Integrand size = 30, antiderivative size = 453 \[ \int (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^{3/2} \, dx=\frac {7 i a^{3/2} e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{8 \sqrt {2} d}-\frac {7 i a^{3/2} e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{8 \sqrt {2} d}-\frac {7 i a^{3/2} e^{5/2} \log \left (a-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{16 \sqrt {2} d}+\frac {7 i a^{3/2} e^{5/2} \log \left (a+\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{16 \sqrt {2} d}+\frac {7 i a^2 (e \sec (c+d x))^{5/2}}{12 d \sqrt {a+i a \tan (c+d x)}}-\frac {7 i a e^2 \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}}{8 d}+\frac {i a (e \sec (c+d x))^{5/2} \sqrt {a+i a \tan (c+d x)}}{3 d} \]
7/16*I*a^(3/2)*e^(5/2)*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))/d*2^(1/2)-7/16*I*a^(3/2)*e^(5/2)*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))/d*2^(1 /2)-7/32*I*a^(3/2)*e^(5/2)*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)))/d*2^(1/2)+7/32* I*a^(3/2)*e^(5/2)*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)))/d*2^(1/2)+7/12*I*a^2*(e* sec(d*x+c))^(5/2)/d/(a+I*a*tan(d*x+c))^(1/2)+1/3*I*a*(e*sec(d*x+c))^(5/2)* (a+I*a*tan(d*x+c))^(1/2)/d-7/8*I*a*e^2*(e*sec(d*x+c))^(1/2)*(a+I*a*tan(d*x +c))^(1/2)/d
Time = 4.19 (sec) , antiderivative size = 376, normalized size of antiderivative = 0.83 \[ \int (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^{3/2} \, dx=-\frac {a (e \sec (c+d x))^{5/2} \left (2 i \sqrt {1+\cos (2 c)+i \sin (2 c)} (-9+7 \cos (2 c+2 d x)+14 i \sin (2 c+2 d x)) \sqrt {i-\tan \left (\frac {d x}{2}\right )}+84 \text {arctanh}\left (\frac {\sqrt {1-i \cos (c)+\sin (c)} \sqrt {i-\tan \left (\frac {d x}{2}\right )}}{\sqrt {-1-i \cos (c)-\sin (c)} \sqrt {i+\tan \left (\frac {d x}{2}\right )}}\right ) \cos ^3(c+d x) \sqrt {-1-i \cos (c)-\sin (c)} \sqrt {1+i \cos (c)-\sin (c)} \sqrt {i+\tan \left (\frac {d x}{2}\right )}-84 \text {arctanh}\left (\frac {\sqrt {1+i \cos (c)-\sin (c)} \sqrt {i-\tan \left (\frac {d x}{2}\right )}}{\sqrt {-1+i \cos (c)+\sin (c)} \sqrt {i+\tan \left (\frac {d x}{2}\right )}}\right ) \cos ^3(c+d x) \sqrt {1-i \cos (c)+\sin (c)} \sqrt {-1+i \cos (c)+\sin (c)} \sqrt {i+\tan \left (\frac {d x}{2}\right )}\right ) \sqrt {a+i a \tan (c+d x)}}{96 d \sqrt {1+\cos (2 c)+i \sin (2 c)} \sqrt {i-\tan \left (\frac {d x}{2}\right )}} \]
-1/96*(a*(e*Sec[c + d*x])^(5/2)*((2*I)*Sqrt[1 + Cos[2*c] + I*Sin[2*c]]*(-9 + 7*Cos[2*c + 2*d*x] + (14*I)*Sin[2*c + 2*d*x])*Sqrt[I - Tan[(d*x)/2]] + 84*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]])]*Cos[c + d*x]^3*Sqrt[-1 - I*Co s[c] - Sin[c]]*Sqrt[1 + I*Cos[c] - Sin[c]]*Sqrt[I + Tan[(d*x)/2]] - 84*Arc Tanh[(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]])]*Cos[c + d*x]^3*Sqrt[1 - I*Cos[c] + Sin[c]]*Sqrt[-1 + I*Cos[c] + Sin[c]]*Sqrt[I + Tan[(d*x)/2]])*Sqrt[a + I*a* Tan[c + d*x]])/(d*Sqrt[1 + Cos[2*c] + I*Sin[2*c]]*Sqrt[I - Tan[(d*x)/2]])
Time = 1.05 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.03, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {3042, 3979, 3042, 3979, 3042, 3982, 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.
