\(\int \frac {e^{-\frac {1}{2} i \arctan (a+b x)}}{x} \, dx\) [243]

Optimal result

Integrand size = 18, antiderivative size = 319 \[ \int \frac {e^{-\frac {1}{2} i \arctan (a+b x)}}{x} \, dx=-\frac {2 \sqrt [4]{i+a} \arctan \left (\frac {\sqrt [4]{i-a} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i+a} \sqrt [4]{1+i (a+b x)}}\right )}{\sqrt [4]{i-a}}-\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )+\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )-\frac {2 \sqrt [4]{i+a} \text {arctanh}\left (\frac {\sqrt [4]{i-a} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i+a} \sqrt [4]{1+i (a+b x)}}\right )}{\sqrt [4]{i-a}}+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)} \left (1+\frac {\sqrt {1-i (a+b x)}}{\sqrt {1+i (a+b x)}}\right )}\right ) \] Output:

-2*(I+a)^(1/4)*arctan((I-a)^(1/4)*(1-I*(b*x+a))^(1/4)/(I+a)^(1/4)/(1+I*(b* 
x+a))^(1/4))/(I-a)^(1/4)-2^(1/2)*arctan(1-2^(1/2)*(1-I*(b*x+a))^(1/4)/(1+I 
*(b*x+a))^(1/4))+2^(1/2)*arctan(1+2^(1/2)*(1-I*(b*x+a))^(1/4)/(1+I*(b*x+a) 
)^(1/4))-2*(I+a)^(1/4)*arctanh((I-a)^(1/4)*(1-I*(b*x+a))^(1/4)/(I+a)^(1/4) 
/(1+I*(b*x+a))^(1/4))/(I-a)^(1/4)+2^(1/2)*arctanh(2^(1/2)*(1-I*(b*x+a))^(1 
/4)/(1+I*(b*x+a))^(1/4)/(1+(1-I*(b*x+a))^(1/2)/(1+I*(b*x+a))^(1/2)))
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.04 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.39 \[ \int \frac {e^{-\frac {1}{2} i \arctan (a+b x)}}{x} \, dx=\frac {2 \sqrt [4]{-i (i+a+b x)} \left (2^{3/4} \sqrt [4]{1+i a+i b x} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{4},\frac {5}{4},-\frac {1}{2} i (i+a+b x)\right )-2 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},\frac {1+a^2-i b x+a b x}{1+a^2+i b x+a b x}\right )\right )}{\sqrt [4]{1+i a+i b x}} \] Input:

Integrate[1/(E^((I/2)*ArcTan[a + b*x])*x),x]
 

Output:

(2*((-I)*(I + a + b*x))^(1/4)*(2^(3/4)*(1 + I*a + I*b*x)^(1/4)*Hypergeomet 
ric2F1[1/4, 1/4, 5/4, (-1/2*I)*(I + a + b*x)] - 2*Hypergeometric2F1[1/4, 1 
, 5/4, (1 + a^2 - I*b*x + a*b*x)/(1 + a^2 + I*b*x + a*b*x)]))/(1 + I*a + I 
*b*x)^(1/4)
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.37, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {5617, 981, 755, 756, 218, 221, 1476, 1082, 217, 1479, 25, 27, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {e^{-\frac {1}{2} i \arctan (a+b x)}}{x} \, dx\)

\(\Big \downarrow \) 5617

\(\displaystyle -8 \int \frac {1-i (a+b x)}{(i (a+b x)+1) \left (\frac {1-i (a+b x)}{i (a+b x)+1}+1\right ) \left (-i a-\frac {(i a+1) (1-i (a+b x))}{i (a+b x)+1}+1\right )}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}\)

\(\Big \downarrow \) 981

\(\displaystyle -8 \left (\frac {1}{2} (1-i a) \int \frac {1}{-i a-\frac {(i a+1) (1-i (a+b x))}{i (a+b x)+1}+1}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}-\frac {1}{2} \int \frac {1}{\frac {1-i (a+b x)}{i (a+b x)+1}+1}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}\right )\)

\(\Big \downarrow \) 755

\(\displaystyle -8 \left (\frac {1}{2} (1-i a) \int \frac {1}{-i a-\frac {(i a+1) (1-i (a+b x))}{i (a+b x)+1}+1}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}+\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {\sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}}{\frac {1-i (a+b x)}{i (a+b x)+1}+1}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}-\frac {1}{2} \int \frac {\frac {\sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}+1}{\frac {1-i (a+b x)}{i (a+b x)+1}+1}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}\right )\right )\)

