Integrand size = 30, antiderivative size = 612 \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2} \, dx=\frac {15 i a^3 (e \sec (c+d x))^{3/2}}{8 d \sqrt {a+i a \tan (c+d x)}}-\frac {15 i a^{7/2} e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right ) \sec (c+d x)}{8 \sqrt {2} d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {15 i a^{7/2} e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right ) \sec (c+d x)}{8 \sqrt {2} d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {15 i a^{7/2} e^{3/2} \log \left (a-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt {2} d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {15 i a^{7/2} e^{3/2} \log \left (a+\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt {2} d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {3 i a^2 (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}{4 d}+\frac {i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}{3 d} \]
15/8*I*a^3*(e*sec(d*x+c))^(3/2)/d/(a+I*a*tan(d*x+c))^(1/2)-15/16*I*a^(7/2) *e^(3/2)*arctan(1-2^(1/2)*e^(1/2)*(a-I*a*tan(d*x+c))^(1/2)/a^(1/2)/(e*sec( d*x+c))^(1/2))*sec(d*x+c)/d*2^(1/2)/(a-I*a*tan(d*x+c))^(1/2)/(a+I*a*tan(d* x+c))^(1/2)+15/16*I*a^(7/2)*e^(3/2)*arctan(1+2^(1/2)*e^(1/2)*(a-I*a*tan(d* x+c))^(1/2)/a^(1/2)/(e*sec(d*x+c))^(1/2))*sec(d*x+c)/d*2^(1/2)/(a-I*a*tan( d*x+c))^(1/2)/(a+I*a*tan(d*x+c))^(1/2)+15/32*I*a^(7/2)*e^(3/2)*ln(a-2^(1/2 )*a^(1/2)*e^(1/2)*(a-I*a*tan(d*x+c))^(1/2)/(e*sec(d*x+c))^(1/2)+cos(d*x+c) *(a-I*a*tan(d*x+c)))*sec(d*x+c)/d*2^(1/2)/(a-I*a*tan(d*x+c))^(1/2)/(a+I*a* tan(d*x+c))^(1/2)-15/32*I*a^(7/2)*e^(3/2)*ln(a+2^(1/2)*a^(1/2)*e^(1/2)*(a- I*a*tan(d*x+c))^(1/2)/(e*sec(d*x+c))^(1/2)+cos(d*x+c)*(a-I*a*tan(d*x+c)))* sec(d*x+c)/d*2^(1/2)/(a-I*a*tan(d*x+c))^(1/2)/(a+I*a*tan(d*x+c))^(1/2)+3/4 *I*a^2*(e*sec(d*x+c))^(3/2)*(a+I*a*tan(d*x+c))^(1/2)/d+1/3*I*a*(e*sec(d*x+ c))^(3/2)*(a+I*a*tan(d*x+c))^(3/2)/d
Time = 3.20 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.63 \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2} \, dx=\frac {\cos ^4(c+d x) (e \sec (c+d x))^{3/2} \left (\frac {1}{6} \sec ^3(c+d x) (63+79 \cos (2 (c+d x))+34 i \sin (2 (c+d x))) (i \cos (3 c+d x)+\sin (3 c+d x))+\frac {15 \left (\text {arctanh}\left (\frac {\sqrt {1+i \cos (c)-\sin (c)} \sqrt {i-\tan \left (\frac {d x}{2}\right )}}{\sqrt {-1+i \cos (c)+\sin (c)} \sqrt {i+\tan \left (\frac {d x}{2}\right )}}\right ) \sqrt {-1-i \cos (c)-\sin (c)} \sqrt {1+i \cos (c)-\sin (c)}-\text {arctanh}\left (\frac {\sqrt {1-i \cos (c)+\sin (c)} \sqrt {i-\tan \left (\frac {d x}{2}\right )}}{\sqrt {-1-i \cos (c)-\sin (c)} \sqrt {i+\tan \left (\frac {d x}{2}\right )}}\right ) \sqrt {1-i \cos (c)+\sin (c)} \sqrt {-1+i \cos (c)+\sin (c)}\right ) (\cos (3 c)-i \sin (3 c)) \sqrt {i+\tan \left (\frac {d x}{2}\right )}}{\sqrt {-1-i \cos (c)-\sin (c)} \sqrt {-1+i \cos (c)+\sin (c)} \sqrt {i-\tan \left (\frac {d x}{2}\right )}}\right ) (a+i a \tan (c+d x))^{5/2}}{8 d (\cos (d x)+i \sin (d x))^2} \]
(Cos[c + d*x]^4*(e*Sec[c + d*x])^(3/2)*((Sec[c + d*x]^3*(63 + 79*Cos[2*(c + d*x)] + (34*I)*Sin[2*(c + d*x)])*(I*Cos[3*c + d*x] + Sin[3*c + d*x]))/6 + (15*(ArcTanh[(Sqrt[1 + I*Cos[c] - Sin[c]]*Sqrt[I - Tan[(d*x)/2]])/(Sqrt[ -1 + I*Cos[c] + Sin[c]]*Sqrt[I + Tan[(d*x)/2]])]*Sqrt[-1 - I*Cos[c] - Sin[ c]]*Sqrt[1 + I*Cos[c] - Sin[c]] - ArcTanh[(Sqrt[1 - I*Cos[c] + Sin[c]]*Sqr t[I - Tan[(d*x)/2]])/(Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[I + Tan[(d*x)/2]]) ]*Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[-1 + I*Cos[c] + Sin[c]])*(Cos[3*c] - I* Sin[3*c])*Sqrt[I + Tan[(d*x)/2]])/(Sqrt[-1 - I*Cos[c] - Sin[c]]*Sqrt[-1 + I*Cos[c] + Sin[c]]*Sqrt[I - Tan[(d*x)/2]]))*(a + I*a*Tan[c + d*x])^(5/2))/ (8*d*(Cos[d*x] + I*Sin[d*x])^2)
