Integrand size = 30, antiderivative size = 719 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx=\frac {i a}{3 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i a^{3/2} e^{7/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right ) \sec (c+d x)}{8 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a^{3/2} e^{7/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right ) \sec (c+d x)}{8 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a^{3/2} e^{7/2} \log \left (a-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i a^{3/2} e^{7/2} \log \left (a+\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}} \]
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Time = 1.03 (sec) , antiderivative size = 719, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {3596, 3579, 3582, 3580, 3576, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx=-\frac {5 i a^{3/2} e^{7/2} \sec (c+d x) \arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{8 \sqrt {2} d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}+\frac {5 i a^{3/2} e^{7/2} \sec (c+d x) \arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{8 \sqrt {2} d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}+\frac {5 i a^{3/2} e^{7/2} \sec (c+d x) \log \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{16 \sqrt {2} d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}-\frac {5 i a^{3/2} e^{7/2} \sec (c+d x) \log \left (\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{16 \sqrt {2} d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}-\frac {5 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac {5 i a \cos ^2(c+d x)}{8 d \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}}+\frac {i a}{3 d \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}} \]
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Rule 210
Rule 303
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3576
Rule 3579
Rule 3580
Rule 3582
Rule 3596
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e \sec (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)} \, dx}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}} \\ & = \frac {i a}{3 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {(5 a) \int \frac {(e \sec (c+d x))^{7/2}}{\sqrt {a+i a \tan (c+d x)}} \, dx}{6 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}} \\ & = \frac {i a}{3 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac {\left (5 e^2\right ) \int (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)} \, dx}{8 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}} \\ & = \frac {i a}{3 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac {\left (5 a e^2\right ) \int \frac {(e \sec (c+d x))^{3/2}}{\sqrt {a+i a \tan (c+d x)}} \, dx}{16 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}} \\ & = \frac {i a}{3 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac {\left (5 a e^3 \sec (c+d x)\right ) \int \sqrt {e \sec (c+d x)} \sqrt {a-i a \tan (c+d x)} \, dx}{16 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \\ & = \frac {i a}{3 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac {\left (5 i a^2 e^5 \sec (c+d x)\right ) \text {Subst}\left (\int \frac {x^2}{a^2+e^2 x^4} \, dx,x,\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{4 d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \\ & = \frac {i a}{3 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}-\frac {\left (5 i a^2 e^4 \sec (c+d x)\right ) \text {Subst}\left (\int \frac {a-e x^2}{a^2+e^2 x^4} \, dx,x,\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{8 d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {\left (5 i a^2 e^4 \sec (c+d x)\right ) \text {Subst}\left (\int \frac {a+e x^2}{a^2+e^2 x^4} \, dx,x,\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{8 d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \\ & = \frac {i a}{3 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac {\left (5 i a^2 e^3 \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\frac {a}{e}-\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}+x^2} \, dx,x,\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{16 d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {\left (5 i a^2 e^3 \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\frac {a}{e}+\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}+x^2} \, dx,x,\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{16 d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {\left (5 i a^{3/2} e^{7/2} \sec (c+d x)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {a}}{\sqrt {e}}+2 x}{-\frac {a}{e}-\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}-x^2} \, dx,x,\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{16 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {\left (5 i a^{3/2} e^{7/2} \sec (c+d x)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {a}}{\sqrt {e}}-2 x}{-\frac {a}{e}+\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}-x^2} \, dx,x,\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{16 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \\ & = \frac {i a}{3 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a^{3/2} e^{7/2} \log \left (a-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i a^{3/2} e^{7/2} \log \left (a+\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac {\left (5 i a^{3/2} e^{7/2} \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{8 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {\left (5 i a^{3/2} e^{7/2} \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{8 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \\ & = \frac {i a}{3 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i a^{3/2} e^{7/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right ) \sec (c+d x)}{8 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a^{3/2} e^{7/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right ) \sec (c+d x)}{8 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a^{3/2} e^{7/2} \log \left (a-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i a^{3/2} e^{7/2} \log \left (a+\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}} \\ \end{align*}
