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

Optimal. Leaf size=689 \[ \frac{(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac{(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac{\sqrt{2-\sqrt{2}} \log \left (\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{64 a^2}-\frac{\sqrt{2-\sqrt{2}} \log \left (\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{64 a^2}+\frac{\sqrt{2+\sqrt{2}} \log \left (\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{64 a^2}-\frac{\sqrt{2+\sqrt{2}} \log \left (\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{64 a^2}-\frac{\sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-\frac{2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt{2+\sqrt{2}}}\right )}{32 a^2}-\frac{\sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-\frac{2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt{2-\sqrt{2}}}\right )}{32 a^2}+\frac{\sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}+\frac{2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt{2+\sqrt{2}}}\right )}{32 a^2}+\frac{\sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}+\frac{2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt{2-\sqrt{2}}}\right )}{32 a^2} \]

[Out]

((1 - I*a*x)^(7/8)*(1 + I*a*x)^(1/8))/(8*a^2) + ((1 - I*a*x)^(7/8)*(1 + I*a*x)^(9/8))/(2*a^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]]])/(32*a^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]]])/(32*a^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]]]
)/(32*a^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]]])/(32*a^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)])/(64*a^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)])/(64*a^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)])/(64*a^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)])/(64
*a^2)

________________________________________________________________________________________

Rubi [A]  time = 0.494931, antiderivative size = 689, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 12, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857, Rules used = {5062, 80, 50, 63, 331, 299, 1122, 1169, 634, 618, 204, 628} \[ \frac{(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac{(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac{\sqrt{2-\sqrt{2}} \log \left (\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{64 a^2}-\frac{\sqrt{2-\sqrt{2}} \log \left (\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{64 a^2}+\frac{\sqrt{2+\sqrt{2}} \log \left (\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{64 a^2}-\frac{\sqrt{2+\sqrt{2}} \log \left (\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{64 a^2}-\frac{\sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-\frac{2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt{2+\sqrt{2}}}\right )}{32 a^2}-\frac{\sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-\frac{2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt{2-\sqrt{2}}}\right )}{32 a^2}+\frac{\sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}+\frac{2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt{2+\sqrt{2}}}\right )}{32 a^2}+\frac{\sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}+\frac{2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt{2-\sqrt{2}}}\right )}{32 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - I*a*x)^(7/8)*(1 + I*a*x)^(1/8))/(8*a^2) + ((1 - I*a*x)^(7/8)*(1 + I*a*x)^(9/8))/(2*a^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]]])/(32*a^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]]])/(32*a^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]]]
)/(32*a^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]]])/(32*a^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)])/(64*a^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)])/(64*a^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)])/(64*a^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)])/(64
*a^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 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(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 50

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] + Dist[(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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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]

Rubi steps

\begin{align*} \int e^{\frac{1}{4} i \tan ^{-1}(a x)} x \, dx &=\int \frac{x \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}} \, dx\\ &=\frac{(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}-\frac{i \int \frac{\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}} \, dx}{8 a}\\ &=\frac{(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac{(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}-\frac{i \int \frac{1}{\sqrt [8]{1-i a x} (1+i a x)^{7/8}} \, dx}{32 a}\\ &=\frac{(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac{(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac{\operatorname{Subst}\left (\int \frac{x^6}{\left (2-x^8\right )^{7/8}} \, dx,x,\sqrt [8]{1-i a x}\right )}{4 a^2}\\ &=\frac{(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac{(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac{\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 )}{4 a^2}\\ &=\frac{(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac{(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac{\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 )}{8 \sqrt{2} a^2}-\frac{\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 )}{8 \sqrt{2} a^2}\\ &=\frac{(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac{(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}-\frac{\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 )}{8 \sqrt{2} a^2}+\frac{\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 )}{8 \sqrt{2} a^2}\\ &=\frac{(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac{(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac{\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 )}{16 \sqrt{2 \left (2-\sqrt{2}\right )} a^2}+\frac{\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 )}{16 \sqrt{2 \left (2-\sqrt{2}\right )} a^2}-\frac{\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 )}{16 \sqrt{2 \left (2+\sqrt{2}\right )} a^2}-\frac{\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 )}{16 \sqrt{2 \left (2+\sqrt{2}\right )} a^2}\\ &=\frac{(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac{(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac{\sqrt{\frac{1}{2} \left (3-2 \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 )}{32 a^2}+\frac{\sqrt{\frac{1}{2} \left (3-2 \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 )}{32 a^2}+\frac{\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 )}{64 a^2}-\frac{\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 )}{64 a^2}+\frac{\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 )}{64 a^2}-\frac{\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 )}{64 a^2}+\frac{\sqrt{\frac{1}{2} \left (3+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 )}{32 a^2}+\frac{\sqrt{\frac{1}{2} \left (3+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 )}{32 a^2}\\ &=\frac{(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac{(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac{\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 )}{64 a^2}-\frac{\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 )}{64 a^2}+\frac{\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 )}{64 a^2}-\frac{\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 )}{64 a^2}-\frac{\sqrt{\frac{1}{2} \left (3-2 \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 )}{16 a^2}-\frac{\sqrt{\frac{1}{2} \left (3-2 \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 )}{16 a^2}-\frac{\sqrt{\frac{1}{2} \left (3+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 )}{16 a^2}-\frac{\sqrt{\frac{1}{2} \left (3+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 )}{16 a^2}\\ &=\frac{(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac{(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}-\frac{\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 )}{32 a^2}-\frac{\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 )}{32 a^2}+\frac{\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 )}{32 a^2}+\frac{\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 )}{32 a^2}+\frac{\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 )}{64 a^2}-\frac{\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 )}{64 a^2}+\frac{\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 )}{64 a^2}-\frac{\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 )}{64 a^2}\\ \end{align*}