Time = 1.25 (sec) , antiderivative size = 503, normalized size of antiderivative = 0.82, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 3979, 3042, 3979, 3042, 3979, 3042, 3980, 3042, 3976, 826, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (a+i a \tan (c+d x))^{5/2} (e \sec (c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+i a \tan (c+d x))^{5/2} (e \sec (c+d x))^{3/2}dx\) |
| \(\Big \downarrow \) 3979 |
| \(\displaystyle \frac {3}{2} a \int (e \sec (c+d x))^{3/2} (i \tan (c+d x) a+a)^{3/2}dx+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{2} a \int (e \sec (c+d x))^{3/2} (i \tan (c+d x) a+a)^{3/2}dx+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3979 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \int (e \sec (c+d x))^{3/2} \sqrt {i \tan (c+d x) a+a}dx+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \int (e \sec (c+d x))^{3/2} \sqrt {i \tan (c+d x) a+a}dx+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3979 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {1}{2} a \int \frac {(e \sec (c+d x))^{3/2}}{\sqrt {i \tan (c+d x) a+a}}dx+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {1}{2} a \int \frac {(e \sec (c+d x))^{3/2}}{\sqrt {i \tan (c+d x) a+a}}dx+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3980 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {a e \sec (c+d x) \int \sqrt {e \sec (c+d x)} \sqrt {a-i a \tan (c+d x)}dx}{2 \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {a e \sec (c+d x) \int \sqrt {e \sec (c+d x)} \sqrt {a-i a \tan (c+d x)}dx}{2 \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3976 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {2 i a^2 e^3 \sec (c+d x) \int \frac {\cos (c+d x) (a-i a \tan (c+d x))}{e \left (a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2\right )}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\int \frac {a+\cos (c+d x) (a-i a \tan (c+d x))}{a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}-\frac {\int \frac {a-\cos (c+d x) (a-i a \tan (c+d x))}{a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\int \frac {1}{\frac {a}{e}-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}+\frac {\int \frac {1}{\frac {a}{e}+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}}{2 e}-\frac {\int \frac {a-\cos (c+d x) (a-i a \tan (c+d x))}{a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\int \frac {1}{-\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}-1}d\left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\int \frac {1}{-\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}-1}d\left (\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\int \frac {a-\cos (c+d x) (a-i a \tan (c+d x))}{a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\int \frac {a-\cos (c+d x) (a-i a \tan (c+d x))}{a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{\sqrt {e} \left (\frac {a}{e}-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}\right )}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{\sqrt {e} \left (\frac {a}{e}+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}\right )}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{\sqrt {e} \left (\frac {a}{e}-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}\right )}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{\sqrt {e} \left (\frac {a}{e}+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}\right )}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{\frac {a}{e}-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} e}+\frac {\int \frac {\sqrt {a}+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{\frac {a}{e}+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {a} e}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{2 \sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{2 \sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}{3 d}\) |
