\(\int \frac {e^{\frac {3}{2} i \arctan (a+b x)}}{x} \, dx\) [238]

Optimal result

Integrand size = 18, antiderivative size = 344 \[ \int \frac {e^{\frac {3}{2} i \arctan (a+b x)}}{x} \, dx=\frac {2 (i-a)^{3/4} \arctan \left (\frac {\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i+a)^{3/4}}+\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )-\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )-\frac {2 (i-a)^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i+a)^{3/4}}-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x} \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}\right )}\right ) \] Output:

2*(I-a)^(3/4)*arctan((I+a)^(1/4)*(1+I*a+I*b*x)^(1/4)/(I-a)^(1/4)/(1-I*a-I* 
b*x)^(1/4))/(I+a)^(3/4)+2^(1/2)*arctan(1-2^(1/2)*(1-I*a-I*b*x)^(1/4)/(1+I* 
a+I*b*x)^(1/4))-2^(1/2)*arctan(1+2^(1/2)*(1-I*a-I*b*x)^(1/4)/(1+I*a+I*b*x) 
^(1/4))-2*(I-a)^(3/4)*arctanh((I+a)^(1/4)*(1+I*a+I*b*x)^(1/4)/(I-a)^(1/4)/ 
(1-I*a-I*b*x)^(1/4))/(I+a)^(3/4)-2^(1/2)*arctanh(2^(1/2)*(1-I*a-I*b*x)^(1/ 
4)/(1+I*a+I*b*x)^(1/4)/(1+(1-I*a-I*b*x)^(1/2)/(1+I*a+I*b*x)^(1/2)))
 

Mathematica [C] (verified)

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

Time = 0.09 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.35 \[ \int \frac {e^{\frac {3}{2} i \arctan (a+b x)}}{x} \, dx=2 \sqrt [4]{-i (i+a+b x)} \left (-2^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{4},\frac {5}{4},-\frac {1}{2} i (i+a+b x)\right )+\frac {2 (-i+a) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},\frac {1+a^2-i b x+a b x}{1+a^2+i b x+a b x}\right )}{(i+a) \sqrt [4]{1+i a+i b x}}\right ) \] Input:

Integrate[E^(((3*I)/2)*ArcTan[a + b*x])/x,x]
 

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.96 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.24, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5618, 140, 27, 73, 104, 25, 770, 755, 827, 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.

\(\displaystyle \int \frac {e^{\frac {3}{2} i \arctan (a+b x)}}{x} \, dx\)

\(\Big \downarrow \) 5618

\(\displaystyle \int \frac {(i a+i b x+1)^{3/4}}{x (-i a-i b x+1)^{3/4}}dx\)

\(\Big \downarrow \) 140

\(\displaystyle i b \int \frac {1}{(-i a-i b x+1)^{3/4} \sqrt [4]{i a+i b x+1}}dx+\int \frac {i a+1}{x (-i a-i b x+1)^{3/4} \sqrt [4]{i a+i b x+1}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle i b \int \frac {1}{(-i a-i b x+1)^{3/4} \sqrt [4]{i a+i b x+1}}dx+(1+i a) \int \frac {1}{x (-i a-i b x+1)^{3/4} \sqrt [4]{i a+i b x+1}}dx\)

\(\Big \downarrow \) 73

\(\displaystyle (1+i a) \int \frac {1}{x (-i a-i b x+1)^{3/4} \sqrt [4]{i a+i b x+1}}dx-4 \int \frac {1}{\sqrt [4]{i a+i b x+1}}d\sqrt [4]{-i a-i b x+1}\)

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 770

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

\(\Big \downarrow \) 755

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

\(\Big \downarrow \) 827

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

Int[E^(((3*I)/2)*ArcTan[a + b*x])/x,x]
 

Output:

4*(1 + I*a)*(((I/2)*ArcTan[((I + a)^(1/4)*(1 + I*a + I*b*x)^(1/4))/((I - a 
)^(1/4)*(1 - I*a - I*b*x)^(1/4))])/((I - a)^(1/4)*(I + a)^(3/4)) - ((I/2)* 
ArcTanh[((I + a)^(1/4)*(1 + I*a + I*b*x)^(1/4))/((I - a)^(1/4)*(1 - I*a - 
I*b*x)^(1/4))])/((I - a)^(1/4)*(I + a)^(3/4))) - 4*((-(ArcTan[1 - (Sqrt[2] 
*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)]/Sqrt[2]) + ArcTan[1 + ( 
Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)]/Sqrt[2])/2 + (-1 
/2*Log[1 + Sqrt[1 - I*a - I*b*x] - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + 
I*a + I*b*x)^(1/4)]/Sqrt[2] + Log[1 + Sqrt[1 - I*a - I*b*x] + (Sqrt[2]*(1 
- I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)]/(2*Sqrt[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 73

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]
 

rule 104

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]
 

rule 140

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]))
 

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 770

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n)   Subst[In 
t[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p + 1 
/n]
 

rule 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ 
[a/b, 0]
 

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 5618

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

Maple [F]

\[\int \frac {{\left (\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}\right )}^{\frac {3}{2}}}{x}d x\]

Input:

int(((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(3/2)/x,x)
 

Output:

int(((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(3/2)/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. 690 vs. \(2 (228) = 456\).

Time = 0.10 (sec) , antiderivative size = 690, normalized size of antiderivative = 2.01 \[ \int \frac {e^{\frac {3}{2} i \arctan (a+b x)}}{x} \, dx =\text {Too large to display} \] Input:

integrate(((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(3/2)/x,x, algorithm="fricas 
")
 

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

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{\frac {3}{2} i \arctan (a+b x)}}{x} \, dx=\text {Timed out} \] Input:

integrate(((1+I*(b*x+a))/(1+(b*x+a)**2)**(1/2))**(3/2)/x,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {e^{\frac {3}{2} i \arctan (a+b x)}}{x} \, dx=\int { \frac {\left (\frac {i \, b x + i \, a + 1}{\sqrt {{\left (b x + a\right )}^{2} + 1}}\right )^{\frac {3}{2}}}{x} \,d x } \] Input:

integrate(((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(3/2)/x,x, algorithm="maxima 
")
 

Output:

integrate(((I*b*x + I*a + 1)/sqrt((b*x + a)^2 + 1))^(3/2)/x, x)
                                                                                    
                                                                                    
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{\frac {3}{2} i \arctan (a+b x)}}{x} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(3/2)/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. \[ \int \frac {e^{\frac {3}{2} i \arctan (a+b x)}}{x} \, dx=\int \frac {{\left (\frac {1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}{\sqrt {{\left (a+b\,x\right )}^2+1}}\right )}^{3/2}}{x} \,d x \] Input:

int(((a*1i + b*x*1i + 1)/((a + b*x)^2 + 1)^(1/2))^(3/2)/x,x)
 

Output:

int(((a*1i + b*x*1i + 1)/((a + b*x)^2 + 1)^(1/2))^(3/2)/x, x)
 

Reduce [F]

\[ \int \frac {e^{\frac {3}{2} i \arctan (a+b x)}}{x} \, dx=\int \frac {{\left (\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}\right )}^{\frac {3}{2}}}{x}d x \] Input:

int(((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(3/2)/x,x)
 

Output:

int(((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(3/2)/x,x)