| \(\displaystyle \int (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{5/2}dx\) |
| \(\Big \downarrow \) 3979 |
| \(\displaystyle \frac {7}{6} a \int (e \sec (c+d x))^{5/2} \sqrt {i \tan (c+d x) a+a}dx+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{6} a \int (e \sec (c+d x))^{5/2} \sqrt {i \tan (c+d x) a+a}dx+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3979 |
| \(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \int \frac {(e \sec (c+d x))^{5/2}}{\sqrt {i \tan (c+d x) a+a}}dx+\frac {i a (e \sec (c+d x))^{5/2}}{2 d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \int \frac {(e \sec (c+d x))^{5/2}}{\sqrt {i \tan (c+d x) a+a}}dx+\frac {i a (e \sec (c+d x))^{5/2}}{2 d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3982 |
| \(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \left (\frac {e^2 \int \sqrt {e \sec (c+d x)} \sqrt {i \tan (c+d x) a+a}dx}{2 a}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} \sqrt {e \sec (c+d x)}}{a d}\right )+\frac {i a (e \sec (c+d x))^{5/2}}{2 d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \left (\frac {e^2 \int \sqrt {e \sec (c+d x)} \sqrt {i \tan (c+d x) a+a}dx}{2 a}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} \sqrt {e \sec (c+d x)}}{a d}\right )+\frac {i a (e \sec (c+d x))^{5/2}}{2 d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3976 |
| \(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \left (-\frac {2 i e^4 \int \frac {\cos (c+d x) (i \tan (c+d x) a+a)}{e \left (a^2+\cos ^2(c+d x) (i \tan (c+d x) a+a)^2\right )}d\frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {e \sec (c+d x)}}}{d}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} \sqrt {e \sec (c+d x)}}{a d}\right )+\frac {i a (e \sec (c+d x))^{5/2}}{2 d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \left (-\frac {2 i e^4 \left (\frac {\int \frac {a+\cos (c+d x) (i \tan (c+d x) a+a)}{a^2+\cos ^2(c+d x) (i \tan (c+d x) a+a)^2}d\frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {e \sec (c+d x)}}}{2 e}-\frac {\int \frac {a-\cos (c+d x) (i \tan (c+d x) a+a)}{a^2+\cos ^2(c+d x) (i \tan (c+d x) a+a)^2}d\frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {e \sec (c+d x)}}}{2 e}\right )}{d}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} \sqrt {e \sec (c+d x)}}{a d}\right )+\frac {i a (e \sec (c+d x))^{5/2}}{2 d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \left (-\frac {2 i e^4 \left (\frac {\frac {\int \frac {1}{\frac {a}{e}-\frac {\sqrt {2} \sqrt {i \tan (c+d x) a+a} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (i \tan (c+d x) a+a)}{e}}d\frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {e \sec (c+d x)}}}{2 e}+\frac {\int \frac {1}{\frac {a}{e}+\frac {\sqrt {2} \sqrt {i \tan (c+d x) a+a} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (i \tan (c+d x) a+a)}{e}}d\frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {e \sec (c+d x)}}}{2 e}}{2 e}-\frac {\int \frac {a-\cos (c+d x) (i \tan (c+d x) a+a)}{a^2+\cos ^2(c+d x) (i \tan (c+d x) a+a)^2}d\frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {e \sec (c+d x)}}}{2 e}\right )}{d}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} \sqrt {e \sec (c+d x)}}{a d}\right )+\frac {i a (e \sec (c+d x))^{5/2}}{2 d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \left (-\frac {2 i e^4 \left (\frac {\frac {\int \frac {1}{-\frac {\cos (c+d x) (i \tan (c+d x) a+a)}{e}-1}d\left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {i \tan (c+d x) a+a}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\int \frac {1}{-\frac {\cos (c+d x) (i \tan (c+d x) a+a)}{e}-1}d\left (\frac {\sqrt {2} \sqrt {e} \sqrt {i \tan (c+d x) a+a}}{\sqrt {a} \sqrt {e \sec (c+d x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\int \frac {a-\cos (c+d x) (i \tan (c+d x) a+a)}{a^2+\cos ^2(c+d x) (i \tan (c+d x) a+a)^2}d\frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {e \sec (c+d x)}}}{2 e}\right )}{d}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} \sqrt {e \sec (c+d x)}}{a d}\right )+\frac {i a (e \sec (c+d x))^{5/2}}{2 d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \left (-\frac {2 i e^4 \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\int \frac {a-\cos (c+d x) (i \tan (c+d x) a+a)}{a^2+\cos ^2(c+d x) (i \tan (c+d x) a+a)^2}d\frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {e \sec (c+d x)}}}{2 e}\right )}{d}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} \sqrt {e \sec (c+d x)}}{a d}\right )+\frac {i a (e \sec (c+d x))^{5/2}}{2 d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \left (-\frac {2 i e^4 \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {e} \sqrt {i \tan (c+d x) a+a}}{\sqrt {e \sec (c+d x)}}}{\sqrt {e} \left (\frac {a}{e}-\frac {\sqrt {2} \sqrt {i \tan (c+d x) a+a} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (i \tan (c+d x) a+a)}{e}\right )}d\frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {e} \sqrt {i \tan (c+d x) a+a}}{\sqrt {e \sec (c+d x)}}\right )}{\sqrt {e} \left (\frac {a}{e}+\frac {\sqrt {2} \sqrt {i \tan (c+d x) a+a} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (i \tan (c+d x) a+a)}{e}\right )}d\frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}\right )}{d}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} \sqrt {e \sec (c+d x)}}{a d}\right )+\frac {i a (e \sec (c+d x))^{5/2}}{2 d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \left (-\frac {2 i e^4 \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {e} \sqrt {i \tan (c+d x) a+a}}{\sqrt {e \sec (c+d x)}}}{\sqrt {e} \left (\frac {a}{e}-\frac {\sqrt {2} \sqrt {i \tan (c+d x) a+a} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (i \tan (c+d x) a+a)}{e}\right )}d\frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {e} \sqrt {i \tan (c+d x) a+a}}{\sqrt {e \sec (c+d x)}}\right )}{\sqrt {e} \left (\frac {a}{e}+\frac {\sqrt {2} \sqrt {i \tan (c+d x) a+a} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (i \tan (c+d x) a+a)}{e}\right )}d\frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}\right )}{d}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} \sqrt {e \sec (c+d x)}}{a d}\right )+\frac {i a (e \sec (c+d x))^{5/2}}{2 d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \left (-\frac {2 i e^4 \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {e} \sqrt {i \tan (c+d x) a+a}}{\sqrt {e \sec (c+d x)}}}{\frac {a}{e}-\frac {\sqrt {2} \sqrt {i \tan (c+d x) a+a} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (i \tan (c+d x) a+a)}{e}}d\frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} e}+\frac {\int \frac {\sqrt {a}+\frac {\sqrt {2} \sqrt {e} \sqrt {i \tan (c+d x) a+a}}{\sqrt {e \sec (c+d x)}}}{\frac {a}{e}+\frac {\sqrt {2} \sqrt {i \tan (c+d x) a+a} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (i \tan (c+d x) a+a)}{e}}d\frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {a} e}}{2 e}\right )}{d}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} \sqrt {e \sec (c+d x)}}{a d}\right )+\frac {i a (e \sec (c+d x))^{5/2}}{2 d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \left (-\frac {2 i e^4 \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right )}{2 \sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right )}{2 \sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}\right )}{d}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} \sqrt {e \sec (c+d x)}}{a d}\right )+\frac {i a (e \sec (c+d x))^{5/2}}{2 d \sqrt {a+i a \tan (c+d x)}}\right )+\frac {i a \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{5/2}}{3 d}\) |
((I/3)*a*(e*Sec[c + d*x])^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/d + (7*a*(((I/ 2)*a*(e*Sec[c + d*x])^(5/2))/(d*Sqrt[a + I*a*Tan[c + d*x]]) + (3*a*(((-2*I )*e^4*((-(ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a] *Sqrt[e*Sec[c + d*x]])]/(Sqrt[2]*Sqrt[a]*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*S qrt[e]*Sqrt[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])]/(Sqrt[2]*Sqrt[a]*Sqrt[e]) + 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])]/(2*Sqrt[2]*Sqrt[a]*Sqrt[e]))/(2*e)))/d - (I*e^2*Sqrt[e*Sec[c + d*x ]]*Sqrt[a + I*a*Tan[c + d*x]])/(a*d)))/4))/6