((I/3)*a*(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(3/2))/d + (3*a*((( I/2)*a*(e*Sec[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/d + (5*a*((I*a*( e*Sec[c + d*x])^(3/2))/(d*Sqrt[a + I*a*Tan[c + d*x]]) + ((2*I)*a^2*e^3*((- (ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*S ec[c + d*x]])]/(Sqrt[2]*Sqrt[a]*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sq rt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]/(Sqrt[2]*Sqrt[a] *Sqrt[e]))/(2*e) - (-1/2*Log[a - (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])]/(S qrt[2]*Sqrt[a]*Sqrt[e]) + Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Ta n[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]/( 2*Sqrt[2]*Sqrt[a]*Sqrt[e]))/(2*e))*Sec[c + d*x])/(d*Sqrt[a - I*a*Tan[c + d *x]]*Sqrt[a + I*a*Tan[c + d*x]])))/4))/2
3.5.8.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[Sqrt[(d_.)*sec[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.) *(x_)]], x_Symbol] :> Simp[-4*b*(d^2/f) Subst[Int[x^2/(a^2 + d^2*x^4), x] , x, Sqrt[a + b*Tan[e + f*x]]/Sqrt[d*Sec[e + f*x]]], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0]
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)])^(n_), x_Symbol] :> Simp[b*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] + Simp[a*((m + 2*n - 2)/(m + n - 1)) Int[(d*Se c[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && GtQ[n, 0] && NeQ[m + n - 1, 0] && IntegersQ [2*m, 2*n]
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(3/2)/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_ .)*(x_)]], x_Symbol] :> Simp[d*(Sec[e + f*x]/(Sqrt[a - b*Tan[e + f*x]]*Sqrt [a + b*Tan[e + f*x]])) Int[Sqrt[d*Sec[e + f*x]]*Sqrt[a - b*Tan[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0]
Time = 9.46 (sec) , antiderivative size = 556, normalized size of antiderivative = 0.91
| method | result | size |
| default | \(\frac {\left (-\frac {1}{96}-\frac {i}{96}\right ) \left (-\tan \left (d x +c \right )+i\right )^{2} \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {e \sec \left (d x +c \right )}\, \left (-45 \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+34 \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+45 i \left (\cos ^{3}\left (d x +c \right )\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 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-79 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (\cos ^{2}\left (d x +c \right )\right )+45 \left (\cos ^{3}\left (d x +c \right )\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 )-45 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (\cos ^{3}\left (d x +c \right )\right )-79 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (\cos ^{2}\left (d x +c \right )\right )-26 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )+8 \sin \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-8 i \sin \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-34 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right ) \sin \left (d x +c \right )-45 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (\cos ^{3}\left (d x +c \right )\right )+45 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-26 