Integrand size = 14, antiderivative size = 523 \[ \int e^{\frac {1}{4} i \arctan (a x)} x \, dx=\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}-\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{32 a^2}-\frac {\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{32 a^2}+\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{32 a^2}+\frac {\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{32 a^2}-\frac {\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x} \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}\right )}{32 a^2}-\frac {\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x} \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}\right )}{32 a^2} \] Output:
1/8*(1-I*a*x)^(7/8)*(1+I*a*x)^(1/8)/a^2+1/2*(1-I*a*x)^(7/8)*(1+I*a*x)^(9/8 )/a^2-1/32*(2+2^(1/2))^(1/2)*arctan(((2-2^(1/2))^(1/2)-2*(1-I*a*x)^(1/8)/( 1+I*a*x)^(1/8))/(2+2^(1/2))^(1/2))/a^2-1/32*(2-2^(1/2))^(1/2)*arctan(((2+2 ^(1/2))^(1/2)-2*(1-I*a*x)^(1/8)/(1+I*a*x)^(1/8))/(2-2^(1/2))^(1/2))/a^2+1/ 32*(2+2^(1/2))^(1/2)*arctan(((2-2^(1/2))^(1/2)+2*(1-I*a*x)^(1/8)/(1+I*a*x) ^(1/8))/(2+2^(1/2))^(1/2))/a^2+1/32*(2-2^(1/2))^(1/2)*arctan(((2+2^(1/2))^ (1/2)+2*(1-I*a*x)^(1/8)/(1+I*a*x)^(1/8))/(2-2^(1/2))^(1/2))/a^2-1/32*(2-2^ (1/2))^(1/2)*arctanh((2-2^(1/2))^(1/2)*(1-I*a*x)^(1/8)/(1+I*a*x)^(1/8)/(1+ (1-I*a*x)^(1/4)/(1+I*a*x)^(1/4)))/a^2-1/32*(2+2^(1/2))^(1/2)*arctanh((2+2^ (1/2))^(1/2)*(1-I*a*x)^(1/8)/(1+I*a*x)^(1/8)/(1+(1-I*a*x)^(1/4)/(1+I*a*x)^ (1/4)))/a^2
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.12 \[ \int e^{\frac {1}{4} i \arctan (a x)} x \, dx=\frac {(1-i a x)^{7/8} \left (7 (1+i a x)^{9/8}+2 \sqrt [8]{2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{8},\frac {7}{8},\frac {15}{8},\frac {1}{2} (1-i a x)\right )\right )}{14 a^2} \] Input:
Integrate[E^((I/4)*ArcTan[a*x])*x,x]
Output:
((1 - I*a*x)^(7/8)*(7*(1 + I*a*x)^(9/8) + 2*2^(1/8)*Hypergeometric2F1[-1/8 , 7/8, 15/8, (1 - I*a*x)/2]))/(14*a^2)
Time = 1.27 (sec) , antiderivative size = 697, normalized size of antiderivative = 1.33, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {5585, 90, 60, 73, 854, 828, 1442, 1483, 1142, 25, 1083, 217, 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.
