Integrand size = 16, antiderivative size = 630 \[ \int \frac {e^{\frac {1}{4} i \arctan (a x)}}{x} \, dx=-2 \arctan \left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )+\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )+\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )-\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )-\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )+\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-2 \text {arctanh}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )+\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x} \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}\right )+\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x} \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x} \left (1+\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )}\right ) \] Output:
-2*arctan((1+I*a*x)^(1/8)/(1-I*a*x)^(1/8))+(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))+(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))-(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))-(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))+2^(1/2)*a rctan(1-2^(1/2)*(1+I*a*x)^(1/8)/(1-I*a*x)^(1/8))-2^(1/2)*arctan(1+2^(1/2)* (1+I*a*x)^(1/8)/(1-I*a*x)^(1/8))-2*arctanh((1+I*a*x)^(1/8)/(1-I*a*x)^(1/8) )+(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)))+(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)))-2^(1/2)*arctanh(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)))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.15 \[ \int \frac {e^{\frac {1}{4} i \arctan (a x)}}{x} \, dx=-\frac {4 (1-i a x)^{7/8} \left (\sqrt [8]{2} (1+i a x)^{7/8} \operatorname {Hypergeometric2F1}\left (\frac {7}{8},\frac {7}{8},\frac {15}{8},\frac {1}{2} (1-i a x)\right )+2 \operatorname {Hypergeometric2F1}\left (\frac {7}{8},1,\frac {15}{8},\frac {i+a x}{i-a x}\right )\right )}{7 (1+i a x)^{7/8}} \] Input:
Integrate[E^((I/4)*ArcTan[a*x])/x,x]
Output:
(-4*(1 - I*a*x)^(7/8)*(2^(1/8)*(1 + I*a*x)^(7/8)*Hypergeometric2F1[7/8, 7/ 8, 15/8, (1 - I*a*x)/2] + 2*Hypergeometric2F1[7/8, 1, 15/8, (I + a*x)/(I - a*x)]))/(7*(1 + I*a*x)^(7/8))
Time = 1.78 (sec) , antiderivative size = 918, normalized size of antiderivative = 1.46, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.562, Rules used = {5585, 140, 73, 104, 758, 755, 756, 216, 219, 854, 828, 1442, 1476, 1082, 217, 1479, 25, 27, 1103, 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.
| \(\displaystyle \int \frac {e^{\frac {1}{4} i \arctan (a x)}}{x} \, dx\) |
| \(\Big \downarrow \) 5585 |
| \(\displaystyle \int \frac {\sqrt [8]{1+i a x}}{x \sqrt [8]{1-i a x}}dx\) |
| \(\Big \downarrow \) 140 |
| \(\displaystyle i a \int \frac {1}{\sqrt [8]{1-i a x} (i a x+1)^{7/8}}dx+\int \frac {1}{x \sqrt [8]{1-i a x} (i a x+1)^{7/8}}dx\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \int \frac {1}{x \sqrt [8]{1-i a x} (i a x+1)^{7/8}}dx-8 \int \frac {(1-i a x)^{3/4}}{(i a x+1)^{7/8}}d\sqrt [8]{1-i a x}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle 8 \int \frac {1}{\frac {i a x+1}{1-i a x}-1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}-8 \int \frac {(1-i a x)^{3/4}}{(i a x+1)^{7/8}}d\sqrt [8]{1-i a x}\) |
| \(\Big \downarrow \) 758 |
| \(\displaystyle 8 \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}-\frac {1}{2} \int \frac {1}{\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )-8 \int \frac {(1-i a x)^{3/4}}{(i a x+1)^{7/8}}d\sqrt [8]{1-i a x}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle 8 \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}}{\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}-\frac {1}{2} \int \frac {\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+1}{\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )-8 \int \frac {(1-i a x)^{3/4}}{(i a x+1)^{7/8}}d\sqrt [8]{1-i a x}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle 8 \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}-\frac {1}{2} \int \frac {1}{\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )+\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}}{\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}-\frac {1}{2} \int \frac {\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+1}{\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )\right )-8 \int \frac {(1-i a x)^{3/4}}{(i a x+1)^{7/8}}d\sqrt [8]{1-i a x}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle 8 \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )\right )+\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}}{\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}-\frac {1}{2} \int \frac {\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+1}{\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )\right )-8 \int \frac {(1-i a x)^{3/4}}{(i a x+1)^{7/8}}d\sqrt [8]{1-i a x}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 8 \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}}{\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}-\frac {1}{2} \int \frac {\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+1}{\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )\right )\right )-8 \int \frac {(1-i a x)^{3/4}}{(i a x+1)^{7/8}}d\sqrt [8]{1-i a x}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle 8 \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}}{\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}-\frac {1}{2} \int \frac {\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+1}{\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )\right )\right )-8 \int \frac {(1-i a x)^{3/4}}{2-i a x}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}\) |
| \(\Big \downarrow \) 828 |
