Integrand size = 30, antiderivative size = 719 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx=\frac {i a}{3 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i a^{3/2} e^{7/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right ) \sec (c+d x)}{8 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a^{3/2} e^{7/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right ) \sec (c+d x)}{8 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a^{3/2} e^{7/2} \log \left (a-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i a^{3/2} e^{7/2} \log \left (a+\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}} \]
1/3*I*a/d/(e*cos(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))^(1/2)+5/8*I*a*cos(d*x+c) ^2/d/(e*cos(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))^(1/2)-5/16*I*a^(3/2)*e^(7/2)* 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/(e*cos(d*x+c))^(7/2)/(e*sec(d*x+c))^(7/2)*2^(1/2)/(a-I* a*tan(d*x+c))^(1/2)/(a+I*a*tan(d*x+c))^(1/2)+5/16*I*a^(3/2)*e^(7/2)*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/(e*cos(d*x+c))^(7/2)/(e*sec(d*x+c))^(7/2)*2^(1/2)/(a-I*a*tan( d*x+c))^(1/2)/(a+I*a*tan(d*x+c))^(1/2)+5/32*I*a^(3/2)*e^(7/2)*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/(e*cos(d*x+c))^(7/2)/(e*sec(d*x+c))^(7/2) *2^(1/2)/(a-I*a*tan(d*x+c))^(1/2)/(a+I*a*tan(d*x+c))^(1/2)-5/32*I*a^(3/2)* e^(7/2)*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/(e*cos(d*x+c))^(7/2)/ (e*sec(d*x+c))^(7/2)*2^(1/2)/(a-I*a*tan(d*x+c))^(1/2)/(a+I*a*tan(d*x+c))^( 1/2)-5/12*I*cos(d*x+c)^2*(a+I*a*tan(d*x+c))^(1/2)/d/(e*cos(d*x+c))^(7/2)
Time = 4.51 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx=\frac {\sqrt {\cos (c+d x)} \left (-\frac {40}{3} i \cos ^{\frac {3}{2}}(c+d x)+\frac {5}{8} i e^{-\frac {7}{2} i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^3 \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )} \left (2 \arctan \left (1-\sqrt {2} e^{\frac {1}{2} i (c+d x)}\right )-2 \arctan \left (1+\sqrt {2} e^{\frac {1}{2} i (c+d x)}\right )+\log \left (1-\sqrt {2} e^{\frac {1}{2} i (c+d x)}+e^{i (c+d x)}\right )-\log \left (1+\sqrt {2} e^{\frac {1}{2} i (c+d x)}+e^{i (c+d x)}\right )\right )+\frac {32}{3} \sqrt {\cos (c+d x)} (i \cos (c+d x)+\sin (c+d x))+20 \cos ^{\frac {5}{2}}(c+d x) (i \cos (c+d x)+\sin (c+d x))\right ) \sqrt {a+i a \tan (c+d x)}}{32 d (e \cos (c+d x))^{7/2}} \]
(Sqrt[Cos[c + d*x]]*(((-40*I)/3)*Cos[c + d*x]^(3/2) + (((5*I)/8)*(1 + E^(( 2*I)*(c + d*x)))^3*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(2*ArcT an[1 - Sqrt[2]*E^((I/2)*(c + d*x))] - 2*ArcTan[1 + Sqrt[2]*E^((I/2)*(c + d *x))] + Log[1 - Sqrt[2]*E^((I/2)*(c + d*x)) + E^(I*(c + d*x))] - Log[1 + S qrt[2]*E^((I/2)*(c + d*x)) + E^(I*(c + d*x))]))/E^(((7*I)/2)*(c + d*x)) + (32*Sqrt[Cos[c + d*x]]*(I*Cos[c + d*x] + Sin[c + d*x]))/3 + 20*Cos[c + d*x ]^(5/2)*(I*Cos[c + d*x] + Sin[c + d*x]))*Sqrt[a + I*a*Tan[c + d*x]])/(32*d *(e*Cos[c + d*x])^(7/2))
Time = 1.47 (sec) , antiderivative size = 538, normalized size of antiderivative = 0.75, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3998, 3042, 3979, 3042, 3982, 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.