| \(\displaystyle \int x e^{\frac {1}{4} i \arctan (a x)} \, dx\) |
| \(\Big \downarrow \) 5585 |
| \(\displaystyle \int \frac {x \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}dx\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}-\frac {i \int \frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}dx}{8 a}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}-\frac {i \left (\frac {1}{4} \int \frac {1}{\sqrt [8]{1-i a x} (i a x+1)^{7/8}}dx+\frac {i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{a}\right )}{8 a}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}-\frac {i \left (\frac {2 i \int \frac {(1-i a x)^{3/4}}{(i a x+1)^{7/8}}d\sqrt [8]{1-i a x}}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{a}\right )}{8 a}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle \frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}-\frac {i \left (\frac {2 i \int \frac {(1-i a x)^{3/4}}{2-i a x}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{a}\right )}{8 a}\) |
| \(\Big \downarrow \) 828 |
| \(\displaystyle \frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}-\frac {i \left (\frac {2 i \left (\frac {\int \frac {\sqrt {1-i a x}}{\sqrt {1-i a x}-\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {1-i a x}}{\sqrt {1-i a x}+\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}\right )}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{a}\right )}{8 a}\) |
| \(\Big \downarrow \) 1442 |
| \(\displaystyle \frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}-\frac {i \left (\frac {2 i \left (\frac {\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}-\int \frac {1-\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt {1-i a x}-\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}-\int \frac {\sqrt {2} \sqrt [4]{1-i a x}+1}{\sqrt {1-i a x}+\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}\right )}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{a}\right )}{8 a}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}-\frac {i \left (\frac {2 i \left (\frac {-\frac {\int \frac {\sqrt {2+\sqrt {2}}-\frac {\left (1+\sqrt {2}\right ) \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt [4]{1-i a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2+\sqrt {2}}}-\frac {\int \frac {\frac {\left (1+\sqrt {2}\right ) \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt [4]{1-i a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{2 \sqrt {2}}-\frac {-\frac {\int \frac {\sqrt {2-\sqrt {2}}-\frac {\left (1-\sqrt {2}\right ) \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt [4]{1-i a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2-\sqrt {2}}}-\frac {\int \frac {\frac {\left (1-\sqrt {2}\right ) \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt [4]{1-i a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{2 \sqrt {2}}\right )}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{a}\right )}{8 a}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {(1-i a x)^{7/8} (i a x+1)^{9/8}}{2 a^2}-\frac {i \left (\frac {2 i \left (\frac {-\frac {-\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{\sqrt [4]{1-i a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}-\frac {1}{2} \left (1+\sqrt {2}\right ) \int -\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt [4]{1-i a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt [4]{1-i a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}-\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{\sqrt [4]{1-i a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}-\frac {-\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{\sqrt [4]{1-i a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}-\frac {1}{2} \left (1-\sqrt {2}\right ) \int -\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt [4]{1-i a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2-\sqrt {2}}}-\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{\sqrt [4]{1-i a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt [4]{1-i a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}\right )}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{i a x+1}}{a}\right )}{8 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(1-i a x)^{7/8} (i a x+1)^{9/8}}{2 a^2}-\frac {i \left (\frac {2 i \left (\frac {-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt [4]{1-i a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}-\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{\sqrt [4]{1-i a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt [4]{1-i a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}-\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{\sqrt [4]{1-i a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}-\frac {-\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{\sqrt [4]{1-i a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt [4]{1-i a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2-\sqrt {2}}}-\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{\sqrt [4]{1-i a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt [4]{1-i a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}\right )}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{i a x+1}}{a}\right )}{8 a}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {(1-i a x)^{7/8} (i a x+1)^{9/8}}{2 a^2}-\frac {i \left (\frac {2 i \left (\frac {-\frac {\sqrt {2-\sqrt {2}} \int \frac {1}{-\sqrt [4]{1-i a x}+\sqrt {2}-2}d\left (\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}-\sqrt {2+\sqrt {2}}\right )+\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt [4]{1-i a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2+\sqrt {2}}}-\frac {\sqrt {2-\sqrt {2}} \int \frac {1}{-\sqrt [4]{1-i a x}+\sqrt {2}-2}d\left (\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2+\sqrt {2}}\right )+\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt [4]{1-i a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}-\frac {-\frac {\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt [4]{1-i a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}-\sqrt {2+\sqrt {2}} \int \frac {1}{-\sqrt [4]{1-i a x}-\sqrt {2}-2}d\left (\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}-\sqrt {2-\sqrt {2}}\right )}{2 \sqrt {2-\sqrt {2}}}-\frac {\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt [4]{1-i a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}-\sqrt {2+\sqrt {2}} \int \frac {1}{-\sqrt [4]{1-i a x}-\sqrt {2}-2}d\left (\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2-\sqrt {2}}\right )}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}\right )}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{i a x+1}}{a}\right )}{8 a}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(1-i a x)^{7/8} (i a x+1)^{9/8}}{2 a^2}-\frac {i \left (\frac {2 i \left (\frac {-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt [4]{1-i a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}-\arctan \left (\frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}-\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt [4]{1-i a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}-\arctan \left (\frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}-\frac {-\frac {\arctan \left (\frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}-\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt [4]{1-i a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2-\sqrt {2}}}-\frac {\arctan \left (\frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt [4]{1-i a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}\right )}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{i a x+1}}{a}\right )}{8 a}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}-\frac {i \left (\frac {2 i \left (\frac {-\frac {-\arctan \left (\frac {-\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{2} \left (1+\sqrt {2}\right ) \log \left (\sqrt [4]{1-i a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \log \left (\sqrt [4]{1-i a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )-\arctan \left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{2 \sqrt {2}}-\frac {-\frac {\arctan \left (\frac {-\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{2} \left (1-\sqrt {2}\right ) \log \left (\sqrt [4]{1-i a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{2 \sqrt {2-\sqrt {2}}}-\frac {\arctan \left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (1-\sqrt {2}\right ) \log \left (\sqrt [4]{1-i a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{2 \sqrt {2}}\right )}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{a}\right )}{8 a}\) |
Input:
Int[E^((I/4)*ArcTan[a*x])*x,x]
Output:
((1 - I*a*x)^(7/8)*(1 + I*a*x)^(9/8))/(2*a^2) - ((I/8)*((I*(1 - I*a*x)^(7/ 8)*(1 + I*a*x)^(1/8))/a + ((2*I)*(-1/2*((1 - I*a*x)^(1/8)/(1 + I*a*x)^(1/8 ) - (ArcTan[(-Sqrt[2 - Sqrt[2]] + (2*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8)) /Sqrt[2 + Sqrt[2]]] - ((1 - Sqrt[2])*Log[1 + (1 - I*a*x)^(1/4) - (Sqrt[2 - Sqrt[2]]*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8)])/2)/(2*Sqrt[2 - Sqrt[2]]) - (ArcTan[(Sqrt[2 - Sqrt[2]] + (2*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8))/Sq rt[2 + Sqrt[2]]] + ((1 - Sqrt[2])*Log[1 + (1 - I*a*x)^(1/4) + (Sqrt[2 - Sq rt[2]]*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8)])/2)/(2*Sqrt[2 - Sqrt[2]]))/Sq rt[2] + ((1 - I*a*x)^(1/8)/(1 + I*a*x)^(1/8) - (-ArcTan[(-Sqrt[2 + Sqrt[2] ] + (2*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8))/Sqrt[2 - Sqrt[2]]] - ((1 + Sq rt[2])*Log[1 + (1 - I*a*x)^(1/4) - (Sqrt[2 + Sqrt[2]]*(1 - I*a*x)^(1/8))/( 1 + I*a*x)^(1/8)])/2)/(2*Sqrt[2 + Sqrt[2]]) - (-ArcTan[(Sqrt[2 + Sqrt[2]] + (2*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8))/Sqrt[2 - Sqrt[2]]] + ((1 + Sqrt [2])*Log[1 + (1 - I*a*x)^(1/4) + (Sqrt[2 + Sqrt[2]]*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8)])/2)/(2*Sqrt[2 + Sqrt[2]]))/(2*Sqrt[2])))/a))/a
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
