Integrand size = 30, antiderivative size = 524 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx=\frac {i a}{d (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}-\frac {i a^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e}}\right ) \sec (c+d x)}{\sqrt {2} d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e}}\right ) \sec (c+d x)}{\sqrt {2} d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a^{3/2} \log \left (a-\frac {\sqrt {2} \sqrt {a} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{2 \sqrt {2} d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {i a^{3/2} \log \left (a+\frac {\sqrt {2} \sqrt {a} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{2 \sqrt {2} d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \]
I*a/d/(e*cos(d*x+c))^(3/2)/(a+I*a*tan(d*x+c))^(1/2)-1/2*I*a^(3/2)*arctan(1 -2^(1/2)*(e*cos(d*x+c))^(1/2)*(a-I*a*tan(d*x+c))^(1/2)/a^(1/2)/e^(1/2))*se c(d*x+c)/d/e^(3/2)*2^(1/2)/(a-I*a*tan(d*x+c))^(1/2)/(a+I*a*tan(d*x+c))^(1/ 2)+1/2*I*a^(3/2)*arctan(1+2^(1/2)*(e*cos(d*x+c))^(1/2)*(a-I*a*tan(d*x+c))^ (1/2)/a^(1/2)/e^(1/2))*sec(d*x+c)/d/e^(3/2)*2^(1/2)/(a-I*a*tan(d*x+c))^(1/ 2)/(a+I*a*tan(d*x+c))^(1/2)+1/4*I*a^(3/2)*ln(a-2^(1/2)*a^(1/2)*(e*cos(d*x+ c))^(1/2)*(a-I*a*tan(d*x+c))^(1/2)/e^(1/2)+cos(d*x+c)*(a-I*a*tan(d*x+c)))* sec(d*x+c)/d/e^(3/2)*2^(1/2)/(a-I*a*tan(d*x+c))^(1/2)/(a+I*a*tan(d*x+c))^( 1/2)-1/4*I*a^(3/2)*ln(a+2^(1/2)*a^(1/2)*(e*cos(d*x+c))^(1/2)*(a-I*a*tan(d* x+c))^(1/2)/e^(1/2)+cos(d*x+c)*(a-I*a*tan(d*x+c)))*sec(d*x+c)/d/e^(3/2)*2^ (1/2)/(a-I*a*tan(d*x+c))^(1/2)/(a+I*a*tan(d*x+c))^(1/2)
Time = 5.15 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx=\frac {i e^{-\frac {1}{2} i (c+d x)} \cos ^2(c+d x) \left (2 \sqrt {2} \cos \left (\frac {1}{2} (c+d x)\right )+2 \arctan \left (1-\sqrt {2} e^{\frac {1}{2} i (c+d x)}\right ) \cos (c+d x)-2 \arctan \left (1+\sqrt {2} e^{\frac {1}{2} i (c+d x)}\right ) \cos (c+d x)+\cos (c+d x) \log \left (1-\sqrt {2} e^{\frac {1}{2} i (c+d x)}+e^{i (c+d x)}\right )-\cos (c+d x) \log \left (1+\sqrt {2} e^{\frac {1}{2} i (c+d x)}+e^{i (c+d x)}\right )-2 i \sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) (\cos (c+d x)+i \sin (c+d x)) \sqrt {a+i a \tan (c+d x)}}{\sqrt {2} d \left (1+e^{2 i (c+d x)}\right ) (e \cos (c+d x))^{3/2}} \]
(I*Cos[c + d*x]^2*(2*Sqrt[2]*Cos[(c + d*x)/2] + 2*ArcTan[1 - Sqrt[2]*E^((I /2)*(c + d*x))]*Cos[c + d*x] - 2*ArcTan[1 + Sqrt[2]*E^((I/2)*(c + d*x))]*C os[c + d*x] + Cos[c + d*x]*Log[1 - Sqrt[2]*E^((I/2)*(c + d*x)) + E^(I*(c + d*x))] - Cos[c + d*x]*Log[1 + Sqrt[2]*E^((I/2)*(c + d*x)) + E^(I*(c + d*x ))] - (2*I)*Sqrt[2]*Sin[(c + d*x)/2])*(Cos[c + d*x] + I*Sin[c + d*x])*Sqrt [a + I*a*Tan[c + d*x]])/(Sqrt[2]*d*E^((I/2)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))*(e*Cos[c + d*x])^(3/2))
Time = 1.03 (sec) , antiderivative size = 438, normalized size of antiderivative = 0.84, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {3042, 3998, 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.
