\(\int \frac {e^{\frac {1}{4} i \arctan (a x)}}{x} \, dx\) [131]

Optimal result

Integrand size = 16, antiderivative size = 859 \[ \int \frac {e^{\frac {1}{4} i \arctan (a x)}}{x} \, dx=-2 \arctan \left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )+\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )+\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )-\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )-\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )+\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-2 \text {arctanh}\left (\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {\log \left (1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )}{\sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )}{\sqrt {2}} \]

[Out]

-2*arctan((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))+1/2*ln(1+(1+I*a*x)^(1/4)
/(1-I*a*x)^(1/4)-(1+I*a*x)^(1/8)*2^(1/2)/(1-I*a*x)^(1/8))*2^(1/2)-1/2*ln(1+(1+I*a*x)^(1/4)/(1-I*a*x)^(1/4)+(1+
I*a*x)^(1/8)*2^(1/2)/(1-I*a*x)^(1/8))*2^(1/2)+arctan(1-(1+I*a*x)^(1/8)*2^(1/2)/(1-I*a*x)^(1/8))*2^(1/2)-arctan
(1+(1+I*a*x)^(1/8)*2^(1/2)/(1-I*a*x)^(1/8))*2^(1/2)+arctan((-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))*(2-2^(1/2))^(1/2)-arctan((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))*(2-2^(1/2))^(1/2)-1/2*ln(1+(1-I*a*x)^(1/4)/(1+I*a*x)^(1/4)-(1-I*a*x)^(1/8)*(2-2^(1/2))^(1/2)/(1+I*a
*x)^(1/8))*(2-2^(1/2))^(1/2)+1/2*ln(1+(1-I*a*x)^(1/4)/(1+I*a*x)^(1/4)+(1-I*a*x)^(1/8)*(2-2^(1/2))^(1/2)/(1+I*a
*x)^(1/8))*(2-2^(1/2))^(1/2)+arctan((-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))*
(2+2^(1/2))^(1/2)-arctan((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))*(2+2^(1/2))^
(1/2)-1/2*ln(1+(1-I*a*x)^(1/4)/(1+I*a*x)^(1/4)-(1-I*a*x)^(1/8)*(2+2^(1/2))^(1/2)/(1+I*a*x)^(1/8))*(2+2^(1/2))^
(1/2)+1/2*ln(1+(1-I*a*x)^(1/4)/(1+I*a*x)^(1/4)+(1-I*a*x)^(1/8)*(2+2^(1/2))^(1/2)/(1+I*a*x)^(1/8))*(2+2^(1/2))^
(1/2)

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 859, normalized size of antiderivative = 1.00, number of steps used = 39, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {5170, 132, 65, 338, 305, 1136, 1183, 648, 632, 210, 642, 95, 220, 218, 212, 209, 217, 1179, 1176, 631} \[ \int \frac {e^{\frac {1}{4} i \arctan (a x)}}{x} \, dx=-2 \arctan \left (\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )+\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt {2+\sqrt {2}}}\right )+\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt {2-\sqrt {2}}}\right )-\sqrt {2+\sqrt {2}} \arctan \left (\frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )-\sqrt {2-\sqrt {2}} \arctan \left (\frac {\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )+\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )-\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )-2 \text {arctanh}\left (\frac {\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1\right )+\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1\right )+\frac {\log \left (\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{\sqrt {2}}-\frac {\log \left (\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+\frac {\sqrt {2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{\sqrt {2}} \]

[In]

Int[E^((I/4)*ArcTan[a*x])/x,x]

[Out]

