Integrand size = 30, antiderivative size = 121 \[ \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {2 i \sqrt {e \sec (c+d x)}}{9 d (a+i a \tan (c+d x))^{5/2}}+\frac {8 i \sqrt {e \sec (c+d x)}}{45 a d (a+i a \tan (c+d x))^{3/2}}+\frac {16 i \sqrt {e \sec (c+d x)}}{45 a^2 d \sqrt {a+i a \tan (c+d x)}} \]
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Time = 0.26 (sec) , antiderivative size = 121, 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 = {3583, 3569} \[ \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {16 i \sqrt {e \sec (c+d x)}}{45 a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {8 i \sqrt {e \sec (c+d x)}}{45 a d (a+i a \tan (c+d x))^{3/2}}+\frac {2 i \sqrt {e \sec (c+d x)}}{9 d (a+i a \tan (c+d x))^{5/2}} \]
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Rule 3569
Rule 3583
Rubi steps \begin{align*} \text {integral}& = \frac {2 i \sqrt {e \sec (c+d x)}}{9 d (a+i a \tan (c+d x))^{5/2}}+\frac {4 \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^{3/2}} \, dx}{9 a} \\ & = \frac {2 i \sqrt {e \sec (c+d x)}}{9 d (a+i a \tan (c+d x))^{5/2}}+\frac {8 i \sqrt {e \sec (c+d x)}}{45 a d (a+i a \tan (c+d x))^{3/2}}+\frac {8 \int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx}{45 a^2} \\ & = \frac {2 i \sqrt {e \sec (c+d x)}}{9 d (a+i a \tan (c+d x))^{5/2}}+\frac {8 i \sqrt {e \sec (c+d x)}}{45 a d (a+i a \tan (c+d x))^{3/2}}+\frac {16 i \sqrt {e \sec (c+d x)}}{45 a^2 d \sqrt {a+i a \tan (c+d x)}} \\ \end{align*}
Time = 1.35 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^{5/2}} \, dx=-\frac {i \sec ^2(c+d x) \sqrt {e \sec (c+d x)} (9+25 \cos (2 (c+d x))+20 i \sin (2 (c+d x)))}{45 a^2 d (-i+\tan (c+d x))^2 \sqrt {a+i a \tan (c+d x)}} \]
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Time = 13.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.57
method | result | size |
default | \(\frac {2 i \sqrt {e \sec \left (d x +c \right )}\, \left (20 i \tan \left (d x +c \right )+25-8 \left (\sec ^{2}\left (d x +c \right )\right )\right )}{45 d \left (1+i \tan \left (d x +c \right )\right )^{2} \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, a^{2}}\) | \(69\) |
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Time = 0.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {\sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (45 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 63 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 23 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 5 i\right )} e^{\left (-\frac {9}{2} i \, d x - \frac {9}{2} i \, c\right )}}{90 \, a^{3} d} \]
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\[ \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^{5/2}} \, dx=\int \frac {\sqrt {e \sec {\left (c + d x \right )}}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.39 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {\sqrt {e} {\left (5 i \, \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 18 i \, \cos \left (\frac {5}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 45 i \, \cos \left (\frac {1}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 5 \, \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 18 \, \sin \left (\frac {5}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 45 \, \sin \left (\frac {1}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right )\right )}}{90 \, a^{\frac {5}{2}} d} \]
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\[ \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^{5/2}} \, dx=\int { \frac {\sqrt {e \sec \left (d x + c\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 5.34 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,\left (\cos \left (2\,c+2\,d\,x\right )\,18{}\mathrm {i}+\cos \left (4\,c+4\,d\,x\right )\,5{}\mathrm {i}+18\,\sin \left (2\,c+2\,d\,x\right )+5\,\sin \left (4\,c+4\,d\,x\right )+45{}\mathrm {i}\right )}{90\,a^2\,d\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,c+2\,d\,x\right )+1}}} \]
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