\(\int e^{-\frac {1}{2} i \arctan (a x)} x \, dx\) [105]

Optimal result

Integrand size = 14, antiderivative size = 226 \[ \int e^{-\frac {1}{2} i \arctan (a x)} x \, dx=\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}+\frac {(1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a^2}+\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt {2} a^2}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt {2} a^2}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x} \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}\right )}\right )}{4 \sqrt {2} a^2} \] Output:

1/4*(1-I*a*x)^(1/4)*(1+I*a*x)^(3/4)/a^2+1/2*(1-I*a*x)^(5/4)*(1+I*a*x)^(3/4 
)/a^2+1/8*arctan(1-2^(1/2)*(1-I*a*x)^(1/4)/(1+I*a*x)^(1/4))*2^(1/2)/a^2-1/ 
8*arctan(1+2^(1/2)*(1-I*a*x)^(1/4)/(1+I*a*x)^(1/4))*2^(1/2)/a^2-1/8*arctan 
h(2^(1/2)*(1-I*a*x)^(1/4)/(1+I*a*x)^(1/4)/(1+(1-I*a*x)^(1/2)/(1+I*a*x)^(1/ 
2)))*2^(1/2)/a^2
 

Mathematica [C] (verified)

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

Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.28 \[ \int e^{-\frac {1}{2} i \arctan (a x)} x \, dx=\frac {(1-i a x)^{5/4} \left (5 (1+i a x)^{3/4}-2^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {5}{4},\frac {9}{4},\frac {1}{2} (1-i a x)\right )\right )}{10 a^2} \] Input:

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

Output:

((1 - I*a*x)^(5/4)*(5*(1 + I*a*x)^(3/4) - 2^(3/4)*Hypergeometric2F1[1/4, 5 
/4, 9/4, (1 - I*a*x)/2]))/(10*a^2)
 

Rubi [A] (warning: unable to verify)

Time = 0.69 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.24, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {5585, 90, 60, 73, 770, 755, 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.

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

\(\Big \downarrow \) 5585

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

\(\Big \downarrow \) 90

\(\displaystyle \frac {i \int \frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}dx}{4 a}+\frac {(1+i a x)^{3/4} (1-i a x)^{5/4}}{2 a^2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {i \left (\frac {1}{2} \int \frac {1}{(1-i a x)^{3/4} \sqrt [4]{i a x+1}}dx-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}\right )}{4 a}+\frac {(1+i a x)^{3/4} (1-i a x)^{5/4}}{2 a^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {i \left (\frac {2 i \int \frac {1}{\sqrt [4]{i a x+1}}d\sqrt [4]{1-i a x}}{a}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}\right )}{4 a}+\frac {(1+i a x)^{3/4} (1-i a x)^{5/4}}{2 a^2}\)

\(\Big \downarrow \) 770

\(\displaystyle \frac {i \left (\frac {2 i \int \frac {1}{2-i a x}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{a}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}\right )}{4 a}+\frac {(1+i a x)^{3/4} (1-i a x)^{5/4}}{2 a^2}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {i \left (\frac {2 i \left (\frac {1}{2} \int \frac {1-\sqrt {1-i a x}}{2-i a x}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+\frac {1}{2} \int \frac {\sqrt {1-i a x}+1}{2-i a x}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}\right )}{a}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}\right )}{4 a}+\frac {(1+i a x)^{3/4} (1-i a x)^{5/4}}{2 a^2}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {i \left (\frac {2 i \left (\frac {1}{2} \int \frac {1-\sqrt {1-i a x}}{2-i a x}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-i a x}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+\frac {1}{2} \int \frac {1}{\sqrt {1-i a x}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}\right )\right )}{a}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}\right )}{4 a}+\frac {(1+i a x)^{3/4} (1-i a x)^{5/4}}{2 a^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {i \left (\frac {2 i \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {1-i a x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {1-i a x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \int \frac {1-\sqrt {1-i a x}}{2-i a x}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}\right )}{a}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}\right )}{4 a}+\frac {(1+i a x)^{3/4} (1-i a x)^{5/4}}{2 a^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {i \left (\frac {2 i \left (\frac {1}{2} \int \frac {1-\sqrt {1-i a x}}{2-i a x}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}\right )\right )}{a}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}\right )}{4 a}+\frac {(1+i a x)^{3/4} (1-i a x)^{5/4}}{2 a^2}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {i \left (\frac {2 i \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{\sqrt {1-i a x}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1\right )}{\sqrt {1-i a x}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}\right )\right )}{a}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}\right )}{4 a}+\frac {(1+i a x)^{3/4} (1-i a x)^{5/4}}{2 a^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {i \left (\frac {2 i \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{\sqrt {1-i a x}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1\right )}{\sqrt {1-i a x}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}\right )\right )}{a}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}\right )}{4 a}+\frac {(1+i a x)^{3/4} (1-i a x)^{5/4}}{2 a^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {i \left (\frac {2 i \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{\sqrt {1-i a x}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}{\sqrt {1-i a x}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+1}d\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}\right )\right )}{a}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}\right )}{4 a}+\frac {(1+i a x)^{3/4} (1-i a x)^{5/4}}{2 a^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {(1+i a x)^{3/4} (1-i a x)^{5/4}}{2 a^2}+\frac {i \left (\frac {2 i \left (\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {1-i a x}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {1-i a x}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{2 \sqrt {2}}\right )\right )}{a}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{a}\right )}{4 a}\)