| \(\displaystyle \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3998 |
| \(\displaystyle \frac {\int (e \sec (c+d x))^{3/2} \sqrt {i \tan (c+d x) a+a}dx}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (e \sec (c+d x))^{3/2} \sqrt {i \tan (c+d x) a+a}dx}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3979 |
| \(\displaystyle \frac {\frac {1}{2} a \int \frac {(e \sec (c+d x))^{3/2}}{\sqrt {i \tan (c+d x) a+a}}dx+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} a \int \frac {(e \sec (c+d x))^{3/2}}{\sqrt {i \tan (c+d x) a+a}}dx+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3980 |
| \(\displaystyle \frac {\frac {a e \sec (c+d x) \int \sqrt {e \sec (c+d x)} \sqrt {a-i a \tan (c+d x)}dx}{2 \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a e \sec (c+d x) \int \sqrt {e \sec (c+d x)} \sqrt {a-i a \tan (c+d x)}dx}{2 \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3976 |
| \(\displaystyle \frac {\frac {2 i a^2 e^3 \sec (c+d x) \int \frac {\cos (c+d x) (a-i a \tan (c+d x))}{e \left (a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2\right )}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\int \frac {a+\cos (c+d x) (a-i a \tan (c+d x))}{a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}-\frac {\int \frac {a-\cos (c+d x) (a-i a \tan (c+d x))}{a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\int \frac {1}{\frac {a}{e}-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}+\frac {\int \frac {1}{\frac {a}{e}+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}}{2 e}-\frac {\int \frac {a-\cos (c+d x) (a-i a \tan (c+d x))}{a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\int \frac {1}{-\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}-1}d\left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\int \frac {1}{-\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}-1}d\left (\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\int \frac {a-\cos (c+d x) (a-i a \tan (c+d x))}{a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\int \frac {a-\cos (c+d x) (a-i a \tan (c+d x))}{a^2+\cos ^2(c+d x) (a-i a \tan (c+d x))^2}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{\sqrt {e} \left (\frac {a}{e}-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}\right )}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{\sqrt {e} \left (\frac {a}{e}+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}\right )}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{\sqrt {e} \left (\frac {a}{e}-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}\right )}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{\sqrt {e} \left (\frac {a}{e}+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}\right )}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{\frac {a}{e}-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {2} \sqrt {a} e}+\frac {\int \frac {\sqrt {a}+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{\frac {a}{e}+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {a}}{\sqrt {e} \sqrt {e \sec (c+d x)}}+\frac {\cos (c+d x) (a-i a \tan (c+d x))}{e}}d\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}}{2 \sqrt {a} e}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {2 i a^2 e^3 \sec (c+d x) \left (\frac {\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{2 \sqrt {2} \sqrt {a} \sqrt {e}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{2 \sqrt {2} \sqrt {a} \sqrt {e}}}{2 e}\right )}{d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a (e \sec (c+d x))^{3/2}}{d \sqrt {a+i a \tan (c+d x)}}}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\) |
((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]*S qrt[e*Sec[c + d*x]])]/(Sqrt[2]*Sqrt[a]*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqr t[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))*Sec[c + d*x])/(d*Sqrt[a - I*a*Ta n[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]))/((e*Cos[c + d*x])^(3/2)*(e*Sec[c + d*x])^(3/2))
3.7.78.3.1 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[(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 = 10.44 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.61
| method | result | size |
| default | \(-\frac {i \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (i \cos \left (d x +c \right ) \operatorname {arctanh}\left (\frac {\cos \left (d x +c \right )-\sin \left (d x +c \right )+1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\right )+i \cos \left (d x +c \right ) \operatorname {arctanh}\left (\frac {\cos \left (d x +c \right )+\sin \left (d x +c \right )+1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\right )-2 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )-2 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+\operatorname {arctanh}\left (\frac {\cos \left (d x +c \right )-\sin \left (d x +c \right )+1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )-\operatorname {arctanh}\left (\frac {\cos \left (d x +c \right )+\sin \left (d x +c \right )+1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )-2 \sin \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\right )}{2 d \left (i \cos \left (d x +c \right )+i-\sin \left (d x +c \right )\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, e \sqrt {e \cos \left (d x +c \right )}}\) | \(319\) |
