3.131 \(\int \frac{e^{\frac{1}{4} i \tan ^{-1}(a x)}}{x} \, dx\)

Optimal. Leaf size=859 \[ -2 \tan ^{-1}\left (\frac{\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )+\sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-\frac{2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt{2+\sqrt{2}}}\right )+\sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-\frac{2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt{2-\sqrt{2}}}\right )-\sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\frac{2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )-\sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\frac{2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )+\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )-\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )-2 \tanh ^{-1}\left (\frac{\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )-\frac{1}{2} \sqrt{2-\sqrt{2}} \log \left (\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}-\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1\right )+\frac{1}{2} \sqrt{2-\sqrt{2}} \log \left (\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1\right )-\frac{1}{2} \sqrt{2+\sqrt{2}} \log \left (\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1\right )+\frac{1}{2} \sqrt{2+\sqrt{2}} \log \left (\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1\right )+\frac{\log \left (\frac{\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac{\sqrt{2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{\sqrt{2}}-\frac{\log \left (\frac{\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+\frac{\sqrt{2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{\sqrt{2}} \]

[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]

________________________________________________________________________________________

Rubi [A]  time = 0.545886, antiderivative size = 859, normalized size of antiderivative = 1., number of steps used = 39, number of rules used = 20, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.25, Rules used = {5062, 105, 63, 331, 299, 1122, 1169, 634, 618, 204, 628, 93, 214, 212, 206, 203, 211, 1165, 1162, 617} \[ -2 \tan ^{-1}\left (\frac{\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )+\sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-\frac{2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt{2+\sqrt{2}}}\right )+\sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-\frac{2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}}{\sqrt{2-\sqrt{2}}}\right )-\sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\frac{2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )-\sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\frac{2 \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )+\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )-\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )-2 \tanh ^{-1}\left (\frac{\sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}\right )-\frac{1}{2} \sqrt{2-\sqrt{2}} \log \left (\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}-\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1\right )+\frac{1}{2} \sqrt{2-\sqrt{2}} \log \left (\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1\right )-\frac{1}{2} \sqrt{2+\sqrt{2}} \log \left (\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1\right )+\frac{1}{2} \sqrt{2+\sqrt{2}} \log \left (\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{i a x+1}}+\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-i a x}}{\sqrt [8]{i a x+1}}+1\right )+\frac{\log \left (\frac{\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac{\sqrt{2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{\sqrt{2}}-\frac{\log \left (\frac{\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}+\frac{\sqrt{2} \sqrt [8]{i a x+1}}{\sqrt [8]{1-i a x}}+1\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[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 5062

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]

Rule 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

Rule 63

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 331

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 299

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 1122

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 1169

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 634

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 618

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 204

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

Rule 628

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 93

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 214

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 212

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] &&
!GtQ[a/b, 0]

Rule 206

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

Rule 203

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

Rule 211

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 1165

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 1162

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 617

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]

Rubi steps

\begin{align*} \int \frac{e^{\frac{1}{4} i \tan ^{-1}(a x)}}{x} \, dx &=\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 \operatorname{Subst}\left (\int \frac{x^6}{\left (2-x^8\right )^{7/8}} \, dx,x,\sqrt [8]{1-i a x}\right )\right )+8 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 ) \operatorname{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 ) \operatorname{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 \tan ^{-1}\left (\frac{\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-2 \tanh ^{-1}\left (\frac{\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )+\frac{\operatorname{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{\operatorname{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 ) \operatorname{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 ) \operatorname{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 )-\operatorname{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 )-\operatorname{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 \tan ^{-1}\left (\frac{\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-2 \tanh ^{-1}\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} \operatorname{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} \operatorname{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}} \operatorname{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}} \operatorname{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}} \operatorname{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}} \operatorname{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 \tan ^{-1}\left (\frac{\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )+\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-2 \tanh ^{-1}\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}} \operatorname{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}} \operatorname{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 ) \operatorname{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 ) \operatorname{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}} \operatorname{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}} \operatorname{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 ) \operatorname{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 ) \operatorname{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 \tan ^{-1}\left (\frac{\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )+\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-2 \tanh ^{-1}\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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 \tan ^{-1}\left (\frac{\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )+\sqrt{2+\sqrt{2}} \tan ^{-1}\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}} \tan ^{-1}\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}} \tan ^{-1}\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}} \tan ^{-1}\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} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-\sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}}\right )-2 \tanh ^{-1}\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]  time = 0.033823, size = 97, normalized size = 0.11 \[ -\frac{4 (1-i a x)^{7/8} \left (\sqrt [8]{2} (1+i a x)^{7/8} \text{Hypergeometric2F1}\left (\frac{7}{8},\frac{7}{8},\frac{15}{8},\frac{1}{2} (1-i a x)\right )+2 \text{Hypergeometric2F1}\left (\frac{7}{8},1,\frac{15}{8},\frac{a x+i}{-a x+i}\right )\right )}{7 (1+i a x)^{7/8}} \]

Warning: Unable to verify antiderivative.

[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))

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}\sqrt [4]{{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}\right )^{\frac{1}{4}}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.8875, size = 1423, normalized size = 1.66 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError