\(\int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^6} \, dx\) [69]

Optimal result

Integrand size = 16, antiderivative size = 240 \[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^6} \, dx=-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}-\frac {9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}+\frac {269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}-\frac {611 a^4 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{1920 x}-\frac {31}{128} i a^5 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {31}{128} i a^5 \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \]

[Out]

-1/5*(1-I*a*x)^(3/4)*(1+I*a*x)^(1/4)/x^5-9/40*I*a*(1-I*a*x)^(3/4)*(1+I*a*x)^(1/4)/x^4+11/48*a^2*(1-I*a*x)^(3/4
)*(1+I*a*x)^(1/4)/x^3+269/960*I*a^3*(1-I*a*x)^(3/4)*(1+I*a*x)^(1/4)/x^2-611/1920*a^4*(1-I*a*x)^(3/4)*(1+I*a*x)
^(1/4)/x-31/128*I*a^5*arctan((1+I*a*x)^(1/4)/(1-I*a*x)^(1/4))-31/128*I*a^5*arctanh((1+I*a*x)^(1/4)/(1-I*a*x)^(
1/4))

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5170, 101, 156, 12, 95, 218, 212, 209} \[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^6} \, dx=-\frac {31}{128} i a^5 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {31}{128} i a^5 \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {611 a^4 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{1920 x}+\frac {269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}-\frac {9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4} \]

[In]

Int[E^((I/2)*ArcTan[a*x])/x^6,x]

[Out]

-1/5*((1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/x^5 - (((9*I)/40)*a*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/x^4 + (11*
a^2*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(48*x^3) + (((269*I)/960)*a^3*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/x^
2 - (611*a^4*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(1920*x) - ((31*I)/128)*a^5*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*
a*x)^(1/4)] - ((31*I)/128)*a^5*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 101

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -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 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 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 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 [4]{1+i a x}}{x^6 \sqrt [4]{1-i a x}} \, dx \\ & = -\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}+\frac {1}{5} \int \frac {\frac {9 i a}{2}-4 a^2 x}{x^5 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx \\ & = -\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}-\frac {9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}-\frac {1}{20} \int \frac {\frac {55 a^2}{4}+\frac {27}{2} i a^3 x}{x^4 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx \\ & = -\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}-\frac {9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}+\frac {1}{60} \int \frac {-\frac {269 i a^3}{8}+\frac {55 a^4 x}{2}}{x^3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx \\ & = -\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}-\frac {9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}+\frac {269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}-\frac {1}{120} \int \frac {-\frac {611 a^4}{16}-\frac {269}{8} i a^5 x}{x^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx \\ & = -\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}-\frac {9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}+\frac {269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}-\frac {611 a^4 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{1920 x}+\frac {1}{120} \int \frac {465 i a^5}{32 x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx \\ & = -\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}-\frac {9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}+\frac {269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}-\frac {611 a^4 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{1920 x}+\frac {1}{256} \left (31 i a^5\right ) \int \frac {1}{x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx \\ & = -\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}-\frac {9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}+\frac {269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}-\frac {611 a^4 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{1920 x}+\frac {1}{64} \left (31 i a^5\right ) \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \\ & = -\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}-\frac {9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}+\frac {269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}-\frac {611 a^4 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{1920 x}-\frac {1}{128} \left (31 i a^5\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {1}{128} \left (31 i a^5\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \\ & = -\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}-\frac {9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}+\frac {269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}-\frac {611 a^4 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{1920 x}-\frac {31}{128} i a^5 \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {31}{128} i a^5 \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \\ \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 = 111, normalized size of antiderivative = 0.46 \[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^6} \, dx=\frac {(1-i a x)^{3/4} \left (-384-816 i a x+872 a^2 x^2+978 i a^3 x^3-1149 a^4 x^4-611 i a^5 x^5-310 i a^5 x^5 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},\frac {i+a x}{i-a x}\right )\right )}{1920 x^5 (1+i a x)^{3/4}} \]

[In]

Integrate[E^((I/2)*ArcTan[a*x])/x^6,x]

[Out]

((1 - I*a*x)^(3/4)*(-384 - (816*I)*a*x + 872*a^2*x^2 + (978*I)*a^3*x^3 - 1149*a^4*x^4 - (611*I)*a^5*x^5 - (310
*I)*a^5*x^5*Hypergeometric2F1[3/4, 1, 7/4, (I + a*x)/(I - a*x)]))/(1920*x^5*(1 + I*a*x)^(3/4))

Maple [F]

\[\int \frac {\sqrt {\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}}}{x^{6}}d x\]

[In]

int(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/2)/x^6,x)

[Out]

int(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/2)/x^6,x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.83 \[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^6} \, dx=\frac {-465 i \, a^{5} x^{5} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) + 465 \, a^{5} x^{5} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) - 465 \, a^{5} x^{5} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) + 465 i \, a^{5} x^{5} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right ) - 2 \, {\left (-611 i \, a^{5} x^{5} + 73 \, a^{4} x^{4} - 98 i \, a^{3} x^{3} - 8 \, a^{2} x^{2} + 48 i \, a x + 384\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{3840 \, x^{5}} \]

[In]

integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/2)/x^6,x, algorithm="fricas")

[Out]

1/3840*(-465*I*a^5*x^5*log(sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I)) + 1) + 465*a^5*x^5*log(sqrt(I*sqrt(a^2*x^2 + 1)
/(a*x + I)) + I) - 465*a^5*x^5*log(sqrt(I*sqrt(a^2*x^2 + 1)/(a*x + I)) - I) + 465*I*a^5*x^5*log(sqrt(I*sqrt(a^
2*x^2 + 1)/(a*x + I)) - 1) - 2*(-611*I*a^5*x^5 + 73*a^4*x^4 - 98*I*a^3*x^3 - 8*a^2*x^2 + 48*I*a*x + 384)*sqrt(
I*sqrt(a^2*x^2 + 1)/(a*x + I)))/x^5

Sympy [F]

\[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^6} \, dx=\int \frac {\sqrt {\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}}{x^{6}}\, dx \]

[In]

integrate(((1+I*a*x)/(a**2*x**2+1)**(1/2))**(1/2)/x**6,x)

[Out]

Integral(sqrt(I*(a*x - I)/sqrt(a**2*x**2 + 1))/x**6, x)

Maxima [F]

\[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^6} \, dx=\int { \frac {\sqrt {\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}}}{x^{6}} \,d x } \]

[In]

integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/2)/x^6,x, algorithm="maxima")

[Out]

integrate(sqrt((I*a*x + 1)/sqrt(a^2*x^2 + 1))/x^6, x)

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^6} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(((1+I*a*x)/(a^2*x^2+1)^(1/2))^(1/2)/x^6,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. \[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^6} \, dx=\int \frac {\sqrt {\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}}}{x^6} \,d x \]

[In]

int(((a*x*1i + 1)/(a^2*x^2 + 1)^(1/2))^(1/2)/x^6,x)

[Out]

int(((a*x*1i + 1)/(a^2*x^2 + 1)^(1/2))^(1/2)/x^6, x)