\(\int e^{\frac {1}{3} i \arctan (x)} x \, dx\) [116]

Optimal result

Integrand size = 12, antiderivative size = 278 \[ \int e^{\frac {1}{3} i \arctan (x)} x \, dx=\frac {1}{6} (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac {1}{2} (1-i x)^{5/6} (1+i x)^{7/6}-\frac {1}{18} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{18} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{9} \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {\log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )}{12 \sqrt {3}}-\frac {\log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )}{12 \sqrt {3}} \]

[Out]

1/6*(1-I*x)^(5/6)*(1+I*x)^(1/6)+1/2*(1-I*x)^(5/6)*(1+I*x)^(7/6)+1/9*arctan((1-I*x)^(1/6)/(1+I*x)^(1/6))+1/18*a
rctan(2*(1-I*x)^(1/6)/(1+I*x)^(1/6)-3^(1/2))+1/18*arctan(2*(1-I*x)^(1/6)/(1+I*x)^(1/6)+3^(1/2))+1/36*ln(1+(1-I
*x)^(1/3)/(1+I*x)^(1/3)-(1-I*x)^(1/6)*3^(1/2)/(1+I*x)^(1/6))*3^(1/2)-1/36*ln(1+(1-I*x)^(1/3)/(1+I*x)^(1/3)+(1-
I*x)^(1/6)*3^(1/2)/(1+I*x)^(1/6))*3^(1/2)

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {5170, 81, 52, 65, 338, 301, 648, 632, 210, 642, 209} \[ \int e^{\frac {1}{3} i \arctan (x)} x \, dx=-\frac {1}{18} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{18} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{9} \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{2} (1-i x)^{5/6} (1+i x)^{7/6}+\frac {1}{6} (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac {\log \left (\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}+1\right )}{12 \sqrt {3}}-\frac {\log \left (\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}+1\right )}{12 \sqrt {3}} \]

[In]

Int[E^((I/3)*ArcTan[x])*x,x]

[Out]

((1 - I*x)^(5/6)*(1 + I*x)^(1/6))/6 + ((1 - I*x)^(5/6)*(1 + I*x)^(7/6))/2 - ArcTan[Sqrt[3] - (2*(1 - I*x)^(1/6
))/(1 + I*x)^(1/6)]/18 + ArcTan[Sqrt[3] + (2*(1 - I*x)^(1/6))/(1 + I*x)^(1/6)]/18 + ArcTan[(1 - I*x)^(1/6)/(1
+ I*x)^(1/6)]/9 + Log[1 + (1 - I*x)^(1/3)/(1 + I*x)^(1/3) - (Sqrt[3]*(1 - I*x)^(1/6))/(1 + I*x)^(1/6)]/(12*Sqr
t[3]) - Log[1 + (1 - I*x)^(1/3)/(1 + I*x)^(1/3) + (Sqrt[3]*(1 - I*x)^(1/6))/(1 + I*x)^(1/6)]/(12*Sqrt[3])

Rule 52

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 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 81

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 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 301

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/
b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(
2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 +
2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x]; 2*(-1)^(m/2)*(r^(m + 2)/(a*n*s^m))*Int[1/(r^2 + s^2*x^2), x] +
Dist[2*(r^(m + 1)/(a*n*s^m)), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] &&
IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]

