Integrand size = 12, antiderivative size = 674 \[ \int e^{\frac {1}{4} i \arctan (a x)} \, dx=\frac {i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{a}-\frac {i \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{4 a}-\frac {i \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{4 a}+\frac {i \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{4 a}+\frac {i \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{4 a}+\frac {i \sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{8 a}-\frac {i \sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{8 a}+\frac {i \sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{8 a}-\frac {i \sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{8 a} \]
I*(1-I*a*x)^(7/8)*(1+I*a*x)^(1/8)/a-1/4*I*arctan((-2*(1-I*a*x)^(1/8)/(1+I* a*x)^(1/8)+(2+2^(1/2))^(1/2))/(2-2^(1/2))^(1/2))*(2-2^(1/2))^(1/2)/a+1/4*I *arctan((2*(1-I*a*x)^(1/8)/(1+I*a*x)^(1/8)+(2+2^(1/2))^(1/2))/(2-2^(1/2))^ (1/2))*(2-2^(1/2))^(1/2)/a+1/8*I*ln(1+(1-I*a*x)^(1/4)/(1+I*a*x)^(1/4)-(1-I *a*x)^(1/8)*(2-2^(1/2))^(1/2)/(1+I*a*x)^(1/8))*(2-2^(1/2))^(1/2)/a-1/8*I*l n(1+(1-I*a*x)^(1/4)/(1+I*a*x)^(1/4)+(1-I*a*x)^(1/8)*(2-2^(1/2))^(1/2)/(1+I *a*x)^(1/8))*(2-2^(1/2))^(1/2)/a-1/4*I*arctan((-2*(1-I*a*x)^(1/8)/(1+I*a*x )^(1/8)+(2-2^(1/2))^(1/2))/(2+2^(1/2))^(1/2))*(2+2^(1/2))^(1/2)/a+1/4*I*ar ctan((2*(1-I*a*x)^(1/8)/(1+I*a*x)^(1/8)+(2-2^(1/2))^(1/2))/(2+2^(1/2))^(1/ 2))*(2+2^(1/2))^(1/2)/a+1/8*I*ln(1+(1-I*a*x)^(1/4)/(1+I*a*x)^(1/4)-(1-I*a* x)^(1/8)*(2+2^(1/2))^(1/2)/(1+I*a*x)^(1/8))*(2+2^(1/2))^(1/2)/a-1/8*I*ln(1 +(1-I*a*x)^(1/4)/(1+I*a*x)^(1/4)+(1-I*a*x)^(1/8)*(2+2^(1/2))^(1/2)/(1+I*a* x)^(1/8))*(2+2^(1/2))^(1/2)/a
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.06 \[ \int e^{\frac {1}{4} i \arctan (a x)} \, dx=-\frac {16 i e^{\frac {9}{4} i \arctan (a x)} \operatorname {Hypergeometric2F1}\left (\frac {9}{8},2,\frac {17}{8},-e^{2 i \arctan (a x)}\right )}{9 a} \]
(((-16*I)/9)*E^(((9*I)/4)*ArcTan[a*x])*Hypergeometric2F1[9/8, 2, 17/8, -E^ ((2*I)*ArcTan[a*x])])/a
Time = 0.70 (sec) , antiderivative size = 656, normalized size of antiderivative = 0.97, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5584, 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.
| \(\displaystyle \int e^{\frac {1}{4} i \arctan (a x)} \, dx\) |
| \(\Big \downarrow \) 5584 |
| \(\displaystyle \int \frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{4} \int \frac {1}{\sqrt [8]{1-i a x} (i a x+1)^{7/8}}dx+\frac {i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{a}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2 i \int \frac {(1-i a x)^{3/4}}{(i a x+1)^{7/8}}d\sqrt [8]{1-i a x}}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{a}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle \frac {2 i \int \frac {(1-i a x)^{3/4}}{2-i a x}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{a}\) |
| \(\Big \downarrow \) 828 |
| \(\displaystyle \frac {2 i \left (\frac {\int \frac {\sqrt {1-i a x}}{\sqrt {1-i a x}-\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {1-i a x}}{\sqrt {1-i a x}+\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}\right )}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{a}\) |
| \(\Big \downarrow \) 1442 |
| \(\displaystyle \frac {2 i \left (\frac {\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}-\int \frac {1-\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt {1-i a x}-\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}-\int \frac {\sqrt {2} \sqrt [4]{1-i a x}+1}{\sqrt {1-i a x}+\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}\right )}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{a}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {2 i \left (\frac {-\frac {\int \frac {\sqrt {2+\sqrt {2}}-\frac {\left (1+\sqrt {2}\right ) \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt [4]{1-i a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2+\sqrt {2}}}-\frac {\int \frac {\frac {\left (1+\sqrt {2}\right ) \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt [4]{1-i a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{2 \sqrt {2}}-\frac {-\frac {\int \frac {\sqrt {2-\sqrt {2}}-\frac {\left (1-\sqrt {2}\right ) \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt [4]{1-i a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2-\sqrt {2}}}-\frac {\int \frac {\frac {\left (1-\sqrt {2}\right ) \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt [4]{1-i a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{2 \sqrt {2}}\right )}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{a}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \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}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \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}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {2 i \left (\frac {-\frac {\sqrt {2-\sqrt {2}} \int \frac {1}{-\sqrt [4]{1-i a x}+\sqrt {2}-2}d\left (\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}-\sqrt {2+\sqrt {2}}\right )+\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt [4]{1-i a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2+\sqrt {2}}}-\frac {\sqrt {2-\sqrt {2}} \int \frac {1}{-\sqrt [4]{1-i a x}+\sqrt {2}-2}d\left (\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2+\sqrt {2}}\right )+\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt [4]{1-i a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}-\frac {-\frac {\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt [4]{1-i a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}-\sqrt {2+\sqrt {2}} \int \frac {1}{-\sqrt [4]{1-i a x}-\sqrt {2}-2}d\left (\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}-\sqrt {2-\sqrt {2}}\right )}{2 \sqrt {2-\sqrt {2}}}-\frac {\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt [4]{1-i