Integrand size = 30, antiderivative size = 164 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{7/2}} \, dx=\frac {12 i a}{35 d e^2 (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}+\frac {32 i a \sqrt {e \sec (c+d x)}}{35 d e^4 \sqrt {a+i a \tan (c+d x)}}-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{7 d (e \sec (c+d x))^{7/2}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{35 d e^2 (e \sec (c+d x))^{3/2}} \]
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Time = 0.37 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3578, 3583, 3569} \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{7/2}} \, dx=\frac {32 i a \sqrt {e \sec (c+d x)}}{35 d e^4 \sqrt {a+i a \tan (c+d x)}}+\frac {12 i a}{35 d e^2 \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{35 d e^2 (e \sec (c+d x))^{3/2}}-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{7 d (e \sec (c+d x))^{7/2}} \]
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Rule 3569
Rule 3578
Rule 3583
Rubi steps \begin{align*} \text {integral}& = -\frac {2 i \sqrt {a+i a \tan (c+d x)}}{7 d (e \sec (c+d x))^{7/2}}+\frac {(6 a) \int \frac {1}{(e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \, dx}{7 e^2} \\ & = \frac {12 i a}{35 d e^2 (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{7 d (e \sec (c+d x))^{7/2}}+\frac {24 \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{3/2}} \, dx}{35 e^2} \\ & = \frac {12 i a}{35 d e^2 (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{7 d (e \sec (c+d x))^{7/2}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{35 d e^2 (e \sec (c+d x))^{3/2}}+\frac {(16 a) \int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx}{35 e^4} \\ & = \frac {12 i a}{35 d e^2 (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}+\frac {32 i a \sqrt {e \sec (c+d x)}}{35 d e^4 \sqrt {a+i a \tan (c+d x)}}-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{7 d (e \sec (c+d x))^{7/2}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{35 d e^2 (e \sec (c+d x))^{3/2}} \\ \end{align*}
Time = 1.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.49 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{7/2}} \, dx=\frac {(35 i \cos (c+d x)+i \cos (3 (c+d x))+70 \sin (c+d x)+6 \sin (3 (c+d x))) \sqrt {a+i a \tan (c+d x)}}{70 d e^3 \sqrt {e \sec (c+d x)}} \]
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Time = 10.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.49
method | result | size |
default | \(-\frac {2 i \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (6 i \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-\left (\cos ^{3}\left (d x +c \right )\right )+16 i \sin \left (d x +c \right )-8 \cos \left (d x +c \right )\right )}{35 d \sqrt {e \sec \left (d x +c \right )}\, e^{3}}\) | \(80\) |
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Time = 0.24 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{7/2}} \, dx=\frac {\sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-5 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 40 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 70 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 112 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i\right )} e^{\left (-\frac {5}{2} i \, d x - \frac {5}{2} i \, c\right )}}{140 \, d e^{4}} \]
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Timed out. \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{7/2}} \, dx=\text {Timed out} \]
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Time = 0.39 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{7/2}} \, dx=\frac {\sqrt {a} {\left (7 i \, \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) - 5 i \, \cos \left (\frac {7}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right ) - 35 i \, \cos \left (\frac {3}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right ) + 105 i \, \cos \left (\frac {1}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right ) + 7 \, \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 5 \, \sin \left (\frac {7}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right ) + 35 \, \sin \left (\frac {3}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right ) + 105 \, \sin \left (\frac {1}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right )\right )}}{140 \, d e^{\frac {7}{2}}} \]
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\[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{7/2}} \, dx=\int { \frac {\sqrt {i \, a \tan \left (d x + c\right ) + a}}{\left (e \sec \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
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Time = 5.78 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{7/2}} \, dx=\frac {\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,c+2\,d\,x\right )+1}}\,\left (\cos \left (2\,c+2\,d\,x\right )\,36{}\mathrm {i}+\cos \left (4\,c+4\,d\,x\right )\,1{}\mathrm {i}+76\,\sin \left (2\,c+2\,d\,x\right )+6\,\sin \left (4\,c+4\,d\,x\right )+35{}\mathrm {i}\right )}{140\,d\,e^4} \]
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