\(\int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^6} \, dx\) [84]

Optimal result

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

-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*a 
rctanh((1+I*a*x)^(1/4)/(1-I*a*x)^(1/4))
 

Mathematica [C] (verified)

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

Time = 0.05 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.46 \[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^6} \, dx=\frac {(1-i a x)^{3/4} \left (-384-816 i a x+872 a^2 x^2+978 i a^3 x^3-1149 a^4 x^4-611 i a^5 x^5-310 i a^5 x^5 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},\frac {i+a x}{i-a x}\right )\right )}{1920 x^5 (1+i a x)^{3/4}} \] Input:

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

Output:

((1 - I*a*x)^(3/4)*(-384 - (816*I)*a*x + 872*a^2*x^2 + (978*I)*a^3*x^3 - 1 
149*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))
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.03, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.938, Rules used = {5585, 110, 27, 168, 27, 168, 27, 168, 27, 168, 27, 104, 756, 216, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 5585

\(\displaystyle \int \frac {\sqrt [4]{1+i a x}}{x^6 \sqrt [4]{1-i a x}}dx\)

\(\Big \downarrow \) 110

\(\displaystyle \frac {1}{5} \int \frac {a (9 i-8 a x)}{2 x^5 \sqrt [4]{1-i a x} (i a x+1)^{3/4}}dx-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} a \int \frac {9 i-8 a x}{x^5 \sqrt [4]{1-i a x} (i a x+1)^{3/4}}dx-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{10} a \left (-\frac {1}{4} \int \frac {a (54 i a x+55)}{2 x^4 \sqrt [4]{1-i a x} (i a x+1)^{3/4}}dx-\frac {9 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}\right )-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} a \left (-\frac {1}{8} a \int \frac {54 i a x+55}{x^4 \sqrt [4]{1-i a x} (i a x+1)^{3/4}}dx-\frac {9 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}\right )-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{10} a \left (-\frac {1}{8} a \left (-\frac {1}{3} \int -\frac {a (269 i-220 a x)}{2 x^3 \sqrt [4]{1-i a x} (i a x+1)^{3/4}}dx-\frac {55 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}\right )-\frac {9 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}\right )-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} a \left (-\frac {1}{8} a \left (\frac {1}{6} a \int \frac {269 i-220 a x}{x^3 \sqrt [4]{1-i a x} (i a x+1)^{3/4}}dx-\frac {55 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}\right )-\frac {9 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}\right )-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{10} a \left (-\frac {1}{8} a \left (\frac {1}{6} a \left (-\frac {1}{2} \int \frac {a (538 i a x+611)}{2 x^2 \sqrt [4]{1-i a x} (i a x+1)^{3/4}}dx-\frac {269 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{2 x^2}\right )-\frac {55 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}\right )-\frac {9 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}\right )-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} a \left (-\frac {1}{8} a \left (\frac {1}{6} a \left (-\frac {1}{4} a \int \frac {538 i a x+611}{x^2 \sqrt [4]{1-i a x} (i a x+1)^{3/4}}dx-\frac {269 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{2 x^2}\right )-\frac {55 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}\right )-\frac {9 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}\right )-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{10} a \left (-\frac {1}{8} a \left (\frac {1}{6} a \left (-\frac {1}{4} a \left (-\int -\frac {465 i a}{2 x \sqrt [4]{1-i a x} (i a x+1)^{3/4}}dx-\frac {611 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{x}\right )-\frac {269 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{2 x^2}\right )-\frac {55 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}\right )-\frac {9 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}\right )-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} a \left (-\frac {1}{8} a \left (\frac {1}{6} a \left (-\frac {1}{4} a \left (\frac {465}{2} i a \int \frac {1}{x \sqrt [4]{1-i a x} (i a x+1)^{3/4}}dx-\frac {611 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{x}\right )-\frac {269 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{2 x^2}\right )-\frac {55 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}\right )-\frac {9 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}\right )-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{10} a \left (-\frac {1}{8} a \left (\frac {1}{6} a \left (-\frac {1}{4} a \left (930 i a \int \frac {1}{\frac {i a x+1}{1-i a x}-1}d\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac {611 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{x}\right )-\frac {269 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{2 x^2}\right )-\frac {55 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}\right )-\frac {9 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}\right )-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {1}{10} a \left (-\frac {1}{8} a \left (\frac {1}{6} a \left (-\frac {1}{4} a \left (930 i a \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}}d\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac {1}{2} \int \frac {1}{\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}+1}d\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}\right )-\frac {611 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{x}\right )-\frac {269 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{2 x^2}\right )-\frac {55 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}\right )-\frac {9 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}\right )-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{10} a \left (-\frac {1}{8} a \left (\frac {1}{6} a \left (-\frac {1}{4} a \left (930 i a \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}}d\frac {\sqrt [4]{i a x+1}}{\sqrt [4]{1-i a x}}-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\right )-\frac {611 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{x}\right )-\frac {269 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{2 x^2}\right )-\frac {55 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}\right )-\frac {9 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}\right )-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{10} a \left (-\frac {1}{8} a \left (\frac {1}{6} a \left (-\frac {1}{4} a \left (930 i a \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\right )-\frac {611 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{x}\right )-\frac {269 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{2 x^2}\right )-\frac {55 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}\right )-\frac {9 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}\right )-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}\)

Input:

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

Output:

-1/5*((1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/x^5 + (a*((((-9*I)/4)*(1 - I*a* 
x)^(3/4)*(1 + I*a*x)^(1/4))/x^4 - (a*((-55*(1 - I*a*x)^(3/4)*(1 + I*a*x)^( 
1/4))/(3*x^3) + (a*((((-269*I)/2)*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/x^2 
 - (a*((-611*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/x + (930*I)*a*(-1/2*ArcT 
an[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] - ArcTanh[(1 + I*a*x)^(1/4)/(1 - I 
*a*x)^(1/4)]/2)))/4))/6))/8))/10
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[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] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 110

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

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] + S 
imp[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 216

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

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 756

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

rule 5585

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] &&  !Intege 
rQ[(I*n - 1)/2]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.83 \[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^6} \, dx=\frac {-465 i \, a^{5} x^{5} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) + 465 \, a^{5} x^{5} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) - 465 \, a^{5} x^{5} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) + 465 i \, a^{5} x^{5} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right ) - 2 \, {\left (-611 i \, a^{5} x^{5} + 73 \, a^{4} x^{4} - 98 i \, a^{3} x^{3} - 8 \, a^{2} x^{2} + 48 i \, a x + 384\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{3840 \, x^{5}} \] Input:

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

Output:

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(sqr 
t(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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Warning, need to choose a branch fo 
r the root of a polynomial with parameters. This might be wrong.The choice 
 was done
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {e^{\frac {1}{2} i \arctan (a x)}}{x^6} \, dx=\frac {-1860 \sqrt {a i x +1}\, \left (a^{2} x^{2}+1\right )^{\frac {3}{4}} a^{5} i \,x^{5}-930 \sqrt {a i x +1}\, \left (a^{2} x^{2}+1\right )^{\frac {3}{4}} a^{4} x^{4}+292 \sqrt {a i x +1}\, \left (a^{2} x^{2}+1\right )^{\frac {3}{4}} a^{3} i \,x^{3}+784 \sqrt {a i x +1}\, \left (a^{2} x^{2}+1\right )^{\frac {3}{4}} a^{2} x^{2}-96 \sqrt {a i x +1}\, \left (a^{2} x^{2}+1\right )^{\frac {3}{4}} a i x -768 \sqrt {a i x +1}\, \left (a^{2} x^{2}+1\right )^{\frac {3}{4}}+3720 \left (\int \frac {\sqrt {a i x +1}\, x}{\left (a^{2} x^{2}+1\right )^{\frac {1}{4}}}d x \right ) a^{7} i \,x^{5}+465 \left (\int \frac {\sqrt {a i x +1}\, \left (a^{2} x^{2}+1\right )^{\frac {3}{4}}}{a^{2} x^{3}+x}d x \right ) a^{5} i \,x^{5}}{3840 x^{5}} \] Input:

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

Output:

( - 1860*sqrt(a*i*x + 1)*(a**2*x**2 + 1)**(3/4)*a**5*i*x**5 - 930*sqrt(a*i 
*x + 1)*(a**2*x**2 + 1)**(3/4)*a**4*x**4 + 292*sqrt(a*i*x + 1)*(a**2*x**2 
+ 1)**(3/4)*a**3*i*x**3 + 784*sqrt(a*i*x + 1)*(a**2*x**2 + 1)**(3/4)*a**2* 
x**2 - 96*sqrt(a*i*x + 1)*(a**2*x**2 + 1)**(3/4)*a*i*x - 768*sqrt(a*i*x + 
1)*(a**2*x**2 + 1)**(3/4) + 3720*int((sqrt(a*i*x + 1)*(a**2*x**2 + 1)**(3/ 
4)*x)/(a**2*x**2 + 1),x)*a**7*i*x**5 + 465*int((sqrt(a*i*x + 1)*(a**2*x**2 
 + 1)**(3/4))/(a**2*x**3 + x),x)*a**5*i*x**5)/(3840*x**5)