\(\Big \downarrow \) 756

\(\displaystyle -8 \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {\sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}}{\frac {1-i (a+b x)}{i (a+b x)+1}+1}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}-\frac {1}{2} \int \frac {\frac {\sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}+1}{\frac {1-i (a+b x)}{i (a+b x)+1}+1}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}\right )+\frac {1}{2} (1-i a) \left (\frac {i \int \frac {1}{\sqrt {a+i}-\frac {\sqrt {i-a} \sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}}{2 \sqrt {a+i}}+\frac {i \int \frac {1}{\sqrt {a+i}+\frac {\sqrt {i-a} \sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}}{2 \sqrt {a+i}}\right )\right )\)

\(\Big \downarrow \) 218

\(\displaystyle -8 \left (\frac {1}{2} (1-i a) \left (\frac {i \int \frac {1}{\sqrt {a+i}-\frac {\sqrt {i-a} \sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}}{2 \sqrt {a+i}}+\frac {i \arctan \left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{2 \sqrt [4]{-a+i} (a+i)^{3/4}}\right )+\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {\sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}}{\frac {1-i (a+b x)}{i (a+b x)+1}+1}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}-\frac {1}{2} \int \frac {\frac {\sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}+1}{\frac {1-i (a+b x)}{i (a+b x)+1}+1}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}\right )\right )\)

\(\Big \downarrow \) 221

\(\displaystyle -8 \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {\sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}}{\frac {1-i (a+b x)}{i (a+b x)+1}+1}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}-\frac {1}{2} \int \frac {\frac {\sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}+1}{\frac {1-i (a+b x)}{i (a+b x)+1}+1}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}\right )+\frac {1}{2} (1-i a) \left (\frac {i \arctan \left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{2 \sqrt [4]{-a+i} (a+i)^{3/4}}+\frac {i \text {arctanh}\left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{2 \sqrt [4]{-a+i} (a+i)^{3/4}}\right )\right )\)

\(\Big \downarrow \) 1476

\(\displaystyle -8 \left (\frac {1}{2} \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{\frac {\sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}-\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}+1}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}-\frac {1}{2} \int \frac {1}{\frac {\sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}+\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}+1}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}\right )-\frac {1}{2} \int \frac {1-\frac {\sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}}{\frac {1-i (a+b x)}{i (a+b x)+1}+1}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}\right )+\frac {1}{2} (1-i a) \left (\frac {i \arctan \left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{2 \sqrt [4]{-a+i} (a+i)^{3/4}}+\frac {i \text {arctanh}\left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{2 \sqrt [4]{-a+i} (a+i)^{3/4}}\right )\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle -8 \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\frac {\sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}-1}d\left (\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}+1\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\frac {\sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\frac {\sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}}{\frac {1-i (a+b x)}{i (a+b x)+1}+1}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}\right )+\frac {1}{2} (1-i a) \left (\frac {i \arctan \left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{2 \sqrt [4]{-a+i} (a+i)^{3/4}}+\frac {i \text {arctanh}\left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{2 \sqrt [4]{-a+i} (a+i)^{3/4}}\right )\right )\)

\(\Big \downarrow \) 217

\(\displaystyle -8 \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )}{\sqrt {2}}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\frac {\sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}}{\frac {1-i (a+b x)}{i (a+b x)+1}+1}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}\right )+\frac {1}{2} (1-i a) \left (\frac {i \arctan \left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{2 \sqrt [4]{-a+i} (a+i)^{3/4}}+\frac {i \text {arctanh}\left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{2 \sqrt [4]{-a+i} (a+i)^{3/4}}\right )\right )\)

\(\Big \downarrow \) 1479

\(\displaystyle -8 \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}}{\frac {\sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}-\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}+1}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}+1\right )}{\frac {\sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}+\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}+1}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )}{\sqrt {2}}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )}{\sqrt {2}}\right )\right )+\frac {1}{2} (1-i a) \left (\frac {i \arctan \left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{2 \sqrt [4]{-a+i} (a+i)^{3/4}}+\frac {i \text {arctanh}\left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{2 \sqrt [4]{-a+i} (a+i)^{3/4}}\right )\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -8 \left (\frac {1}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}}{\frac {\sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}-\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}+1}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}+1\right )}{\frac {\sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}+\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}+1}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )}{\sqrt {2}}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )}{\sqrt {2}}\right )\right )+\frac {1}{2} (1-i a) \left (\frac {i \arctan \left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{2 \sqrt [4]{-a+i} (a+i)^{3/4}}+\frac {i \text {arctanh}\left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{2 \sqrt [4]{-a+i} (a+i)^{3/4}}\right )\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -8 \left (\frac {1}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}}{\frac {\sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}-\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}+1}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}+1}{\frac {\sqrt {1-i (a+b x)}}{\sqrt {i (a+b x)+1}}+\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}+1}d\frac {\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i (a+b x)+1}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )}{\sqrt {2}}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )}{\sqrt {2}}\right )\right )+\frac {1}{2} (1-i a) \left (\frac {i \arctan \left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{2 \sqrt [4]{-a+i} (a+i)^{3/4}}+\frac {i \text {arctanh}\left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{2 \sqrt [4]{-a+i} (a+i)^{3/4}}\right )\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle -8 \left (\frac {1}{2} (1-i a) \left (\frac {i \arctan \left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{2 \sqrt [4]{-a+i} (a+i)^{3/4}}+\frac {i \text {arctanh}\left (\frac {\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{2 \sqrt [4]{-a+i} (a+i)^{3/4}}\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )}{\sqrt {2}}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\frac {\sqrt {1-i (a+b x)}}{\sqrt {1+i (a+b x)}}-\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\frac {\sqrt {1-i (a+b x)}}{\sqrt {1+i (a+b x)}}+\frac {\sqrt {2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}+1\right )}{2 \sqrt {2}}\right )\right )\right )\)