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )+8 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\right ) \left (4 i \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+2 i \cos \left (d x +c \right ) \sin \left (d x +c \right )-4 \left (\cos ^{3}\left (d x +c \right )\right )-i \sin \left (d x +c \right )-2 \left (\cos ^{2}\left (d x +c \right )\right )+3 \cos \left (d x +c \right )+1\right ) a^{2} e}{d \left (\cos \left (d x +c \right )+1\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\) | \(556\) |
(-1/96-1/96*I)/d*(-tan(d*x+c)+I)^2*(a*(1+I*tan(d*x+c)))^(1/2)*(e*sec(d*x+c ))^(1/2)*(-45*sin(d*x+c)*cos(d*x+c)^2*(1/(cos(d*x+c)+1))^(1/2)+34*sin(d*x+ c)*cos(d*x+c)*(1/(cos(d*x+c)+1))^(1/2)+45*I*cos(d*x+c)^3*arctanh(1/2*(cos( d*x+c)+sin(d*x+c)+1)/(cos(d*x+c)+1)/(1/(cos(d*x+c)+1))^(1/2))+8*I*(1/(cos( d*x+c)+1))^(1/2)-79*I*(1/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)^2+45*cos(d*x+c)^ 3*arctanh(1/2*(cos(d*x+c)-sin(d*x+c)+1)/(cos(d*x+c)+1)/(1/(cos(d*x+c)+1))^ (1/2))-45*(1/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)^3-79*(1/(cos(d*x+c)+1))^(1/2 )*cos(d*x+c)^2-26*(1/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)+8*sin(d*x+c)*(1/(cos (d*x+c)+1))^(1/2)-8*I*sin(d*x+c)*(1/(cos(d*x+c)+1))^(1/2)-34*I*(1/(cos(d*x +c)+1))^(1/2)*cos(d*x+c)*sin(d*x+c)-45*I*(1/(cos(d*x+c)+1))^(1/2)*cos(d*x+ c)^3+45*I*(1/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)^2*sin(d*x+c)-26*I*(1/(cos(d* x+c)+1))^(1/2)*cos(d*x+c)+8*(1/(cos(d*x+c)+1))^(1/2))*(4*I*cos(d*x+c)^2*si n(d*x+c)+2*I*cos(d*x+c)*sin(d*x+c)-4*cos(d*x+c)^3-I*sin(d*x+c)-2*cos(d*x+c )^2+3*cos(d*x+c)+1)*a^2*e/(cos(d*x+c)+1)/(1/(cos(d*x+c)+1))^(1/2)
Time = 0.26 (sec) , antiderivative size = 635, normalized size of antiderivative = 1.04 \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2} \, dx=\frac {{\left (113 i \, a^{2} e e^{\left (4 i \, d x + 4 i \, c\right )} + 126 i \, a^{2} e e^{\left (2 i \, d x + 2 i \, c\right )} + 45 i \, a^{2} e\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 )} + 6 \, \sqrt {\frac {225 i \, a^{5} e^{3}}{64 \, d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {2 \, {\left (15 \, {\left (a^{2} e e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} e\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 )} + 8 i \, \sqrt {\frac {225 i \, a^{5} e^{3}}{64 \, d^{2}}} d\right )}}{15 \, a^{2} e}\right ) - 6 \, \sqrt {\frac {225 i \, a^{5} e^{3}}{64 \, d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {2 \, {\left (15 \, {\left (a^{2} e e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} e\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 )} - 8 i \, \sqrt {\frac {225 i \, a^{5} e^{3}}{64 \, d^{2}}} d\right )}}{15 \, a^{2} e}\right ) + 6 \, \sqrt {-\frac {225 i \, a^{5} e^{3}}{64 \, d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {2 \, {\left (15 \, {\left (a^{2} e e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} e\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 )} + 8 i \, \sqrt {-\frac {225 i \, a^{5} e^{3}}{64 \, d^{2}}} d\right )}}{15 \, a^{2} e}\right ) - 6 \, \sqrt {-\frac {225 i \, a^{5} e^{3}}{64 \, d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {2 \, {\left (15 \, {\left (a^{2} e e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} e\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 )} - 8 i \, \sqrt {-\frac {225 i \, a^{5} e^{3}}{64 \, d^{2}}} d\right )}}{15 \, a^{2} e}\right )}{12 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