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 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 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 [6]{1+i x} x}{\sqrt [6]{1-i x}} \, dx \\ & = \frac {1}{2} (1-i x)^{5/6} (1+i x)^{7/6}-\frac {1}{6} i \int \frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}} \, dx \\ & = \frac {1}{6} (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac {1}{2} (1-i x)^{5/6} (1+i x)^{7/6}-\frac {1}{18} i \int \frac {1}{\sqrt [6]{1-i x} (1+i x)^{5/6}} \, dx \\ & = \frac {1}{6} (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac {1}{2} (1-i x)^{5/6} (1+i x)^{7/6}+\frac {1}{3} \text {Subst}\left (\int \frac {x^4}{\left (2-x^6\right )^{5/6}} \, dx,x,\sqrt [6]{1-i x}\right ) \\ & = \frac {1}{6} (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac {1}{2} (1-i x)^{5/6} (1+i x)^{7/6}+\frac {1}{3} \text {Subst}\left (\int \frac {x^4}{1+x^6} \, dx,x,\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right ) \\ & = \frac {1}{6} (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac {1}{2} (1-i x)^{5/6} (1+i x)^{7/6}+\frac {1}{9} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{9} \text {Subst}\left (\int \frac {-\frac {1}{2}+\frac {\sqrt {3} x}{2}}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{9} \text {Subst}\left (\int \frac {-\frac {1}{2}-\frac {\sqrt {3} x}{2}}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right ) \\ & = \frac {1}{6} (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac {1}{2} (1-i x)^{5/6} (1+i x)^{7/6}+\frac {1}{9} \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{36} \text {Subst}\left (\int \frac {1}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{36} \text {Subst}\left (\int \frac {1}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {\text {Subst}\left (\int \frac {-\sqrt {3}+2 x}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )}{12 \sqrt {3}}-\frac {\text {Subst}\left (\int \frac {\sqrt {3}+2 x}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )}{12 \sqrt {3}} \\ & = \frac {1}{6} (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac {1}{2} (1-i x)^{5/6} (1+i x)^{7/6}+\frac {1}{9} \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {\log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )}{12 \sqrt {3}}-\frac {\log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )}{12 \sqrt {3}}-\frac {1}{18} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\frac {1}{18} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right ) \\ & = \frac {1}{6} (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac {1}{2} (1-i x)^{5/6} (1+i x)^{7/6}-\frac {1}{18} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{18} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{9} \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {\log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )}{12 \sqrt {3}}-\frac {\log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )}{12 \sqrt {3}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.21 \[ \int e^{\frac {1}{3} i \arctan (x)} x \, dx=\frac {1}{10} (1-i x)^{5/6} \left (5 (1+i x)^{7/6}+2 \sqrt [6]{2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {5}{6},\frac {11}{6},\frac {1}{2}-\frac {i x}{2}\right )\right ) \]

[In]

Integrate[E^((I/3)*ArcTan[x])*x,x]

[Out]

((1 - I*x)^(5/6)*(5*(1 + I*x)^(7/6) + 2*2^(1/6)*Hypergeometric2F1[-1/6, 5/6, 11/6, 1/2 - (I/2)*x]))/10

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.70 \[ \int e^{\frac {1}{3} i \arctan (x)} x \, dx=-\frac {1}{36} \, {\left (\sqrt {3} + i\right )} \log \left (\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2} i\right ) - \frac {1}{36} \, {\left (\sqrt {3} - i\right )} \log \left (\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2} i\right ) + \frac {1}{36} \, {\left (\sqrt {3} - i\right )} \log \left (-\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2} i\right ) + \frac {1}{36} \, {\left (\sqrt {3} + i\right )} \log \left (-\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2} i\right ) + \frac {1}{6} \, {\left (3 \, x^{2} - i \, x + 4\right )} \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{18} i \, \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + i\right ) + \frac {1}{18} i \, \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - i\right ) \]

[In]

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

[Out]

-1/36*(sqrt(3) + I)*log(1/2*sqrt(3) + (I*sqrt(x^2 + 1)/(x + I))^(1/3) + 1/2*I) - 1/36*(sqrt(3) - I)*log(1/2*sq
rt(3) + (I*sqrt(x^2 + 1)/(x + I))^(1/3) - 1/2*I) + 1/36*(sqrt(3) - I)*log(-1/2*sqrt(3) + (I*sqrt(x^2 + 1)/(x +
I))^(1/3) + 1/2*I) + 1/36*(sqrt(3) + I)*log(-1/2*sqrt(3) + (I*sqrt(x^2 + 1)/(x + I))^(1/3) - 1/2*I) + 1/6*(3*
x^2 - I*x + 4)*(I*sqrt(x^2 + 1)/(x + I))^(1/3) - 1/18*I*log((I*sqrt(x^2 + 1)/(x + I))^(1/3) + I) + 1/18*I*log(
(I*sqrt(x^2 + 1)/(x + I))^(1/3) - I)

Sympy [F]

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

[In]

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

[Out]

Integral(x*(I*(x - I)/sqrt(x**2 + 1))**(1/3), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int e^{\frac {1}{3} i \arctan (x)} x \, dx=\int x\,{\left (\frac {1+x\,1{}\mathrm {i}}{\sqrt {x^2+1}}\right )}^{1/3} \,d x \]

[In]

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

[Out]

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