Integrand size = 30, antiderivative size = 125 \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{7/2}} \, dx=\frac {16 i a^2 \sqrt {e \sec (c+d x)}}{21 d e^4 \sqrt {a+i a \tan (c+d x)}}-\frac {8 i a \sqrt {a+i a \tan (c+d x)}}{21 d e^2 (e \sec (c+d x))^{3/2}}-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{7 d (e \sec (c+d x))^{7/2}} \]
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Time = 0.28 (sec) , antiderivative size = 125, 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 = {3578, 3569} \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{7/2}} \, dx=\frac {16 i a^2 \sqrt {e \sec (c+d x)}}{21 d e^4 \sqrt {a+i a \tan (c+d x)}}-\frac {8 i a \sqrt {a+i a \tan (c+d x)}}{21 d e^2 (e \sec (c+d x))^{3/2}}-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{7 d (e \sec (c+d x))^{7/2}} \]
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Rule 3569
Rule 3578
Rubi steps \begin{align*} \text {integral}& = -\frac {2 i (a+i a \tan (c+d x))^{3/2}}{7 d (e \sec (c+d x))^{7/2}}+\frac {(4 a) \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{3/2}} \, dx}{7 e^2} \\ & = -\frac {8 i a \sqrt {a+i a \tan (c+d x)}}{21 d e^2 (e \sec (c+d x))^{3/2}}-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{7 d (e \sec (c+d x))^{7/2}}+\frac {\left (8 a^2\right ) \int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx}{21 e^4} \\ & = \frac {16 i a^2 \sqrt {e \sec (c+d x)}}{21 d e^4 \sqrt {a+i a \tan (c+d x)}}-\frac {8 i a \sqrt {a+i a \tan (c+d x)}}{21 d e^2 (e \sec (c+d x))^{3/2}}-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{7 d (e \sec (c+d x))^{7/2}} \\ \end{align*}
Time = 1.45 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.78 \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{7/2}} \, dx=\frac {a (\cos (d x)-i \sin (d x)) (-7 i+9 i \cos (2 (c+d x))+12 \sin (2 (c+d x))) (\cos (c+2 d x)+i \sin (c+2 d x)) \sqrt {a+i a \tan (c+d x)}}{21 d e^3 \sqrt {e \sec (c+d x)}} \]
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Time = 9.70 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.62
method | result | size |
default | \(-\frac {2 \cos \left (d x +c \right ) \left (\tan \left (d x +c \right )-i\right ) a \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (-12 i \cos \left (d x +c \right ) \sin \left (d x +c \right )+9 \left (\cos ^{2}\left (d x +c \right )\right )-8\right )}{21 d \sqrt {e \sec \left (d x +c \right )}\, e^{3}}\) | \(77\) |
risch | \(-\frac {i a \sqrt {\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left (3 \,{\mathrm e}^{3 i \left (d x +c \right )}-7 \cos \left (d x +c \right )+35 i \sin \left (d x +c \right )\right )}{42 e^{3} \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, d}\) | \(92\) |
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Time = 0.25 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.73 \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{7/2}} \, dx=\frac {{\left (-3 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} - 17 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 7 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 21 i \, a\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{42 \, d e^{4}} \]
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Timed out. \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{7/2}} \, dx=\text {Timed out} \]
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Time = 0.41 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.67 \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{7/2}} \, dx=\frac {{\left (-3 i \, a \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) - 14 i \, a \cos \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 21 i \, a \cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 14 \, a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 21 \, a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{42 \, d e^{\frac {7}{2}}} \]
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\[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{7/2}} \, dx=\int { \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\left (e \sec \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
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Time = 5.85 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.88 \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{7/2}} \, dx=\frac {a\,\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,c+2\,d\,x\right )+1}}\,\left (\cos \left (2\,c+2\,d\,x\right )\,4{}\mathrm {i}-\cos \left (4\,c+4\,d\,x\right )\,3{}\mathrm {i}+38\,\sin \left (2\,c+2\,d\,x\right )+3\,\sin \left (4\,c+4\,d\,x\right )+7{}\mathrm {i}\right )}{84\,d\,e^4} \]
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