| \(\displaystyle \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3998 |
| \(\displaystyle \frac {\int (e \sec (c+d x))^{7/2} \sqrt {i \tan (c+d x) a+a}dx}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (e \sec (c+d x))^{7/2} \sqrt {i \tan (c+d x) a+a}dx}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3979 |
| \(\displaystyle \frac {\frac {5}{6} a \int \frac {(e \sec (c+d x))^{7/2}}{\sqrt {i \tan (c+d x) a+a}}dx+\frac {i a (e \sec (c+d x))^{7/2}}{3 d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5}{6} a \int \frac {(e \sec (c+d x))^{7/2}}{\sqrt {i \tan (c+d x) a+a}}dx+\frac {i a (e \sec (c+d x))^{7/2}}{3 d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3982 |
| \(\displaystyle \frac {\frac {5}{6} a \left (\frac {3 e^2 \int (e \sec (c+d x))^{3/2} \sqrt {i \tan (c+d x) a+a}dx}{4 a}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 a d}\right )+\frac {i a (e \sec (c+d x))^{7/2}}{3 d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5}{6} a \left (\frac {3 e^2 \int (e \sec (c+d x))^{3/2} \sqrt {i \tan (c+d x) a+a}dx}{4 a}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 a d}\right )+\frac {i a (e \sec (c+d x))^{7/2}}{3 d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3979 |
| \(\displaystyle \frac {\frac {5}{6} a \left (\frac {3 e^2 \left (\frac {1}{2} a \int \frac {(e \sec (c+d x))^{3/2}}{\sqrt {i \tan (c+d x) a+a}}dx+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )}{4 a}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 a d}\right )+\frac {i a (e \sec (c+d x))^{7/2}}{3 d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5}{6} a \left (\frac {3 e^2 \left (\frac {1}{2} a \int \frac {(e \sec (c+d x))^{3/2}}{\sqrt {i \tan (c+d x) a+a}}dx+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )}{4 a}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 a d}\right )+\frac {i a (e \sec (c+d x))^{7/2}}{3 d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3980 |
| \(\displaystyle \frac {\frac {5}{6} a \left (\frac {3 e^2 \left (\frac {a e \sec (c+d x) \int \sqrt {e \sec (c+d x)} \sqrt {a-i a \tan (c+d x)}dx}{2 \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )}{4 a}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 a d}\right )+\frac {i a (e \sec (c+d x))^{7/2}}{3 d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5}{6} a \left (\frac {3 e^2 \left (\frac {a e \sec (c+d x) \int \sqrt {e \sec (c+d x)} \sqrt {a-i a \tan (c+d x)}dx}{2 \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )}{4 a}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 a d}\right )+\frac {i a (e \sec (c+d x))^{7/2}}{3 d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3976 |
| \(\displaystyle \frac {\frac {5}{6} a \left (\frac {3 e^2 \left (\frac {2 i a^2 e^3 \sec (c+d x) \int \frac {\cos (c+d x) (a-i a \tan (c+d x))}{e \left (a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2\right )}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )}{4 a}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 a d}\right )+\frac {i a (e \sec (c+d x))^{7/2}}{3 d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {\frac {5}{6} a \left (\frac {3 e^2 \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\int \frac {a+\cos (c+d x) (a-i a \tan (c+d x))}{a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}-\frac {\int \frac {a-\cos (c+d x) (a-i a \tan (c+d x))}{a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )}{4 a}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 a d}\right )+\frac {i a (e \sec (c+d x))^{7/2}}{3 d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {5}{6} a \left (\frac {3 e^2 \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\int \frac {1}{\frac {a}{e}-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}+\frac {\int \frac {1}{\frac {a}{e}+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}}{2 e}-\frac {\int \frac {a-\cos (c+d x) (a-i a \tan (c+d x))}{a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )}{4 a}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 a d}\right )+\frac {i a (e \sec (c+d x))^{7/2}}{3 d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {5}{6} a \left (\frac {3 e^2 \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\int \frac {1}{-\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}-1}d\left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\int \frac {1}{-\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}-1}d\left (\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\int \frac {a-\cos (c+d x) (a-i a \tan (c+d x))}{a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )}{4 a}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 a d}\right )+\frac {i a (e \sec (c+d x))^{7/2}}{3 d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {5}{6} a \left (\frac {3 e^2 \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\int \frac {a-\cos (c+d x) (a-i a \tan (c+d x))}{a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )}{4 a}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 a d}\right )+\frac {i a (e \sec (c+d x))^{7/2}}{3 d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {5}{6} a \left (\frac {3 e^2 \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{\sqrt {e} \left (\frac {a}{e}-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}\right )}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{\sqrt {e} \left (\frac {a}{e}+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}\right )}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )}{4 a}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 a d}\right )+\frac {i a (e \sec (c+d x))^{7/2}}{3 d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {5}{6} a \left (\frac {3 e^2 \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{\sqrt {e} \left (\frac {a}{e}-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}\right )}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{\sqrt {e} \left (\frac {a}{e}+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}\right )}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )}{4 a}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 a d}\right )+\frac {i a (e \sec (c+d x))^{7/2}}{3 d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {5}{6} a \left (\frac {3 e^2 \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{\frac {a}{e}-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} e}+\frac {\int \frac {\sqrt {a}+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{\frac {a}{e}+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {a} e}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )}{4 a}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 a d}\right )+\frac {i a (e \sec (c+d x))^{7/2}}{3 d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {5}{6} a \left (\frac {3 e^2 \left (\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{2 \sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{2 \sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}\right )}{4 a}-\frac {i e^2 \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}{2 a d}\right )+\frac {i a (e \sec (c+d x))^{7/2}}{3 d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\) |
(((I/3)*a*(e*Sec[c + d*x])^(7/2))/(d*Sqrt[a + I*a*Tan[c + d*x]]) + (5*a*(( (-1/2*I)*e^2*(e*Sec[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(a*d) + (3 *e^2*((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*Sec[c + d*x]])]/(Sqrt[2]*Sqrt[a]*Sqrt[e])) + ArcTan[1 + (Sqrt[2 ]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]/(Sqr t[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]*Sqr t[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))*Sec[c + d*x])/(d*Sqrt[a - I *a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])))/(4*a)))/6)/((e*Cos[c + d*x] )^(7/2)*(e*Sec[c + d*x])^(7/2))
3.7.80.3.1 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]
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]
Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)])^(n_.), x_Symbol] :> Simp[(d*Cos[e + f*x])^m*(d*Sec[e + f*x])^m Int[( a + b*Tan[e + f*x])^n/(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m , n}, x] && !IntegerQ[m]
Time = 8.77 (sec) , antiderivative size = 459, normalized size of antiderivative = 0.64
| method | result | size |
| default | \(\frac {\left (\frac {1}{48}-\frac {i}{48}\right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (10 i \tan \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+15 i \cos \left (d x +c \right ) \operatorname {arctanh}\left (\frac {\cos \left (d x +c \right )+\sin \left (d x +c \right )+1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\right )-8 i \left (\sec ^{2}\left (d x +c \right )\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-5 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+8 i \tan \left (d x +c \right ) \sec \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-15 \,\operatorname {arctanh}\left (\frac {-\cos \left (d x +c \right )+\sin \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )-15 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )-15 \sin \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+15 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sin \left (d x +c \right )+2 i \sec \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-5 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-10 \tan \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-15 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )+2 \sec \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-8 \tan \left (d x +c \right ) \sec \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-8 \left (\sec ^{2}\left (d x +c \right )\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\right )}{d \left (i \cos \left (d x +c \right )+i-\sin \left (d x +c \right )\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, e^{3} \sqrt {e \cos \left (d x +c \right )}}\) | \(459\) |