Input:

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

Output:

((1 - I*a*x)^(5/4)*(1 + I*a*x)^(3/4))/(2*a^2) + ((I/4)*(((-I)*(1 - I*a*x)^ 
(1/4)*(1 + I*a*x)^(3/4))/a + ((2*I)*((-(ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1 
/4))/(1 + I*a*x)^(1/4)]/Sqrt[2]) + ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/ 
(1 + I*a*x)^(1/4)]/Sqrt[2])/2 + (-1/2*Log[1 + Sqrt[1 - I*a*x] - (Sqrt[2]*( 
1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/Sqrt[2] + Log[1 + Sqrt[1 - I*a*x] + ( 
Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/(2*Sqrt[2]))/2))/a))/a
 

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 60

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]
 

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 90

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]
 

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 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 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 5585

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]
 

Maple [F]

\[\int \frac {x}{\sqrt {\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}}}d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.05 \[ \int e^{-\frac {1}{2} i \arctan (a x)} x \, dx=\frac {2 \, a^{2} \sqrt {\frac {i}{16 \, a^{4}}} \log \left (4 i \, a^{2} \sqrt {\frac {i}{16 \, a^{4}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - 2 \, a^{2} \sqrt {\frac {i}{16 \, a^{4}}} \log \left (-4 i \, a^{2} \sqrt {\frac {i}{16 \, a^{4}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - 2 \, a^{2} \sqrt {-\frac {i}{16 \, a^{4}}} \log \left (4 i \, a^{2} \sqrt {-\frac {i}{16 \, a^{4}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + 2 \, a^{2} \sqrt {-\frac {i}{16 \, a^{4}}} \log \left (-4 i \, a^{2} \sqrt {-\frac {i}{16 \, a^{4}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + \sqrt {a^{2} x^{2} + 1} {\left (-2 i \, a x + 3\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{4 \, a^{2}} \] Input:

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

Output:

1/4*(2*a^2*sqrt(1/16*I/a^4)*log(4*I*a^2*sqrt(1/16*I/a^4) + sqrt(I*sqrt(a^2 
*x^2 + 1)/(a*x + I))) - 2*a^2*sqrt(1/16*I/a^4)*log(-4*I*a^2*sqrt(1/16*I/a^ 
4) + sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I))) - 2*a^2*sqrt(-1/16*I/a^4)*log(4* 
I*a^2*sqrt(-1/16*I/a^4) + sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I))) + 2*a^2*sqr 
t(-1/16*I/a^4)*log(-4*I*a^2*sqrt(-1/16*I/a^4) + sqrt(I*sqrt(a^2*x^2 + 1)/( 
a*x + I))) + sqrt(a^2*x^2 + 1)*(-2*I*a*x + 3)*sqrt(I*sqrt(a^2*x^2 + 1)/(a* 
x + I)))/a^2
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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