Mathematica [C]  time = 0.0219907, size = 63, normalized size = 0.09 \[ \frac{(1-i a x)^{7/8} \left (2 \sqrt [8]{2} \text{Hypergeometric2F1}\left (-\frac{1}{8},\frac{7}{8},\frac{15}{8},\frac{1}{2} (1-i a x)\right )+7 (1+i a x)^{9/8}\right )}{14 a^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1 - I*a*x)^(7/8)*(7*(1 + I*a*x)^(9/8) + 2*2^(1/8)*Hypergeometric2F1[-1/8, 7/8, 15/8, (1 - I*a*x)/2]))/(14*a^
2)

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Maple [F]  time = 0.068, size = 0, normalized size = 0. \begin{align*} \int \sqrt [4]{{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}}x\, 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 x \left (\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}\right )^{\frac{1}{4}}\,{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(x*((I*a*x + 1)/sqrt(a^2*x^2 + 1))^(1/4), x)

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Fricas [A]  time = 1.89723, size = 1304, normalized size = 1.89 \begin{align*} -\frac{8 \, a^{2} \left (\frac{i}{1048576 \, a^{8}}\right )^{\frac{1}{4}} \log \left (32 \, a^{2} \left (\frac{i}{1048576 \, a^{8}}\right )^{\frac{1}{4}} + \left (\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}\right )^{\frac{1}{4}}\right ) + 8 i \, a^{2} \left (\frac{i}{1048576 \, a^{8}}\right )^{\frac{1}{4}} \log \left (32 i \, a^{2} \left (\frac{i}{1048576 \, a^{8}}\right )^{\frac{1}{4}} + \left (\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}\right )^{\frac{1}{4}}\right ) - 8 i \, a^{2} \left (\frac{i}{1048576 \, a^{8}}\right )^{\frac{1}{4}} \log \left (-32 i \, a^{2} \left (\frac{i}{1048576 \, a^{8}}\right )^{\frac{1}{4}} + \left (\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}\right )^{\frac{1}{4}}\right ) - 8 \, a^{2} \left (\frac{i}{1048576 \, a^{8}}\right )^{\frac{1}{4}} \log \left (-32 \, a^{2} \left (\frac{i}{1048576 \, a^{8}}\right )^{\frac{1}{4}} + \left (\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}\right )^{\frac{1}{4}}\right ) + 8 \, a^{2} \left (-\frac{i}{1048576 \, a^{8}}\right )^{\frac{1}{4}} \log \left (32 \, a^{2} \left (-\frac{i}{1048576 \, a^{8}}\right )^{\frac{1}{4}} + \left (\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}\right )^{\frac{1}{4}}\right ) + 8 i \, a^{2} \left (-\frac{i}{1048576 \, a^{8}}\right )^{\frac{1}{4}} \log \left (32 i \, a^{2} \left (-\frac{i}{1048576 \, a^{8}}\right )^{\frac{1}{4}} + \left (\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}\right )^{\frac{1}{4}}\right ) - 8 i \, a^{2} \left (-\frac{i}{1048576 \, a^{8}}\right )^{\frac{1}{4}} \log \left (-32 i \, a^{2} \left (-\frac{i}{1048576 \, a^{8}}\right )^{\frac{1}{4}} + \left (\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}\right )^{\frac{1}{4}}\right ) - 8 \, a^{2} \left (-\frac{i}{1048576 \, a^{8}}\right )^{\frac{1}{4}} \log \left (-32 \, a^{2} \left (-\frac{i}{1048576 \, a^{8}}\right )^{\frac{1}{4}} + \left (\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}\right )^{\frac{1}{4}}\right ) -{\left (4 \, a^{2} x^{2} - i \, a x + 5\right )} \left (\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}\right )^{\frac{1}{4}}}{8 \, a^{2}} \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]

-1/8*(8*a^2*(1/1048576*I/a^8)^(1/4)*log(32*a^2*(1/1048576*I/a^8)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4)
) + 8*I*a^2*(1/1048576*I/a^8)^(1/4)*log(32*I*a^2*(1/1048576*I/a^8)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/
4)) - 8*I*a^2*(1/1048576*I/a^8)^(1/4)*log(-32*I*a^2*(1/1048576*I/a^8)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^
(1/4)) - 8*a^2*(1/1048576*I/a^8)^(1/4)*log(-32*a^2*(1/1048576*I/a^8)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(
1/4)) + 8*a^2*(-1/1048576*I/a^8)^(1/4)*log(32*a^2*(-1/1048576*I/a^8)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I))^(
1/4)) + 8*I*a^2*(-1/1048576*I/a^8)^(1/4)*log(32*I*a^2*(-1/1048576*I/a^8)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*x + I
))^(1/4)) - 8*I*a^2*(-1/1048576*I/a^8)^(1/4)*log(-32*I*a^2*(-1/1048576*I/a^8)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a*
x + I))^(1/4)) - 8*a^2*(-1/1048576*I/a^8)^(1/4)*log(-32*a^2*(-1/1048576*I/a^8)^(1/4) + (I*sqrt(a^2*x^2 + 1)/(a
*x + I))^(1/4)) - (4*a^2*x^2 - I*a*x + 5)*(I*sqrt(a^2*x^2 + 1)/(a*x + I))^(1/4))/a^2

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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