3.4.100.3.1 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_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)])^(n_), x_Symbol] :> Simp[d^2*(d*Sec[e + f*x])^(m - 2)*((a + b*Tan[e + f*x])^(n + 1)/(b*f*(m + n - 1))), x] + Simp[d^2*((m - 2)/(a*(m + n - 1))) Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ [{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, 0] && GtQ[m, 1] && !IL tQ[m + n, 0] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
Time = 9.60 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.16
| method | result | size |
| default | \(\frac {\left (-\frac {1}{48}+\frac {i}{48}\right ) \sec \left (d x +c \right ) \left (-\tan \left (d x +c \right )+i\right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {e \sec \left (d x +c \right )}\, a \,e^{2} \left (-8 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+8 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+21 \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}}-7 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (\cos ^{2}\left (d x +c \right )\right )+14 \sin \left (d x +c \right ) \cos \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}}+21 \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 )+14 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right ) \sin \left (d x +c \right )-21 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (\cos ^{3}\left (d x +c \right )\right )+21 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (\cos ^{3}\left (d x +c \right )\right )+7 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (\cos ^{2}\left (d x +c \right )\right )+21 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 )+22 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )-22 \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}}+21 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 )\right )}{d \left (-2 i \left (\cos ^{2}\left (d x +c \right )\right )+2 \sin \left (d x +c \right ) \cos \left (d x +c \right )-i \cos \left (d x +c \right )+\sin \left (d x +c \right )+i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\) | \(524\) |
(-1/48+1/48*I)/d*sec(d*x+c)*(-tan(d*x+c)+I)*(a*(1+I*tan(d*x+c)))^(1/2)*(e* sec(d*x+c))^(1/2)*a*e^2*(-8*(1/(cos(d*x+c)+1))^(1/2)+8*I*(1/(cos(d*x+c)+1) )^(1/2)+21*sin(d*x+c)*cos(d*x+c)^2*(1/(cos(d*x+c)+1))^(1/2)-7*I*(1/(cos(d* x+c)+1))^(1/2)*cos(d*x+c)^2+14*sin(d*x+c)*cos(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)+21*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))+14*I*(1/ (cos(d*x+c)+1))^(1/2)*cos(d*x+c)*sin(d*x+c)-21*I*(1/(cos(d*x+c)+1))^(1/2)* cos(d*x+c)^3+21*(1/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)^3+7*(1/(cos(d*x+c)+1)) ^(1/2)*cos(d*x+c)^2+21*I*(1/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)^2*sin(d*x+c)+ 22*I*(1/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)-22*(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)+21*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)))/(-2*I* cos(d*x+c)^2+2*sin(d*x+c)*cos(d*x+c)-I*cos(d*x+c)+sin(d*x+c)+I)/(1/(cos(d* x+c)+1))^(1/2)
Time = 0.26 (sec) , antiderivative size = 644, normalized size of antiderivative = 1.42 \[ \int (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^{3/2} \, dx=\frac {{\left (-21 i \, a e^{2} e^{\left (5 i \, d x + 5 i \, c\right )} + 18 i \, a e^{2} e^{\left (3 i \, d x + 3 i \, c\right )} + 7 i \, a e^{2} e^{\left (i \, d x + i \, c\right )}\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 {49 i \, a^{3} e^{5}}{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 (7 \, {\left (a e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + a e^{2}\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 \, \sqrt {\frac {49 i \, a^{3} e^{5}}{64 \, d^{2}}} d\right )}}{7 \, a e^{2}}\right ) - 6 \, \sqrt {\frac {49 i \, a^{3} e^{5}}{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 (7 \, {\left (a e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + a e^{2}\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 \, \sqrt {\frac {49 i \, a^{3} e^{5}}{64 \, d^{2}}} d\right )}}{7 \, a e^{2}}\right ) - 6 \, \sqrt {-\frac {49 i \, a^{3} e^{5}}{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 (7 \, {\left (a e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + a e^{2}\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 \, \sqrt {-\frac {49 i \, a^{3} e^{5}}{64 \, d^{2}}} d\right )}}{7 \, a e^{2}}\right ) + 6 \, \sqrt {-\frac {49 i \, a^{3} e^{5}}{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 (7 \, {\left (a e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + a e^{2}\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 \, \sqrt {-\frac {49 i \, a^{3} e^{5}}{64 \, d^{2}}} d\right )}}{7 \, a e^{2}}\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*((-21*I*a*e^2*e^(5*I*d*x + 5*I*c) + 18*I*a*e^2*e^(3*I*d*x + 3*I*c) + 7*I*a*e^2*e^(I*d*x + I*c))*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(49/64*I*a^3*e^5/d^2) *(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)*log(2/7*(7*(a*e^2*e ^(2*I*d*x + 2*I*c) + a*e^2)*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*sqrt(49/64*I*a^3*e^5/d^2 )*d)/(a*e^2)) - 6*sqrt(49/64*I*a^3*e^5/d^2)*(d*e^(4*I*d*x + 4*I*c) + 2*d*e ^(2*I*d*x + 2*I*c) + d)*log(2/7*(7*(a*e^2*e^(2*I*d*x + 2*I*c) + a*e^2)*sqr t(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*sqrt(49/64*I*a^3*e^5/d^2)*d)/(a*e^2)) - 6*sqrt(-49/64*I *a^3*e^5/d^2)*(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)*log(2/ 7*(7*(a*e^2*e^(2*I*d*x + 2*I*c) + a*e^2)*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*sqrt(-49/64 *I*a^3*e^5/d^2)*d)/(a*e^2)) + 6*sqrt(-49/64*I*a^3*e^5/d^2)*(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)*log(2/7*(7*(a*e^2*e^(2*I*d*x + 2*I* c) + a*e^2)*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*sqrt(-49/64*I*a^3*e^5/d^2)*d)/(a*e^2)))/ (d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)
Timed out. \[ \int (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^{3/2} \, dx=\text {Timed out} \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3005 vs. \(2 (331) = 662\).
Time = 0.62 (sec) , antiderivative size = 3005, normalized size of antiderivative = 6.63 \[ \int (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^{3/2} \, dx=\text {Too large to display} \]
-192*(336*a*e^2*cos(11/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 28 8*a*e^2*cos(7/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 112*a*e^2*c os(3/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 336*I*a*e^2*sin(11/4 *arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 288*I*a*e^2*sin(7/4*arctan 2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 112*I*a*e^2*sin(3/4*arctan2(sin(2 *d*x + 2*c), cos(2*d*x + 2*c))) + 42*(sqrt(2)*a*e^2*cos(6*d*x + 6*c) + 3*s qrt(2)*a*e^2*cos(4*d*x + 4*c) + 3*sqrt(2)*a*e^2*cos(2*d*x + 2*c) + I*sqrt( 2)*a*e^2*sin(6*d*x + 6*c) + 3*I*sqrt(2)*a*e^2*sin(4*d*x + 4*c) + 3*I*sqrt( 2)*a*e^2*sin(2*d*x + 2*c) + sqrt(2)*a*e^2)*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*arctan2(sin(2*d *x + 2*c), cos(2*d*x + 2*c))) + 1) + 42*(sqrt(2)*a*e^2*cos(6*d*x + 6*c) + 3*sqrt(2)*a*e^2*cos(4*d*x + 4*c) + 3*sqrt(2)*a*e^2*cos(2*d*x + 2*c) + I*sq rt(2)*a*e^2*sin(6*d*x + 6*c) + 3*I*sqrt(2)*a*e^2*sin(4*d*x + 4*c) + 3*I*sq rt(2)*a*e^2*sin(2*d*x + 2*c) + sqrt(2)*a*e^2)*arctan2(sqrt(2)*cos(1/4*arct an2(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) + 42*(sqrt(2)*a*e^2*cos(6*d*x + 6*c ) + 3*sqrt(2)*a*e^2*cos(4*d*x + 4*c) + 3*sqrt(2)*a*e^2*cos(2*d*x + 2*c) + I*sqrt(2)*a*e^2*sin(6*d*x + 6*c) + 3*I*sqrt(2)*a*e^2*sin(4*d*x + 4*c) + 3* I*sqrt(2)*a*e^2*sin(2*d*x + 2*c) + sqrt(2)*a*e^2)*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*arcta...
\[ \int (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^{3/2} \, dx=\int { \left (e \sec \left (d x + c\right )\right )^{\frac {5}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^{3/2} \, dx=\int {\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2} \,d x \]