1/12*((113*I*a^2*e*e^(4*I*d*x + 4*I*c) + 126*I*a^2*e*e^(2*I*d*x + 2*I*c) + 45*I*a^2*e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^(1/2*I*d*x + 1/2*I*c) + 6*sqrt(225/64*I*a^5*e^3/d^2)*(d*e^(4*I*d* x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)*log(2/15*(15*(a^2*e*e^(2*I*d*x + 2*I*c) + a^2*e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(e/(e^(2*I*d*x + 2* I*c) + 1))*e^(1/2*I*d*x + 1/2*I*c) + 8*I*sqrt(225/64*I*a^5*e^3/d^2)*d)/(a^ 2*e)) - 6*sqrt(225/64*I*a^5*e^3/d^2)*(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d *x + 2*I*c) + d)*log(2/15*(15*(a^2*e*e^(2*I*d*x + 2*I*c) + a^2*e)*sqrt(a/( e^(2*I*d*x + 2*I*c) + 1))*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^(1/2*I*d*x + 1/2*I*c) - 8*I*sqrt(225/64*I*a^5*e^3/d^2)*d)/(a^2*e)) + 6*sqrt(-225/64*I* a^5*e^3/d^2)*(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)*log(2/1 5*(15*(a^2*e*e^(2*I*d*x + 2*I*c) + a^2*e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1) )*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^(1/2*I*d*x + 1/2*I*c) + 8*I*sqrt(-22 5/64*I*a^5*e^3/d^2)*d)/(a^2*e)) - 6*sqrt(-225/64*I*a^5*e^3/d^2)*(d*e^(4*I* d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)*log(2/15*(15*(a^2*e*e^(2*I*d*x + 2*I*c) + a^2*e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^(1/2*I*d*x + 1/2*I*c) - 8*I*sqrt(-225/64*I*a^5*e^3/d^2)*d)/ (a^2*e)))/(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)
Timed out. \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2} \, dx=\text {Timed out} \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3005 vs. \(2 (458) = 916\).
Time = 0.65 (sec) , antiderivative size = 3005, normalized size of antiderivative = 4.91 \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2} \, dx=\text {Too large to display} \]
192*(1808*a^2*e*cos(9/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 201 6*a^2*e*cos(5/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 720*a^2*e*c os(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1808*I*a^2*e*sin(9/4 *arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2016*I*a^2*e*sin(5/4*arcta n2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 720*I*a^2*e*sin(1/4*arctan2(sin( 2*d*x + 2*c), cos(2*d*x + 2*c))) - 90*(sqrt(2)*a^2*e*cos(6*d*x + 6*c) + 3* sqrt(2)*a^2*e*cos(4*d*x + 4*c) + 3*sqrt(2)*a^2*e*cos(2*d*x + 2*c) + I*sqrt (2)*a^2*e*sin(6*d*x + 6*c) + 3*I*sqrt(2)*a^2*e*sin(4*d*x + 4*c) + 3*I*sqrt (2)*a^2*e*sin(2*d*x + 2*c) + sqrt(2)*a^2*e)*arctan2(sqrt(2)*cos(1/4*arctan 2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1, sqrt(2)*sin(1/4*arctan2(sin(2* d*x + 2*c), cos(2*d*x + 2*c))) + 1) - 90*(sqrt(2)*a^2*e*cos(6*d*x + 6*c) + 3*sqrt(2)*a^2*e*cos(4*d*x + 4*c) + 3*sqrt(2)*a^2*e*cos(2*d*x + 2*c) + I*s qrt(2)*a^2*e*sin(6*d*x + 6*c) + 3*I*sqrt(2)*a^2*e*sin(4*d*x + 4*c) + 3*I*s qrt(2)*a^2*e*sin(2*d*x + 2*c) + sqrt(2)*a^2*e)*arctan2(sqrt(2)*cos(1/4*arc tan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1, -sqrt(2)*sin(1/4*arctan2(si n(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) - 90*(sqrt(2)*a^2*e*cos(6*d*x + 6* c) + 3*sqrt(2)*a^2*e*cos(4*d*x + 4*c) + 3*sqrt(2)*a^2*e*cos(2*d*x + 2*c) + I*sqrt(2)*a^2*e*sin(6*d*x + 6*c) + 3*I*sqrt(2)*a^2*e*sin(4*d*x + 4*c) + 3 *I*sqrt(2)*a^2*e*sin(2*d*x + 2*c) + sqrt(2)*a^2*e)*arctan2(sqrt(2)*cos(1/4 *arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 1, sqrt(2)*sin(1/4*arct...
\[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2} \, dx=\int { \left (e \sec \left (d x + c\right )\right )^{\frac {3}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2} \, dx=\int {\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \]