}+\cos \left (4\,e+4\,f\,x\right )\,9{}\mathrm {i}+16\,\sin \left (2\,e+2\,f\,x\right )+9\,\sin \left (4\,e+4\,f\,x\right )+35{}\mathrm {i}\right )}{14\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]
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