-2*ArcTan[(1 + I*a*x)^(1/8)/(1 - I*a*x)^(1/8)] + 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]]] + 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]]] - 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]]] - 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]]] + Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 + I*a*x)^(1/8))/(1 - I*a*x)^(1/8)] - S
qrt[2]*ArcTan[1 + (Sqrt[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)] - (Sqrt[2 - Sqrt[2]]*Log[1 + (1 - I*a*x)^(1/4)/(1 + I*a*x)^(1/4) - (Sqrt[2 - Sqrt[2]]*(1 - I*a*x)^(1/8))/(
1 + I*a*x)^(1/8)])/2 + (Sqrt[2 - Sqrt[2]]*Log[1 + (1 - I*a*x)^(1/4)/(1 + I*a*x)^(1/4) + (Sqrt[2 - Sqrt[2]]*(1
- I*a*x)^(1/8))/(1 + I*a*x)^(1/8)])/2 - (Sqrt[2 + Sqrt[2]]*Log[1 + (1 - I*a*x)^(1/4)/(1 + I*a*x)^(1/4) - (Sqrt
[2 + Sqrt[2]]*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8)])/2 + (Sqrt[2 + Sqrt[2]]*Log[1 + (1 - I*a*x)^(1/4)/(1 + I*a
*x)^(1/4) + (Sqrt[2 + Sqrt[2]]*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8)])/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)]/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)]/Sqrt[2]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[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] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 132

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[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)*ExpandTo
Sum[(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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 210

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 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[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 218

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

Rule 220

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

Rule 305

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[Rt[a/b, 4]], s = Denominator[Rt[a/b,
4]]}, Dist[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] - Dist[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]] /; FreeQ[{a, b}, x] && IGt
Q[n/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && GtQ[a/b, 0]

Rule 338

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1136

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] - Dist[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] && Gt
Q[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[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 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1183

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]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rule 5170

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] &&  !IntegerQ[(I*n - 1)/2]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt [8]{1+i a x}}{x \sqrt [8]{1-i a x}} \, dx \\ & = (i a) \int \frac {1}{\sqrt [8]{1-i a x} (1+i a x)^{7/8}} \, dx+\int \frac {1}{x \sqrt [8]{1-i a x} (1+i a x)^{7/8}} \, dx \\ & = -\left (8 \text {Subst}\left (\int \frac {x^6}{\left (2-x^8\right )^{7/8}} \, dx,x,\sqrt [8]{1-i a x}\right )\right )+8 \text {Subst}\left (\int \frac {1}{-1+x^8} \, dx,x,\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right ) \\ & = -\left (4 \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )\right )-4 \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-8 \text {Subst}\left (\int \frac {x^6}{1+x^8} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right ) \\ & = -\left (2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )\right )-2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-2 \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-2 \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\left (2 \sqrt {2}\right ) \text {Subst}\left (\int \frac {x^4}{1-\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\left (2 \sqrt {2}\right ) \text {Subst}\left (\int \frac {x^4}{1+\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right ) \\ & = -2 \arctan \left (\frac {\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 )+\frac {\text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )}{\sqrt {2}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )}{\sqrt {2}}+\left (2 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1-\sqrt {2} x^2}{1-\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )-\left (2 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1+\sqrt {2} x^2}{1+\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )-\text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right ) \\ & = -2 \arctan \left (\frac {\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 )+\frac {\log \left (1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )}{\sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )}{\sqrt {2}}-\sqrt {2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )+\sqrt {2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )+\sqrt {2-\sqrt {2}} \text {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}-\left (1+\sqrt {2}\right ) x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\sqrt {2-\sqrt {2}} \text {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}+\left (1+\sqrt {2}\right ) x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )-\sqrt {2+\sqrt {2}} \text {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}-\left (1-\sqrt {2}\right ) x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )-\sqrt {2+\sqrt {2}} \text {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}+\left (1-\sqrt {2}\right ) x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right ) \\ & = -2 \arctan \left (\frac {\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 )-\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 )+\frac {\log \left (1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )}{\sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )}{\sqrt {2}}-\frac {1}{2} \sqrt {2-\sqrt {2}} \text {Subst}\left (\int \frac {-\sqrt {2-\sqrt {2}}+2 x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \text {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}+2 x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {1}{2} \left (-2+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {1}{2} \left (-2+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \text {Subst}\left (\int \frac {-\sqrt {2+\sqrt {2}}+2 x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {1}{2} \sqrt {2+\sqrt {2}} \text {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}+2 x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )-\frac {1}{2} \left (2+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )-\frac {1}{2} \left (2+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right ) \\ & = -2 \arctan \left (\frac {\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 )-\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 )-\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {\log \left (1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )}{\sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )}{\sqrt {2}}+\left (2-\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,-\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\left (2-\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\left (2+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,-\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\left (2+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right ) \\ & = -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 )-\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )+\frac {\log \left (1-\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )}{\sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}+\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )}{\sqrt {2}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.11 \[ \int \frac {e^{\frac {1}{4} i \arctan (a x)}}{x} \, dx=-\frac {4 (1-i a x)^{7/8} \left (\sqrt [8]{2} (1+i a x)^{7/8} \operatorname {Hypergeometric2F1}\left (\frac {7}{8},\frac {7}{8},\frac {15}{8},\frac {1}{2} (1-i a x)\right )+2 \operatorname {Hypergeometric2F1}\left (\frac {7}{8},1,\frac {15}{8},\frac {i+a x}{i-a x}\right )\right )}{7 (1+i a x)^{7/8}} \]