a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}-\sqrt {2+\sqrt {2}} \int \frac {1}{-\sqrt [4]{1-i a x}-\sqrt {2}-2}d\left (\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2-\sqrt {2}}\right )}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}\right )}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{i a x+1}}{a}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 i \left (\frac {-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt [4]{1-i a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}-\arctan \left (\frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}-\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt [4]{1-i a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}-\arctan \left (\frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}-\frac {-\frac {\arctan \left (\frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}-\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt [4]{1-i a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2-\sqrt {2}}}-\frac {\arctan \left (\frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt [4]{1-i a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}\right )}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{i a x+1}}{a}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 i \left (\frac {-\frac {-\arctan \left (\frac {-\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{2} \left (1+\sqrt {2}\right ) \log \left (\sqrt [4]{1-i a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \log \left (\sqrt [4]{1-i a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )-\arctan \left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{2 \sqrt {2}}-\frac {-\frac {\arctan \left (\frac {-\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{2} \left (1-\sqrt {2}\right ) \log \left (\sqrt [4]{1-i a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{2 \sqrt {2-\sqrt {2}}}-\frac {\arctan \left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (1-\sqrt {2}\right ) \log \left (\sqrt [4]{1-i a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{2 \sqrt {2}}\right )}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{a}\) |
(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*S qrt[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]]))/Sqrt[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 - Sq rt[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))/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
3.2.30.3.1 Defintions of rubi rules used
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_)^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_Symbol] :> Int[(1 - I*a*x)^(I*(n/2))/(1 + I*a*x)^(I*(n/2)), x] /; FreeQ[{a, n}, x] && !IntegerQ[(I*n - 1)/2]
\[\int {\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}^{\frac {1}{4}}d x\]
Time = 0.29 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.57 \[ \int e^{\frac {1}{4} i \arctan (a x)} \, dx=\frac {-i \, a \left (\frac {i}{256 \, a^{4}}\right )^{\frac {1}{4}} \log \left (4 \, a \left (\frac {i}{256 \, a^{4}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + a \left (\frac {i}{256 \, a^{4}}\right )^{\frac {1}{4}} \log \left (4 i \, a \left (\frac {i}{256 \, a^{4}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - a \left (\frac {i}{256 \, a^{4}}\right )^{\frac {1}{4}} \log \left (-4 i \, a \left (\frac {i}{256 \, a^{4}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + i \, a \left (\frac {i}{256 \, a^{4}}\right )^{\frac {1}{4}} \log \left (-4 \, a \left (\frac {i}{256 \, a^{4}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - i \, a \left (-\frac {i}{256 \, a^{4}}\right )^{\frac {1}{4}} \log \left (4 \, a \left (-\frac {i}{256 \, a^{4}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + a \left (-\frac {i}{256 \, a^{4}}\right )^{\frac {1}{4}} \log \left (4 i \, a \left (-\frac {i}{256 \, a^{4}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - a \left (-\frac {i}{256 \, a^{4}}\right )^{\frac {1}{4}} \log \left (-4 i \, a \left (-\frac {i}{256 \, a^{4}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + i \, a \left (-\frac {i}{256 \, a^{4}}\right )^{\frac {1}{4}} \log \left (-4 \, a \left (-\frac {i}{256 \, a^{4}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + {\left (a x + i\right )} \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}}{a} \]
(-I*a*(1/256*I/a^4)^(1/4)*log(4*a*(1/256*I/a^4)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) + a*(1/256*I/a^4)^(1/4)*log(4*I*a*(1/256*I/a^4)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) - a*(1/256*I/a^4)^(1/4)*log(-4*I *a*(1/256*I/a^4)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) + I*a*(1/2 56*I/a^4)^(1/4)*log(-4*a*(1/256*I/a^4)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) - I*a*(-1/256*I/a^4)^(1/4)*log(4*a*(-1/256*I/a^4)^(1/4) + (I*s qrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) + a*(-1/256*I/a^4)^(1/4)*log(4*I*a*(-1/ 256*I/a^4)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) - a*(-1/256*I/a^ 4)^(1/4)*log(-4*I*a*(-1/256*I/a^4)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I)) ^(1/4)) + I*a*(-1/256*I/a^4)^(1/4)*log(-4*a*(-1/256*I/a^4)^(1/4) + (I*sqrt (a^2*x^2 + 1)/(a*x + I))^(1/4)) + (a*x + I)*(I*sqrt(a^2*x^2 + 1)/(a*x + I) )^(1/4))/a
\[ \int e^{\frac {1}{4} i \arctan (a x)} \, dx=\int \sqrt [4]{\frac {i a x + 1}{\sqrt {a^{2} x^{2} + 1}}}\, dx \]
\[ \int e^{\frac {1}{4} i \arctan (a x)} \, dx=\int { \left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {1}{4}} \,d x } \]
Exception generated. \[ \int e^{\frac {1}{4} i \arctan (a x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:The choice was done assuming 0=[0,0 ]Warning, replacing 0 by -28, a substitution variable should perhaps be pu rged.Warn
Timed out. \[ \int e^{\frac {1}{4} i \arctan (a x)} \, dx=\int {\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{1/4} \,d x \]