Input:

Int[1/(E^((I/2)*ArcTan[a + b*x])*x),x]
 

Output:

-8*(((1 - I*a)*(((I/2)*ArcTan[((I - a)^(1/4)*(1 - I*(a + b*x))^(1/4))/((I 
+ a)^(1/4)*(1 + I*(a + b*x))^(1/4))])/((I - a)^(1/4)*(I + a)^(3/4)) + ((I/ 
2)*ArcTanh[((I - a)^(1/4)*(1 - I*(a + b*x))^(1/4))/((I + a)^(1/4)*(1 + I*( 
a + b*x))^(1/4))])/((I - a)^(1/4)*(I + a)^(3/4))))/2 + ((ArcTan[1 - (Sqrt[ 
2]*(1 - I*(a + b*x))^(1/4))/(1 + I*(a + b*x))^(1/4)]/Sqrt[2] - ArcTan[1 + 
(Sqrt[2]*(1 - I*(a + b*x))^(1/4))/(1 + I*(a + b*x))^(1/4)]/Sqrt[2])/2 + (L 
og[1 + Sqrt[1 - I*(a + b*x)]/Sqrt[1 + I*(a + b*x)] - (Sqrt[2]*(1 - I*(a + 
b*x))^(1/4))/(1 + I*(a + b*x))^(1/4)]/(2*Sqrt[2]) - Log[1 + Sqrt[1 - I*(a 
+ b*x)]/Sqrt[1 + I*(a + b*x)] + (Sqrt[2]*(1 - I*(a + b*x))^(1/4))/(1 + I*( 
a + b*x))^(1/4)]/(2*Sqrt[2]))/2)/2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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])
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 755

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]]))
 

rule 756

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]
 

rule 981

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), 
 x_Symbol] :> Simp[(-a)*(e^n/(b*c - a*d))   Int[(e*x)^(m - n)/(a + b*x^n), 
x], x] + Simp[c*(e^n/(b*c - a*d))   Int[(e*x)^(m - n)/(c + d*x^n), x], x] / 
; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, 
 m, 2*n - 1]
 

rule 1082

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]
 

rule 1103

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]
 

rule 1476

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]
 

rule 1479

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]
 

rule 5617

Int[E^(ArcTan[(c_.)*((a_) + (b_.)*(x_))]*(n_))*(x_)^(m_), x_Symbol] :> Simp 
[4/(I^m*n*b^(m + 1)*c^(m + 1))   Subst[Int[x^(2/(I*n))*((1 - I*a*c - (1 + I 
*a*c)*x^(2/(I*n)))^m/(1 + x^(2/(I*n)))^(m + 2)), x], x, (1 - I*c*(a + b*x)) 
^(I*(n/2))/(1 + I*c*(a + b*x))^(I*(n/2))], x] /; FreeQ[{a, b, c}, x] && ILt 
Q[m, 0] && LtQ[-1, I*n, 1]
 

Maple [F]

\[\int \frac {1}{\sqrt {\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}\, x}d x\]

Input:

int(1/((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(1/2)/x,x)
 

Output:

int(1/((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(1/2)/x,x)
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 470 vs. \(2 (227) = 454\).

