Integrand size = 16, antiderivative size = 571 \[ \int e^{\frac {1}{4} i \arctan (a x)} x^2 \, dx=-\frac {11 i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{32 a^3}-\frac {i (1-i a x)^{7/8} (1+i a x)^{9/8}}{24 a^3}+\frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}+\frac {11 i \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{128 a^3}+\frac {11 i \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{128 a^3}-\frac {11 i \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{128 a^3}-\frac {11 i \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{128 a^3}+\frac {11 i \sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x} \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}\right )}{128 a^3}+\frac {11 i \sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x} \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}\right )}{128 a^3} \] Output:
-11/32*I*(1-I*a*x)^(7/8)*(1+I*a*x)^(1/8)/a^3-1/24*I*(1-I*a*x)^(7/8)*(1+I*a *x)^(9/8)/a^3+1/3*x*(1-I*a*x)^(7/8)*(1+I*a*x)^(9/8)/a^2+11/128*I*(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^3+11/128*I*(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^3-11/128*I*(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^3-11/128*I*(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^3+11/128*I*(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^3+11/128*I*(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^3
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.15 \[ \int e^{\frac {1}{4} i \arctan (a x)} x^2 \, dx=\frac {(1-i a x)^{7/8} \left (7 \sqrt [8]{1+i a x} \left (-i+9 a x+8 i a^2 x^2\right )-66 i \sqrt [8]{2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{8},\frac {7}{8},\frac {15}{8},\frac {1}{2} (1-i a x)\right )\right )}{168 a^3} \] Input:
Integrate[E^((I/4)*ArcTan[a*x])*x^2,x]
Output:
((1 - I*a*x)^(7/8)*(7*(1 + I*a*x)^(1/8)*(-I + 9*a*x + (8*I)*a^2*x^2) - (66 *I)*2^(1/8)*Hypergeometric2F1[-1/8, 7/8, 15/8, (1 - I*a*x)/2]))/(168*a^3)
Time = 1.40 (sec) , antiderivative size = 734, normalized size of antiderivative = 1.29, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.938, Rules used = {5585, 101, 27, 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.
| \(\displaystyle \int x^2 e^{\frac {1}{4} i \arctan (a x)} \, dx\) |
| \(\Big \downarrow \) 5585 |
| \(\displaystyle \int \frac {x^2 \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}dx\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {\int -\frac {\sqrt [8]{i a x+1} (i a x+4)}{4 \sqrt [8]{1-i a x}}dx}{3 a^2}+\frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}-\frac {\int \frac {\sqrt [8]{i a x+1} (i a x+4)}{\sqrt [8]{1-i a x}}dx}{12 a^2}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}-\frac {\frac {33}{8} \int \frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}dx+\frac {i (1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a}}{12 a^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}-\frac {\frac {33}{8} \left (\frac {1}{4} \int \frac {1}{\sqrt [8]{1-i a x} (i a x+1)^{7/8}}dx+\frac {i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{a}\right )+\frac {i (1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a}}{12 a^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}-\frac {\frac {33}{8} \left (\frac {2 i \int \frac {(1-i a x)^{3/4}}{(i a x+1)^{7/8}}d\sqrt [8]{1-i a x}}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{a}\right )+\frac {i (1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a}}{12 a^2}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle \frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}-\frac {\frac {33}{8} \left (\frac {2 i \int \frac {(1-i a x)^{3/4}}{2-i a x}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{a}\right )+\frac {i (1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a}}{12 a^2}\) |
| \(\Big \downarrow \) 828 |
| \(\displaystyle \frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}-\frac {\frac {33}{8} \left (\frac {2 i \left (\frac {\int \frac {\sqrt {1-i a x}}{\sqrt {1-i a x}-\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {1-i a x}}{\sqrt {1-i a x}+\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}\right )}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{a}\right )+\frac {i (1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a}}{12 a^2}\) |
| \(\Big \downarrow \) 1442 |
| \(\displaystyle \frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}-\frac {\frac {33}{8} \left (\frac {2 i \left (\frac {\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}-\int \frac {1-\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt {1-i a x}-\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}-\int \frac {\sqrt {2} \sqrt [4]{1-i a x}+1}{\sqrt {1-i a x}+\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}\right )}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{a}\right )+\frac {i (1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a}}{12 a^2}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}-\frac {\frac {33}{8} \left (\frac {2 i \left (\frac {-\frac {\int \frac {\sqrt {2+\sqrt {2}}-\frac {\left (1+\sqrt {2}\right ) \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt [4]{1-i a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2+\sqrt {2}}}-\frac {\int \frac {\frac {\left (1+\sqrt {2}\right ) \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt [4]{1-i a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{2 \sqrt {2}}-\frac {-\frac {\int \frac {\sqrt {2-\sqrt {2}}-\frac {\left (1-\sqrt {2}\right ) \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt [4]{1-i a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2-\sqrt {2}}}-\frac {\int \frac {\frac {\left (1-\sqrt {2}\right ) \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt [4]{1-i a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{2 \sqrt {2}}\right )}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{a}\right )+\frac {i (1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a}}{12 a^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {x (1-i a x)^{7/8} (i a x+1)^{9/8}}{3 a^2}-\frac {\frac {i (1-i a x)^{7/8} (i a x+1)^{9/8}}{2 a}+\frac {33}{8} \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 )}{12 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x (1-i a x)^{7/8} (i a x+1)^{9/8}}{3 a^2}-\frac {\frac {i (1-i a x)^{7/8} (i a x+1)^{9/8}}{2 a}+\frac {33}{8} \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 )}{12 a^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {x (1-i a x)^{7/8} (i a x+1)^{9/8}}{3 a^2}-\frac {\frac {i (1-i a x)^{7/8} (i a x+1)^{9/8}}{2 a}+\frac {33}{8} \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 )}{12 a^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {x (1-i a x)^{7/8} (i a x+1)^{9/8}}{3 a^2}-\frac {\frac {i (1-i a x)^{7/8} (i a x+1)^{9/8}}{2 a}+\frac {33}{8} \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 )}{12 a^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}-\frac {\frac {33}{8} \left (\frac {2 i \left (\frac {-\frac {-\arctan \left (\frac {-\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{2} \left (1+\sqrt {2}\right ) \log \left (\sqrt [4]{1-i a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \log \left (\sqrt [4]{1-i a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )-\arctan \left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{2 \sqrt {2}}-\frac {-\frac {\arctan \left (\frac {-\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{2} \left (1-\sqrt {2}\right ) \log \left (\sqrt [4]{1-i a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{2 \sqrt {2-\sqrt {2}}}-\frac {\arctan \left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (1-\sqrt {2}\right ) \log \left (\sqrt [4]{1-i a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{2 \sqrt {2}}\right )}{a}+\frac {i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{a}\right )+\frac {i (1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a}}{12 a^2}\) |
Input:
Int[E^((I/4)*ArcTan[a*x])*x^2,x]
Output:
(x*(1 - I*a*x)^(7/8)*(1 + I*a*x)^(9/8))/(3*a^2) - (((I/2)*(1 - I*a*x)^(7/8 )*(1 + I*a*x)^(9/8))/a + (33*((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 - Sqr t[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))/Sqrt[2 + Sqrt[2]]] + ((1 - S qrt[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 - 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*Sqr t[2 + Sqrt[2]]))/(2*Sqrt[2])))/a))/8)/(12*a^2)
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_))^(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*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 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]
\[\int {\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}^{\frac {1}{4}} x^{2}d x\]
Input:
int(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/4)*x^2,x)
Output:
int(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/4)*x^2,x)
Time = 0.15 (sec) , antiderivative size = 435, normalized size of antiderivative = 0.76 \[ \int e^{\frac {1}{4} i \arctan (a x)} x^2 \, dx =\text {Too large to display} \] Input:
integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/4)*x^2,x, algorithm="fricas")
Output:
1/96*(96*I*a^3*(14641/268435456*I/a^12)^(1/4)*log(128/11*a^3*(14641/268435 456*I/a^12)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) - 96*a^3*(14641 /268435456*I/a^12)^(1/4)*log(128/11*I*a^3*(14641/268435456*I/a^12)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) + 96*a^3*(14641/268435456*I/a^12)^ (1/4)*log(-128/11*I*a^3*(14641/268435456*I/a^12)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) - 96*I*a^3*(14641/268435456*I/a^12)^(1/4)*log(-128/1 1*a^3*(14641/268435456*I/a^12)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/ 4)) + 96*I*a^3*(-14641/268435456*I/a^12)^(1/4)*log(128/11*a^3*(-14641/2684 35456*I/a^12)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) - 96*a^3*(-14 641/268435456*I/a^12)^(1/4)*log(128/11*I*a^3*(-14641/268435456*I/a^12)^(1/ 4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) + 96*a^3*(-14641/268435456*I/a ^12)^(1/4)*log(-128/11*I*a^3*(-14641/268435456*I/a^12)^(1/4) + (I*sqrt(a^2 *x^2 + 1)/(a*x + I))^(1/4)) - 96*I*a^3*(-14641/268435456*I/a^12)^(1/4)*log (-128/11*a^3*(-14641/268435456*I/a^12)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) + (32*a^3*x^3 - 4*I*a^2*x^2 - a*x - 37*I)*(I*sqrt(a^2*x^2 + 1) /(a*x + I))^(1/4))/a^3
\[ \int e^{\frac {1}{4} i \arctan (a x)} x^2 \, dx=\int x^{2} \sqrt [4]{\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}\, dx \] Input:
integrate(((1+I*a*x)/(a**2*x**2+1)**(1/2))**(1/4)*x**2,x)
Output:
Integral(x**2*(I*(a*x - I)/sqrt(a**2*x**2 + 1))**(1/4), x)
\[ \int e^{\frac {1}{4} i \arctan (a x)} x^2 \, dx=\int { x^{2} \left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {1}{4}} \,d x } \] Input:
integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/4)*x^2,x, algorithm="maxima")
Output:
integrate(x^2*((I*a*x + 1)/sqrt(a^2*x^2 + 1))^(1/4), x)
Exception generated. \[ \int e^{\frac {1}{4} i \arctan (a x)} x^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/4)*x^2,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. \[ \int e^{\frac {1}{4} i \arctan (a x)} x^2 \, dx=\int x^2\,{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{1/4} \,d x \] Input:
int(x^2*((a*x*1i + 1)/(a^2*x^2 + 1)^(1/2))^(1/4),x)
Output:
int(x^2*((a*x*1i + 1)/(a^2*x^2 + 1)^(1/2))^(1/4), x)
\[ \int e^{\frac {1}{4} i \arctan (a x)} x^2 \, dx=\int \frac {\left (a i x +1\right )^{\frac {1}{4}} x^{2}}{\left (a^{2} x^{2}+1\right )^{\frac {1}{8}}}d x \] Input:
int(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/4)*x^2,x)
Output:
int(((a*i*x + 1)**(1/4)*x**2)/(a**2*x**2 + 1)**(1/8),x)