[In]

Integrate[E^((I/4)*ArcTan[a*x])/x,x]

[Out]

(-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*Hypergeo
metric2F1[7/8, 1, 15/8, (I + a*x)/(I - a*x)]))/(7*(1 + I*a*x)^(7/8))

Maple [F]

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

[In]

int(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/4)/x,x)

[Out]

int(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/4)/x,x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 509, normalized size of antiderivative = 0.59 \[ \int \frac {e^{\frac {1}{4} i \arctan (a x)}}{x} \, dx=-\frac {1}{2} \, \sqrt {4 i} \log \left (\frac {1}{2} \, \sqrt {4 i} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + \frac {1}{2} \, \sqrt {4 i} \log \left (-\frac {1}{2} \, \sqrt {4 i} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - \frac {1}{2} \, \sqrt {-4 i} \log \left (\frac {1}{2} \, \sqrt {-4 i} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + \frac {1}{2} \, \sqrt {-4 i} \log \left (-\frac {1}{2} \, \sqrt {-4 i} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + i^{\frac {1}{4}} \log \left (i^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + i \, i^{\frac {1}{4}} \log \left (i \, i^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - i \, i^{\frac {1}{4}} \log \left (-i \, i^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - i^{\frac {1}{4}} \log \left (-i^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + \left (-i\right )^{\frac {1}{4}} \log \left (\left (-i\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + i \, \left (-i\right )^{\frac {1}{4}} \log \left (i \, \left (-i\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - i \, \left (-i\right )^{\frac {1}{4}} \log \left (-i \, \left (-i\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - \left (-i\right )^{\frac {1}{4}} \log \left (-\left (-i\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - \log \left (\left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}} + 1\right ) - i \, \log \left (\left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}} + i\right ) + i \, \log \left (\left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}} - i\right ) + \log \left (\left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}} - 1\right ) \]

[In]

integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/4)/x,x, algorithm="fricas")

[Out]

-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)

Sympy [F]

\[ \int \frac {e^{\frac {1}{4} i \arctan (a x)}}{x} \, dx=\int \frac {\sqrt [4]{\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}}{x}\, dx \]

[In]

integrate(((1+I*a*x)/(a**2*x**2+1)**(1/2))**(1/4)/x,x)

[Out]

Integral((I*(a*x - I)/sqrt(a**2*x**2 + 1))**(1/4)/x, x)

Maxima [F]

\[ \int \frac {e^{\frac {1}{4} i \arctan (a x)}}{x} \, dx=\int { \frac {\left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {1}{4}}}{x} \,d x } \]

[In]

integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/4)/x,x, algorithm="maxima")

[Out]

integrate(((I*a*x + 1)/sqrt(a^2*x^2 + 1))^(1/4)/x, x)

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{\frac {1}{4} i \arctan (a x)}}{x} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/4)/x,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:The choice was done assuming 0=[0,0]Warning, replacing 0 by -28, a substitution variable should perhaps be
purged.Warn

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{\frac {1}{4} i \arctan (a x)}}{x} \, dx=\int \frac {{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{1/4}}{x} \,d x \]

[In]

int(((a*x*1i + 1)/(a^2*x^2 + 1)^(1/2))^(1/4)/x,x)

[Out]

int(((a*x*1i + 1)/(a^2*x^2 + 1)^(1/2))^(1/4)/x, x)