Time = 0.10 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.47 \[ \int \frac {e^{-\frac {1}{2} i \arctan (a+b x)}}{x} \, dx=-\frac {1}{2} \, \sqrt {4 i} \log \left (\frac {1}{2} i \, \sqrt {4 i} + \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}\right ) + \frac {1}{2} \, \sqrt {4 i} \log \left (-\frac {1}{2} i \, \sqrt {4 i} + \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}\right ) + \frac {1}{2} \, \sqrt {-4 i} \log \left (\frac {1}{2} i \, \sqrt {-4 i} + \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}\right ) - \frac {1}{2} \, \sqrt {-4 i} \log \left (-\frac {1}{2} i \, \sqrt {-4 i} + \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}\right ) + \left (-\frac {a + i}{a - i}\right )^{\frac {1}{4}} \log \left (\frac {{\left (a - i\right )} \left (-\frac {a + i}{a - i}\right )^{\frac {3}{4}} + {\left (a + i\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{a + i}\right ) - \left (-\frac {a + i}{a - i}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (a - i\right )} \left (-\frac {a + i}{a - i}\right )^{\frac {3}{4}} - {\left (a + i\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{a + i}\right ) - i \, \left (-\frac {a + i}{a - i}\right )^{\frac {1}{4}} \log \left (\frac {{\left (i \, a + 1\right )} \left (-\frac {a + i}{a - i}\right )^{\frac {3}{4}} + {\left (a + i\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{a + i}\right ) + i \, \left (-\frac {a + i}{a - i}\right )^{\frac {1}{4}} \log \left (\frac {{\left (-i \, a - 1\right )} \left (-\frac {a + i}{a - i}\right )^{\frac {3}{4}} + {\left (a + i\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{a + i}\right ) \] Input:

integrate(1/((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(1/2)/x,x, algorithm="fric 
as")
 

Output:

-1/2*sqrt(4*I)*log(1/2*I*sqrt(4*I) + sqrt(I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 
 1)/(b*x + a + I))) + 1/2*sqrt(4*I)*log(-1/2*I*sqrt(4*I) + sqrt(I*sqrt(b^2 
*x^2 + 2*a*b*x + a^2 + 1)/(b*x + a + I))) + 1/2*sqrt(-4*I)*log(1/2*I*sqrt( 
-4*I) + sqrt(I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/(b*x + a + I))) - 1/2*sqr 
t(-4*I)*log(-1/2*I*sqrt(-4*I) + sqrt(I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/( 
b*x + a + I))) + (-(a + I)/(a - I))^(1/4)*log(((a - I)*(-(a + I)/(a - I))^ 
(3/4) + (a + I)*sqrt(I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/(b*x + a + I)))/( 
a + I)) - (-(a + I)/(a - I))^(1/4)*log(-((a - I)*(-(a + I)/(a - I))^(3/4) 
- (a + I)*sqrt(I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/(b*x + a + I)))/(a + I) 
) - I*(-(a + I)/(a - I))^(1/4)*log(((I*a + 1)*(-(a + I)/(a - I))^(3/4) + ( 
a + I)*sqrt(I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/(b*x + a + I)))/(a + I)) + 
 I*(-(a + I)/(a - I))^(1/4)*log(((-I*a - 1)*(-(a + I)/(a - I))^(3/4) + (a 
+ I)*sqrt(I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/(b*x + a + I)))/(a + I))
 

Sympy [F]

\[ \int \frac {e^{-\frac {1}{2} i \arctan (a+b x)}}{x} \, dx=\int \frac {1}{x \sqrt {\frac {i \left (a + b x - i\right )}{\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}}}\, dx \] Input:

integrate(1/((1+I*(b*x+a))/(1+(b*x+a)**2)**(1/2))**(1/2)/x,x)
 

Output:

Integral(1/(x*sqrt(I*(a + b*x - I)/sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1))), 
 x)
 

Maxima [F]

\[ \int \frac {e^{-\frac {1}{2} i \arctan (a+b x)}}{x} \, dx=\int { \frac {1}{x \sqrt {\frac {i \, b x + i \, a + 1}{\sqrt {{\left (b x + a\right )}^{2} + 1}}}} \,d x } \] Input:

integrate(1/((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(1/2)/x,x, algorithm="maxi 
ma")
 

Output:

integrate(1/(x*sqrt((I*b*x + I*a + 1)/sqrt((b*x + a)^2 + 1))), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{-\frac {1}{2} i \arctan (a+b x)}}{x} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(1/2)/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
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-\frac {1}{2} i \arctan (a+b x)}}{x} \, dx=\int \frac {1}{x\,\sqrt {\frac {1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}{\sqrt {{\left (a+b\,x\right )}^2+1}}}} \,d x \] Input:

int(1/(x*((a*1i + b*x*1i + 1)/((a + b*x)^2 + 1)^(1/2))^(1/2)),x)
 

Output:

int(1/(x*((a*1i + b*x*1i + 1)/((a + b*x)^2 + 1)^(1/2))^(1/2)), x)
 

Reduce [F]

\[ \int \frac {e^{-\frac {1}{2} i \arctan (a+b x)}}{x} \, dx=\int \frac {1}{\sqrt {\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}\, x}d x \] Input:

int(1/((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(1/2)/x,x)
 

Output:

int(1/((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(1/2)/x,x)