| \(\displaystyle 8 \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}}{\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}-\frac {1}{2} \int \frac {\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+1}{\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )\right )\right )-8 \left (\frac {\int \frac {\sqrt {1-i a x}}{\sqrt {1-i a x}-\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {1-i a x}}{\sqrt {1-i a x}+\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 1442 |
| \(\displaystyle 8 \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}}{\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}-\frac {1}{2} \int \frac {\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+1}{\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )\right )\right )-8 \left (\frac {\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}-\int \frac {1-\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt {1-i a x}-\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}-\int \frac {\sqrt {2} \sqrt [4]{1-i a x}+1}{\sqrt {1-i a x}+\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 8 \left (\frac {1}{2} \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}-\frac {1}{2} \int \frac {1}{\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \int \frac {1-\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}}{\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )\right )\right )-8 \left (\frac {\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}-\int \frac {1-\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt {1-i a x}-\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}-\int \frac {\sqrt {2} \sqrt [4]{1-i a x}+1}{\sqrt {1-i a x}+\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 8 \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-1}d\left (\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-1}d\left (1-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}}{\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )\right )\right )-8 \left (\frac {\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}-\int \frac {1-\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt {1-i a x}-\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}-\int \frac {\sqrt {2} \sqrt [4]{1-i a x}+1}{\sqrt {1-i a x}+\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 8 \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}}{\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )\right )\right )-8 \left (\frac {\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}-\int \frac {1-\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt {1-i a x}-\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}-\int \frac {\sqrt {2} \sqrt [4]{1-i a x}+1}{\sqrt {1-i a x}+\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 8 \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}}{\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )}{\sqrt {2}}\right )\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )\right )\right )-8 \left (\frac {\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}-\int \frac {1-\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt {1-i a x}-\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}-\int \frac {\sqrt {2} \sqrt [4]{1-i a x}+1}{\sqrt {1-i a x}+\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 8 \left (\frac {1}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}}{\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )}{\sqrt {2}}\right )\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )\right )\right )-8 \left (\frac {\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}-\int \frac {1-\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt {1-i a x}-\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}-\int \frac {\sqrt {2} \sqrt [4]{1-i a x}+1}{\sqrt {1-i a x}+\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 8 \left (\frac {1}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}}{\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1}{\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1}d\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )}{\sqrt {2}}\right )\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )\right )\right )-8 \left (\frac {\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}-\int \frac {1-\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt {1-i a x}-\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}-\int \frac {\sqrt {2} \sqrt [4]{1-i a x}+1}{\sqrt {1-i a x}+\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 8 \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+1\right )}{2 \sqrt {2}}\right )\right )\right )-8 \left (\frac {\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}-\int \frac {1-\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt {1-i a x}-\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}-\int \frac {\sqrt {2} \sqrt [4]{1-i a x}+1}{\sqrt {1-i a x}+\sqrt {2} \sqrt [4]{1-i a x}+1}d\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle 8 \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{2 \sqrt {2}}\right )\right )\right )-8 \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]{i a x+1}}}{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]{i a x+1}}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle 8 \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{2 \sqrt {2}}\right )\right )\right )-8 \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 )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 8 \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{2 \sqrt {2}}\right )\right )\right )-8 \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 )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle 8 \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{2 \sqrt {2}}\right )\right )\right )-8 \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 )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 8 \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{2 \sqrt {2}}\right )\right )\right )-8 \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 )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 8 \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{2 \sqrt {2}}\right )\right )\right )-8 \left (\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 ) \log \left (\sqrt [4]{1-i a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+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]{i a x+1}}+1\right )-\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 ) \log \left (\sqrt [4]{1-i a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1\right )}{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 ) \log \left (\sqrt [4]{1-i a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1\right )}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{2 \sqrt {2}}\right )\) |
Input:
Int[E^((I/4)*ArcTan[a*x])/x,x]
Output:
-8*(-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 - 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]]))/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*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])) + 8*((-1/2*ArcTan[(1 + I*a*x)^(1/8)/(1 - I*a*x)^ (1/8)] - ArcTanh[(1 + I*a*x)^(1/8)/(1 - I*a*x)^(1/8)]/2)/2 + ((ArcTan[1 - (Sqrt[2]*(1 + I*a*x)^(1/8))/(1 - I*a*x)^(1/8)]/Sqrt[2] - ArcTan[1 + (Sqrt[ 2]*(1 + I*a*x)^(1/8))/(1 - I*a*x)^(1/8)]/Sqrt[2])/2 + (Log[1 - (Sqrt[2]*(1 + I*a*x)^(1/8))/(1 - I*a*x)^(1/8) + (1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]/ (2*Sqrt[2]) - Log[1 + (Sqrt[2]*(1 + I*a*x)^(1/8))/(1 - I*a*x)^(1/8) + (1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]/(2*Sqrt[2]))/2)/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] :> 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*d^(m + n)*f^p Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] , x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x )*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 0] || SumSimplerQ[m, -1] || !(GtQ[n, 0] || SumSimplerQ[n, -1]))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) 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_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b , 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^(n/2)), x], x] + Simp[r/(2*a) Int[1/(r + s*x^(n/2)), x], x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 1] && !GtQ[a/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] :> 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[((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_) + (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[((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]
\[\int \frac {{\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}^{\frac {1}{4}}}{x}d x\]
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 = 0.08 (sec) , antiderivative size = 509, normalized size of antiderivative = 0.81 \[ \int \frac {e^{\frac {1}{4} i \arctan (a x)}}{x} \, dx =\text {Too large to display} \] Input:
integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/4)/x,x, algorithm="fricas")
Output:
-1/2*sqrt(4*I)*log(1/2*sqrt(4*I) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) + 1/2*sqrt(4*I)*log(-1/2*sqrt(4*I) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4) ) - 1/2*sqrt(-4*I)*log(1/2*sqrt(-4*I) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1 /4)) + 1/2*sqrt(-4*I)*log(-1/2*sqrt(-4*I) + (I*sqrt(a^2*x^2 + 1)/(a*x + I) )^(1/4)) + I^(1/4)*log(I^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) + I*I^(1/4)*log(I*I^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) - I*I^(1/ 4)*log(-I*I^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) - I^(1/4)*log(- I^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) + (-I)^(1/4)*log((-I)^(1/ 4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) + I*(-I)^(1/4)*log(I*(-I)^(1/4 ) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) - I*(-I)^(1/4)*log(-I*(-I)^(1/4 ) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) - (-I)^(1/4)*log(-(-I)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)) - log((I*sqrt(a^2*x^2 + 1)/(a*x + I ))^(1/4) + 1) - I*log((I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4) + I) + I*log(( I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4) - I) + log((I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4) - 1)
\[ \int \frac {e^{\frac {1}{4} i \arctan (a x)}}{x} \, dx=\int \frac {\sqrt [4]{\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}}{x}\, dx \] Input:
integrate(((1+I*a*x)/(a**2*x**2+1)**(1/2))**(1/4)/x,x)
Output:
Integral((I*(a*x - I)/sqrt(a**2*x**2 + 1))**(1/4)/x, x)
\[ \int \frac {e^{\frac {1}{4} i \arctan (a x)}}{x} \, dx=\int { \frac {\left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {1}{4}}}{x} \,d x } \] Input:
integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/4)/x,x, algorithm="maxima")
Output:
integrate(((I*a*x + 1)/sqrt(a^2*x^2 + 1))^(1/4)/x, x)
Exception generated. \[ \int \frac {e^{\frac {1}{4} i \arctan (a x)}}{x} \, dx=\text {Exception raised: TypeError} \] 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. \[ \int \frac {e^{\frac {1}{4} i \arctan (a x)}}{x} \, dx=\int \frac {{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{1/4}}{x} \,d x \] Input:
int(((a*x*1i + 1)/(a^2*x^2 + 1)^(1/2))^(1/4)/x,x)
Output:
int(((a*x*1i + 1)/(a^2*x^2 + 1)^(1/2))^(1/4)/x, x)
\[ \int \frac {e^{\frac {1}{4} i \arctan (a x)}}{x} \, dx=\int \frac {\left (a i x +1\right )^{\frac {1}{4}}}{\left (a^{2} x^{2}+1\right )^{\frac {1}{8}} x}d x \] Input:
int(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/4)/x,x)
Output:
int((a*i*x + 1)**(1/4)/((a**2*x**2 + 1)**(1/8)*x),x)