Time = 4.51 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx=\frac {\sqrt {\cos (c+d x)} \left (-\frac {40}{3} i \cos ^{\frac {3}{2}}(c+d x)+\frac {5}{8} i e^{-\frac {7}{2} i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^3 \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )} \left (2 \arctan \left (1-\sqrt {2} e^{\frac {1}{2} i (c+d x)}\right )-2 \arctan \left (1+\sqrt {2} e^{\frac {1}{2} i (c+d x)}\right )+\log \left (1-\sqrt {2} e^{\frac {1}{2} i (c+d x)}+e^{i (c+d x)}\right )-\log \left (1+\sqrt {2} e^{\frac {1}{2} i (c+d x)}+e^{i (c+d x)}\right )\right )+\frac {32}{3} \sqrt {\cos (c+d x)} (i \cos (c+d x)+\sin (c+d x))+20 \cos ^{\frac {5}{2}}(c+d x) (i \cos (c+d x)+\sin (c+d x))\right ) \sqrt {a+i a \tan (c+d x)}}{32 d (e \cos (c+d x))^{7/2}} \]
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Time = 8.77 (sec) , antiderivative size = 459, normalized size of antiderivative = 0.64
method | result | size |
default | \(\frac {\left (\frac {1}{48}-\frac {i}{48}\right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (10 i \tan \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+15 i \cos \left (d x +c \right ) \operatorname {arctanh}\left (\frac {\cos \left (d x +c \right )+\sin \left (d x +c \right )+1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\right )-8 i \left (\sec ^{2}\left (d x +c \right )\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-5 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+8 i \tan \left (d x +c \right ) \sec \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-15 \,\operatorname {arctanh}\left (\frac {-\cos \left (d x +c \right )+\sin \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )-15 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )-15 \sin \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+15 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sin \left (d x +c \right )+2 i \sec \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-5 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-10 \tan \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-15 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )+2 \sec \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-8 \tan \left (d x +c \right ) \sec \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-8 \left (\sec ^{2}\left (d x +c \right )\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\right )}{d \left (i \cos \left (d x +c \right )+i-\sin \left (d x +c \right )\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, e^{3} \sqrt {e \cos \left (d x +c \right )}}\) | \(459\) |
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Time = 0.28 (sec) , antiderivative size = 657, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx=\frac {\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-5 i \, e^{\left (5 i \, d x + 5 i \, c\right )} + 42 i \, e^{\left (3 i \, d x + 3 i \, c\right )} + 15 i \, e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} - 6 \, {\left (d e^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{4}\right )} \sqrt {\frac {25 i \, a}{64 \, d^{2} e^{7}}} \log \left (\frac {8}{5} \, d e^{4} \sqrt {\frac {25 i \, a}{64 \, d^{2} e^{7}}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{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 )}\right ) + 6 \, {\left (d e^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{4}\right )} \sqrt {\frac {25 i \, a}{64 \, d^{2} e^{7}}} \log \left (-\frac {8}{5} \, d e^{4} \sqrt {\frac {25 i \, a}{64 \, d^{2} e^{7}}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{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 )}\right ) + 6 \, {\left (d e^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{4}\right )} \sqrt {-\frac {25 i \, a}{64 \, d^{2} e^{7}}} \log \left (\frac {8}{5} \, d e^{4} \sqrt {-\frac {25 i \, a}{64 \, d^{2} e^{7}}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{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 )}\right ) - 6 \, {\left (d e^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{4}\right )} \sqrt {-\frac {25 i \, a}{64 \, d^{2} e^{7}}} \log \left (-\frac {8}{5} \, d e^{4} \sqrt {-\frac {25 i \, a}{64 \, d^{2} e^{7}}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{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 )}\right )}{12 \, {\left (d e^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{4}\right )}} \]
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Timed out. \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2661 vs. \(2 (549) = 1098\).
Time = 0.59 (sec) , antiderivative size = 2661, normalized size of antiderivative = 3.70 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx=\int { \frac {\sqrt {i \, a \tan \left (d x + c\right ) + a}}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx=\int \frac {\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \]
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