Integrand size = 30, antiderivative size = 206 \[ \int \frac {1}{(e \sec (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}} \, dx=\frac {2 i}{9 d (e \sec (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {32 i}{105 d e^2 (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}+\frac {256 i \sqrt {e \sec (c+d x)}}{315 d e^4 \sqrt {a+i a \tan (c+d x)}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{63 a d (e \sec (c+d x))^{7/2}}-\frac {128 i \sqrt {a+i a \tan (c+d x)}}{315 a d e^2 (e \sec (c+d x))^{3/2}} \]
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Time = 0.47 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3583, 3578, 3569} \[ \int \frac {1}{(e \sec (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}} \, dx=\frac {256 i \sqrt {e \sec (c+d x)}}{315 d e^4 \sqrt {a+i a \tan (c+d x)}}-\frac {128 i \sqrt {a+i a \tan (c+d x)}}{315 a d e^2 (e \sec (c+d x))^{3/2}}+\frac {32 i}{105 d e^2 \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{63 a d (e \sec (c+d x))^{7/2}}+\frac {2 i}{9 d \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{7/2}} \]
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Rule 3569
Rule 3578
Rule 3583
Rubi steps \begin{align*} \text {integral}& = \frac {2 i}{9 d (e \sec (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {8 \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{7/2}} \, dx}{9 a} \\ & = \frac {2 i}{9 d (e \sec (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{63 a d (e \sec (c+d x))^{7/2}}+\frac {16 \int \frac {1}{(e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \, dx}{21 e^2} \\ & = \frac {2 i}{9 d (e \sec (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {32 i}{105 d e^2 (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{63 a d (e \sec (c+d x))^{7/2}}+\frac {64 \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{3/2}} \, dx}{105 a e^2} \\ & = \frac {2 i}{9 d (e \sec (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {32 i}{105 d e^2 (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{63 a d (e \sec (c+d x))^{7/2}}-\frac {128 i \sqrt {a+i a \tan (c+d x)}}{315 a d e^2 (e \sec (c+d x))^{3/2}}+\frac {128 \int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx}{315 e^4} \\ & = \frac {2 i}{9 d (e \sec (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {32 i}{105 d e^2 (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}+\frac {256 i \sqrt {e \sec (c+d x)}}{315 d e^4 \sqrt {a+i a \tan (c+d x)}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{63 a d (e \sec (c+d x))^{7/2}}-\frac {128 i \sqrt {a+i a \tan (c+d x)}}{315 a d e^2 (e \sec (c+d x))^{3/2}} \\ \end{align*}
Time = 1.40 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.42 \[ \int \frac {1}{(e \sec (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}} \, dx=\frac {\sqrt {e \sec (c+d x)} (945 i-84 i \cos (2 (c+d x))-5 i \cos (4 (c+d x))+336 \sin (2 (c+d x))+40 \sin (4 (c+d x)))}{1260 d e^4 \sqrt {a+i a \tan (c+d x)}} \]
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Time = 10.60 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.43
method | result | size |
default | \(\frac {-\frac {2 i \left (\cos ^{3}\left (d x +c \right )\right )}{63}+\frac {16 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{63}-\frac {32 i \cos \left (d x +c \right )}{315}+\frac {128 \sin \left (d x +c \right )}{315}+\frac {256 i \sec \left (d x +c \right )}{315}}{d \sqrt {e \sec \left (d x +c \right )}\, \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, e^{3}}\) | \(88\) |
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Time = 0.24 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.54 \[ \int \frac {1}{(e \sec (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}} \, dx=\frac {\sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-45 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 465 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 1470 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 2142 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 287 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 35 i\right )} e^{\left (-\frac {9}{2} i \, d x - \frac {9}{2} i \, c\right )}}{2520 \, a d e^{4}} \]
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Timed out. \[ \int \frac {1}{(e \sec (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}} \, dx=\text {Timed out} \]
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Time = 0.80 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(e \sec (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}} \, dx=\frac {35 i \, \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) - 45 i \, \cos \left (\frac {7}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 252 i \, \cos \left (\frac {5}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) - 420 i \, \cos \left (\frac {1}{3} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 1890 i \, \cos \left (\frac {1}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 35 \, \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 45 \, \sin \left (\frac {7}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 252 \, \sin \left (\frac {5}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 420 \, \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 1890 \, \sin \left (\frac {1}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right )}{2520 \, \sqrt {a} d e^{\frac {7}{2}}} \]
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\[ \int \frac {1}{(e \sec (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {1}{\left (e \sec \left (d x + c\right )\right )^{\frac {7}{2}} \sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]
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Time = 4.94 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.53 \[ \int \frac {1}{(e \sec (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}} \, dx=\frac {\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,\left (336\,\sin \left (2\,c+2\,d\,x\right )-\cos \left (4\,c+4\,d\,x\right )\,5{}\mathrm {i}-\cos \left (2\,c+2\,d\,x\right )\,84{}\mathrm {i}+40\,\sin \left (4\,c+4\,d\,x\right )+945{}\mathrm {i}\right )}{1260\,d\,e^4\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,c+2\,d\,x\right )+1}}} \]
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