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_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[R t[a/b, 4]], s = Denominator[Rt[a/b, 4]]}, Simp[s^3/(2*Sqrt[2]*b*r) Int[x^ (m - n/4)/(r^2 - Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x], x] - Simp[s^3/(2*S qrt[2]*b*r) Int[x^(m - n/4)/(r^2 + Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x] , x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && GtQ[a/b, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n )^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 2^(-1)] && IntegersQ[m, p + (m + 1)/n]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{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_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d^3*(d*x)^(m - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), x] - Simp[d^4/(c*(m + 4*p + 1)) Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x ] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2* p] && (IntegerQ[p] || IntegerQ[m])
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*((1 - I*a *x)^(I*(n/2))/(1 + I*a*x)^(I*(n/2))), x] /; FreeQ[{a, m, n}, x] && !Intege rQ[(I*n - 1)/2]
\[\int {\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}^{\frac {1}{4}} x d x\]
Input:
int(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/4)*x,x)
Output:
int(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/4)*x,x)
Time = 0.14 (sec) , antiderivative size = 428, normalized size of antiderivative = 0.82 \[ \int e^{\frac {1}{4} i \arctan (a x)} x \, dx=-\frac {8 \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (32 \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + 8 i \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (32 i \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - 8 i \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (-32 i \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - 8 \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (-32 \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + 8 \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (32 \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + 8 i \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (32 i \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - 8 i \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (-32 i \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - 8 \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (-32 \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - {\left (4 \, a^{2} x^{2} - i \, a x + 5\right )} \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}}{8 \, a^{2}} \] Input:
integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/4)*x,x, algorithm="fricas")
Output:
-1/8*(8*a^2*(1/1048576*I/a^8)^(1/4)*log(32*a^2*(1/1048576*I/a^8)^(1/4) + ( I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) + 8*I*a^2*(1/1048576*I/a^8)^(1/4)*lo g(32*I*a^2*(1/1048576*I/a^8)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4) ) - 8*I*a^2*(1/1048576*I/a^8)^(1/4)*log(-32*I*a^2*(1/1048576*I/a^8)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) - 8*a^2*(1/1048576*I/a^8)^(1/4)*l og(-32*a^2*(1/1048576*I/a^8)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4) ) + 8*a^2*(-1/1048576*I/a^8)^(1/4)*log(32*a^2*(-1/1048576*I/a^8)^(1/4) + ( I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) + 8*I*a^2*(-1/1048576*I/a^8)^(1/4)*l og(32*I*a^2*(-1/1048576*I/a^8)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/ 4)) - 8*I*a^2*(-1/1048576*I/a^8)^(1/4)*log(-32*I*a^2*(-1/1048576*I/a^8)^(1 /4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) - 8*a^2*(-1/1048576*I/a^8)^(1 /4)*log(-32*a^2*(-1/1048576*I/a^8)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I)) ^(1/4)) - (4*a^2*x^2 - I*a*x + 5)*(I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4))/a ^2
\[ \int e^{\frac {1}{4} i \arctan (a x)} x \, dx=\int x \sqrt [4]{\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}\, dx \] Input:
integrate(((1+I*a*x)/(a**2*x**2+1)**(1/2))**(1/4)*x,x)
Output:
Integral(x*(I*(a*x - I)/sqrt(a**2*x**2 + 1))**(1/4), x)
\[ \int e^{\frac {1}{4} i \arctan (a x)} x \, dx=\int { x \left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {1}{4}} \,d x } \] Input:
integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/4)*x,x, algorithm="maxima")
Output:
integrate(x*((I*a*x + 1)/sqrt(a^2*x^2 + 1))^(1/4), x)
Exception generated. \[ \int e^{\frac {1}{4} i \arctan (a x)} x \, dx=\text {Exception raised: TypeError} \] Input:
integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/4)*x,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Warning, need to choose a branch fo r the root of a polynomial with parameters. This might be wrong.The choice was done
Timed out. \[ \int e^{\frac {1}{4} i \arctan (a x)} x \, dx=\int x\,{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{1/4} \,d x \] Input:
int(x*((a*x*1i + 1)/(a^2*x^2 + 1)^(1/2))^(1/4),x)
Output:
int(x*((a*x*1i + 1)/(a^2*x^2 + 1)^(1/2))^(1/4), x)
Time = 0.17 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.06 \[ \int e^{\frac {1}{4} i \arctan (a x)} x \, dx=\frac {4 \left (a i x +1\right )^{\frac {1}{8}} \left (a^{2} x^{2}+1\right )}{9 \left (-a i x +1\right )^{\frac {1}{8}} a^{2}} \] Input:
int(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/4)*x,x)
Output:
(4*(a*i*x + 1)**(1/8)*( - a*i*x + 1)**(1/8)*(a**2*x**2 + 1))/(9*( - a*i*x + 1)**(1/4)*a**2)