-1/2*I/d*(a*(1+I*tan(d*x+c)))^(1/2)*(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))+I*cos(d*x+c)*arctan h(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*(1/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)-2*I*(1/(cos(d*x+c)+1))^(1/2)+arctan h(1/2*(cos(d*x+c)-sin(d*x+c)+1)/(cos(d*x+c)+1)/(1/(cos(d*x+c)+1))^(1/2))*c os(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))*cos(d*x+c)-2*sin(d*x+c)*(1/(cos(d*x+c)+1))^(1/2))/(I*cos(d* x+c)+I-sin(d*x+c))/(1/(cos(d*x+c)+1))^(1/2)/e/(e*cos(d*x+c))^(1/2)
Time = 0.27 (sec) , antiderivative size = 470, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx=\frac {4 i \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} - {\left (d e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{2}\right )} \sqrt {\frac {i \, a}{d^{2} e^{3}}} \log \left (d e^{2} \sqrt {\frac {i \, a}{d^{2} e^{3}}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right ) + {\left (d e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{2}\right )} \sqrt {\frac {i \, a}{d^{2} e^{3}}} \log \left (-d e^{2} \sqrt {\frac {i \, a}{d^{2} e^{3}}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right ) + {\left (d e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{2}\right )} \sqrt {-\frac {i \, a}{d^{2} e^{3}}} \log \left (d e^{2} \sqrt {-\frac {i \, a}{d^{2} e^{3}}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right ) - {\left (d e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{2}\right )} \sqrt {-\frac {i \, a}{d^{2} e^{3}}} \log \left (-d e^{2} \sqrt {-\frac {i \, a}{d^{2} e^{3}}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right )}{2 \, {\left (d e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{2}\right )}} \]
1/2*(4*I*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^2*e^(2*I*d*x + 2*I*c) + d*e^2)*sqrt(I*a/(d^2*e^3))*log(d*e^2*sqrt(I*a/(d^2*e^3)) + 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^2*e^(2*I*d*x + 2*I*c) + d*e^2)*sqrt(I*a/(d^2*e ^3))*log(-d*e^2*sqrt(I*a/(d^2*e^3)) + 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^2*e^(2*I*d*x + 2*I*c) + d*e^2)*sqrt(-I*a/(d^2*e^3))*log(d*e^2*sqrt( -I*a/(d^2*e^3)) + 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^2*e^(2*I*d*x + 2*I*c) + d*e^2)*sqrt(-I*a/(d^2*e^3))*log(-d*e^2*sqrt(-I*a/(d^2*e^3)) + s qrt(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^2*e^(2*I*d*x + 2*I*c) + d*e^2)
\[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx=\int \frac {\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}{\left (e \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1836 vs. \(2 (396) = 792\).
Time = 0.47 (sec) , antiderivative size = 1836, normalized size of antiderivative = 3.50 \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx=\text {Too large to display} \]
-8*(2*(sqrt(2)*cos(2*d*x + 2*c) + I*sqrt(2)*sin(2*d*x + 2*c) + sqrt(2))*ar ctan2(sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1, sq rt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) + 2*(sqrt( 2)*cos(2*d*x + 2*c) + 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) + 2*(sqrt(2)*cos(2*d* x + 2*c) + I*sqrt(2)*sin(2*d*x + 2*c) + sqrt(2))*arctan2(sqrt(2)*cos(1/4*a rctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 1, sqrt(2)*sin(1/4*arctan2(s in(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) + 2*(sqrt(2)*cos(2*d*x + 2*c) + 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) - 2*(-I*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2)*s in(2*d*x + 2*c) - I*sqrt(2))*arctan2(sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2 *c), cos(2*d*x + 2*c))) + sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2* c))), sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + cos(1 /2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) - 2*(I*sqrt(2)*cos(2* d*x + 2*c) - sqrt(2)*sin(2*d*x + 2*c) + I*sqrt(2))*arctan2(-sqrt(2)*sin(1/ 4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))), -sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), co s(2*d*x + 2*c))) + cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))...
\[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx=\int { \frac {\sqrt {i \, a \tan \left (d x + c\right ) + a}}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx=\int \frac {\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]