(1/48-1/48*I)/d*(a*(1+I*tan(d*x+c)))^(1/2)/(I*cos(d*x+c)+I-sin(d*x+c))/(1/ (cos(d*x+c)+1))^(1/2)/e^3/(e*cos(d*x+c))^(1/2)*(10*I*tan(d*x+c)*(1/(cos(d* x+c)+1))^(1/2)+15*I*cos(d*x+c)*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*sec(d*x+c)^2*(1/(cos(d*x+c)+1))^(1 /2)-5*I*(1/(cos(d*x+c)+1))^(1/2)+8*I*tan(d*x+c)*sec(d*x+c)*(1/(cos(d*x+c)+ 1))^(1/2)-15*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))*cos(d*x+c)-15*(1/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)-15*si n(d*x+c)*(1/(cos(d*x+c)+1))^(1/2)+15*I*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c) +2*I*sec(d*x+c)*(1/(cos(d*x+c)+1))^(1/2)-5*(1/(cos(d*x+c)+1))^(1/2)-10*tan (d*x+c)*(1/(cos(d*x+c)+1))^(1/2)-15*I*cos(d*x+c)*(1/(cos(d*x+c)+1))^(1/2)+ 2*sec(d*x+c)*(1/(cos(d*x+c)+1))^(1/2)-8*tan(d*x+c)*sec(d*x+c)*(1/(cos(d*x+ c)+1))^(1/2)-8*sec(d*x+c)^2*(1/(cos(d*x+c)+1))^(1/2))
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 )}} \]
1/12*(sqrt(2)*sqrt(1/2)*sqrt(e*e^(2*I*d*x + 2*I*c) + e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(-5*I*e^(5*I*d*x + 5*I*c) + 42*I*e^(3*I*d*x + 3*I*c) + 15* I*e^(I*d*x + I*c))*e^(-1/2*I*d*x - 1/2*I*c) - 6*(d*e^4*e^(6*I*d*x + 6*I*c) + 3*d*e^4*e^(4*I*d*x + 4*I*c) + 3*d*e^4*e^(2*I*d*x + 2*I*c) + d*e^4)*sqrt (25/64*I*a/(d^2*e^7))*log(8/5*d*e^4*sqrt(25/64*I*a/(d^2*e^7)) + sqrt(2)*sq rt(1/2)*sqrt(e*e^(2*I*d*x + 2*I*c) + e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))* e^(-1/2*I*d*x - 1/2*I*c)) + 6*(d*e^4*e^(6*I*d*x + 6*I*c) + 3*d*e^4*e^(4*I* d*x + 4*I*c) + 3*d*e^4*e^(2*I*d*x + 2*I*c) + d*e^4)*sqrt(25/64*I*a/(d^2*e^ 7))*log(-8/5*d*e^4*sqrt(25/64*I*a/(d^2*e^7)) + sqrt(2)*sqrt(1/2)*sqrt(e*e^ (2*I*d*x + 2*I*c) + e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(-1/2*I*d*x - 1 /2*I*c)) + 6*(d*e^4*e^(6*I*d*x + 6*I*c) + 3*d*e^4*e^(4*I*d*x + 4*I*c) + 3* d*e^4*e^(2*I*d*x + 2*I*c) + d*e^4)*sqrt(-25/64*I*a/(d^2*e^7))*log(8/5*d*e^ 4*sqrt(-25/64*I*a/(d^2*e^7)) + sqrt(2)*sqrt(1/2)*sqrt(e*e^(2*I*d*x + 2*I*c ) + e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(-1/2*I*d*x - 1/2*I*c)) - 6*(d* e^4*e^(6*I*d*x + 6*I*c) + 3*d*e^4*e^(4*I*d*x + 4*I*c) + 3*d*e^4*e^(2*I*d*x + 2*I*c) + d*e^4)*sqrt(-25/64*I*a/(d^2*e^7))*log(-8/5*d*e^4*sqrt(-25/64*I *a/(d^2*e^7)) + sqrt(2)*sqrt(1/2)*sqrt(e*e^(2*I*d*x + 2*I*c) + e)*sqrt(a/( e^(2*I*d*x + 2*I*c) + 1))*e^(-1/2*I*d*x - 1/2*I*c)))/(d*e^4*e^(6*I*d*x + 6 *I*c) + 3*d*e^4*e^(4*I*d*x + 4*I*c) + 3*d*e^4*e^(2*I*d*x + 2*I*c) + d*e^4)
Timed out. \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2661 vs. \(2 (549) = 1098\).
Time = 0.59 (sec) , antiderivative size = 2661, normalized size of antiderivative = 3.70 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx=\text {Too large to display} \]
-192*(30*(sqrt(2)*cos(6*d*x + 6*c) + 3*sqrt(2)*cos(4*d*x + 4*c) + 3*sqrt(2 )*cos(2*d*x + 2*c) + I*sqrt(2)*sin(6*d*x + 6*c) + 3*I*sqrt(2)*sin(4*d*x + 4*c) + 3*I*sqrt(2)*sin(2*d*x + 2*c) + sqrt(2))*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(sin (2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) + 30*(sqrt(2)*cos(6*d*x + 6*c) + 3* sqrt(2)*cos(4*d*x + 4*c) + 3*sqrt(2)*cos(2*d*x + 2*c) + I*sqrt(2)*sin(6*d* x + 6*c) + 3*I*sqrt(2)*sin(4*d*x + 4*c) + 3*I*sqrt(2)*sin(2*d*x + 2*c) + s qrt(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) + 30*(sqrt(2)*cos(6*d*x + 6*c) + 3*sqrt(2)*cos(4*d*x + 4*c) + 3*sqrt(2)*c os(2*d*x + 2*c) + I*sqrt(2)*sin(6*d*x + 6*c) + 3*I*sqrt(2)*sin(4*d*x + 4*c ) + 3*I*sqrt(2)*sin(2*d*x + 2*c) + sqrt(2))*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) + 30*(sqrt(2)*cos(6*d*x + 6*c) + 3*sqr t(2)*cos(4*d*x + 4*c) + 3*sqrt(2)*cos(2*d*x + 2*c) + I*sqrt(2)*sin(6*d*x + 6*c) + 3*I*sqrt(2)*sin(4*d*x + 4*c) + 3*I*sqrt(2)*sin(2*d*x + 2*c) + sqrt (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) - 30*(-I*sqrt(2)*cos(6*d*x + 6*c) - 3*I*sqrt(2)*cos(4*d*x + 4*c) - 3*I*sqrt( 2)*cos(2*d*x + 2*c) + sqrt(2)*sin(6*d*x + 6*c) + 3*sqrt(2)*sin(4*d*x + ...
\[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx=\int { \frac {\sqrt {i \, a \tan \left (d x + c\right ) + a}}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx=\int \frac {\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \]