\(\int \frac {e^{i \arctan (a+b x)}}{x^4} \, dx\) [184]

Optimal result

Integrand size = 16, antiderivative size = 283 \[ \int \frac {e^{i \arctan (a+b x)}}{x^4} \, dx=-\frac {\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{3 (1-i a) x^3}-\frac {(3 i-2 a) b \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{6 (1-i a) \left (1+a^2\right ) x^2}+\frac {\left (4+9 i a-2 a^2\right ) b^2 \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{6 (1-i a) \left (1+a^2\right )^2 x}+\frac {\left (2 a-i \left (1-2 a^2\right )\right ) b^3 \text {arctanh}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{(i-a)^{5/2} (i+a)^{7/2}} \] Output:

-1/3*(1-I*a-I*b*x)^(1/2)*(1+I*a+I*b*x)^(1/2)/(1-I*a)/x^3-1/6*(3*I-2*a)*b*( 
1-I*a-I*b*x)^(1/2)*(1+I*a+I*b*x)^(1/2)/(1-I*a)/(a^2+1)/x^2+1/6*(4+9*I*a-2* 
a^2)*b^2*(1-I*a-I*b*x)^(1/2)*(1+I*a+I*b*x)^(1/2)/(1-I*a)/(a^2+1)^2/x+(2*a- 
I*(-2*a^2+1))*b^3*arctanh((I+a)^(1/2)*(1+I*a+I*b*x)^(1/2)/(I-a)^(1/2)/(1-I 
*a-I*b*x)^(1/2))/(I-a)^(5/2)/(I+a)^(7/2)
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.83 \[ \int \frac {e^{i \arctan (a+b x)}}{x^4} \, dx=\frac {\frac {2 (1-i a) (-i+a) (-i+a+b x) \sqrt {1+a^2+2 a b x+b^2 x^2}}{x^3}+\frac {(1+4 i a) b (-i+a+b x) \sqrt {1+a^2+2 a b x+b^2 x^2}}{x^2}+3 i \left (1+2 i a-2 a^2\right ) b^2 \left (\frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{i x+a x}+\frac {2 b \text {arctanh}\left (\frac {\sqrt {-1-i a} \sqrt {-i (i+a+b x)}}{\sqrt {-1+i a} \sqrt {1+i a+i b x}}\right )}{\sqrt {-1-i a} (-1+i a)^{3/2}}\right )}{6 \left (1+a^2\right )^2} \] Input:

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

Output:

((2*(1 - I*a)*(-I + a)*(-I + a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])/x 
^3 + ((1 + (4*I)*a)*b*(-I + a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])/x^ 
2 + (3*I)*(1 + (2*I)*a - 2*a^2)*b^2*(Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]/(I* 
x + a*x) + (2*b*ArcTanh[(Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)])/(Sqrt[-1 
 + I*a]*Sqrt[1 + I*a + I*b*x])])/(Sqrt[-1 - I*a]*(-1 + I*a)^(3/2))))/(6*(1 
 + a^2)^2)
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {5618, 110, 27, 168, 27, 168, 27, 104, 221}

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.

\(\displaystyle \int \frac {e^{i \arctan (a+b x)}}{x^4} \, dx\)

\(\Big \downarrow \) 5618

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

\(\Big \downarrow \) 110

\(\displaystyle \frac {\int \frac {b (-2 a-2 b x+3 i)}{x^3 \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}dx}{3 (1-i a)}-\frac {\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{3 (1-i a) x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \int \frac {-2 a-2 b x+3 i}{x^3 \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}dx}{3 (1-i a)}-\frac {\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{3 (1-i a) x^3}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {b \left (-\frac {\int \frac {b \left (-2 a^2+9 i a+(3 i-2 a) b x+4\right )}{x^2 \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}dx}{2 \left (a^2+1\right )}-\frac {(-2 a+3 i) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 \left (a^2+1\right ) x^2}\right )}{3 (1-i a)}-\frac {\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{3 (1-i a) x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (-\frac {b \int \frac {-2 a^2+9 i a+(3 i-2 a) b x+4}{x^2 \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}dx}{2 \left (a^2+1\right )}-\frac {(-2 a+3 i) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 \left (a^2+1\right ) x^2}\right )}{3 (1-i a)}-\frac {\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{3 (1-i a) x^3}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {b \left (-\frac {b \left (-\frac {\int -\frac {3 \left (-2 i a^2-2 a+i\right ) b}{x \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}dx}{a^2+1}-\frac {\left (-2 a^2+9 i a+4\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{\left (a^2+1\right ) x}\right )}{2 \left (a^2+1\right )}-\frac {(-2 a+3 i) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 \left (a^2+1\right ) x^2}\right )}{3 (1-i a)}-\frac {\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{3 (1-i a) x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (-\frac {b \left (\frac {3 \left (-2 i a^2-2 a+i\right ) b \int \frac {1}{x \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}dx}{a^2+1}-\frac {\left (-2 a^2+9 i a+4\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{\left (a^2+1\right ) x}\right )}{2 \left (a^2+1\right )}-\frac {(-2 a+3 i) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 \left (a^2+1\right ) x^2}\right )}{3 (1-i a)}-\frac {\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{3 (1-i a) x^3}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {b \left (-\frac {b \left (\frac {6 \left (-2 i a^2-2 a+i\right ) b \int \frac {1}{-i a+\frac {(1-i a) (i a+i b x+1)}{-i a-i b x+1}-1}d\frac {\sqrt {i a+i b x+1}}{\sqrt {-i a-i b x+1}}}{a^2+1}-\frac {\left (-2 a^2+9 i a+4\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{\left (a^2+1\right ) x}\right )}{2 \left (a^2+1\right )}-\frac {(-2 a+3 i) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 \left (a^2+1\right ) x^2}\right )}{3 (1-i a)}-\frac {\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{3 (1-i a) x^3}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {b \left (-\frac {b \left (-\frac {6 i \left (-2 i a^2-2 a+i\right ) b \text {arctanh}\left (\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-a+i} \sqrt {-i a-i b x+1}}\right )}{\sqrt {-a+i} \sqrt {a+i} \left (a^2+1\right )}-\frac {\left (-2 a^2+9 i a+4\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{\left (a^2+1\right ) x}\right )}{2 \left (a^2+1\right )}-\frac {(-2 a+3 i) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 \left (a^2+1\right ) x^2}\right )}{3 (1-i a)}-\frac {\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{3 (1-i a) x^3}\)

Input:

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

Output:

-1/3*(Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/((1 - I*a)*x^3) + (b*(- 
1/2*((3*I - 2*a)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/((1 + a^2)*x 
^2) - (b*(-(((4 + (9*I)*a - 2*a^2)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I* 
b*x])/((1 + a^2)*x)) - ((6*I)*(I - 2*a - (2*I)*a^2)*b*ArcTanh[(Sqrt[I + a] 
*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a - I*b*x])])/(Sqrt[I - a] 
*Sqrt[I + a]*(1 + a^2))))/(2*(1 + a^2))))/(3*(1 - I*a))
 

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 221

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

rule 5618

Int[E^(ArcTan[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.), 
 x_Symbol] :> Int[(d + e*x)^m*((1 - I*a*c - I*b*c*x)^(I*(n/2))/(1 + I*a*c + 
 I*b*c*x)^(I*(n/2))), x] /; FreeQ[{a, b, c, d, e, m, n}, x]
 

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.99

method	result	size
risch	\(-\frac {i \left (2 a^{2} b^{4} x^{4}-9 i a \,b^{4} x^{4}+2 a^{3} b^{3} x^{3}-15 i a^{2} b^{3} x^{3}-3 i a^{3} b^{2} x^{2}-4 b^{4} x^{4}+2 a^{5} b x +3 i a^{4} b x -10 a \,b^{3} x^{3}+3 i b^{3} x^{3}+2 a^{6}-2 a^{2} b^{2} x^{2}-3 i a \,b^{2} x^{2}+4 a^{3} b x +6 i a^{2} b x +6 a^{4}-2 b^{2} x^{2}+2 a b x +3 i b x +6 a^{2}+2\right )}{6 x^{3} \left (-i+a \right )^{2} \left (i+a \right )^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {b^{3} \left (2 a^{2}-2 i a -1\right ) \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{2 \left (a^{2}+1\right )^{\frac {5}{2}} \left (i+a \right )}\)	\(281\)
default	\(\left (i a +1\right ) \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{3 \left (a^{2}+1\right ) x^{3}}-\frac {5 a b \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 \left (a^{2}+1\right ) x^{2}}-\frac {3 a b \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) x}+\frac {a b \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+1\right )}+\frac {b^{2} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{2 \left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{3 \left (a^{2}+1\right )}-\frac {2 b^{2} \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) x}+\frac {a b \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{3 \left (a^{2}+1\right )}\right )+i b \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 \left (a^{2}+1\right ) x^{2}}-\frac {3 a b \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) x}+\frac {a b \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+1\right )}+\frac {b^{2} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{2 \left (a^{2}+1\right )^{\frac {3}{2}}}\right )\)	\(529\)

Input:

int((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2)/x^4,x,method=_RETURNVERBOSE)
 

Output:

-1/6*I*(-3*I*a^3*b^2*x^2+2*a^2*b^4*x^4-3*I*a*b^2*x^2+2*a^3*b^3*x^3+3*I*b^3 
*x^3-4*b^4*x^4+3*I*b*x+3*I*a^4*b*x+2*a^5*b*x-10*a*b^3*x^3+6*I*a^2*b*x+2*a^ 
6-2*a^2*b^2*x^2-15*I*a^2*b^3*x^3+4*a^3*b*x+6*a^4-2*b^2*x^2-9*I*a*b^4*x^4+2 
*a*b*x+6*a^2+2)/x^3/(-I+a)^2/(I+a)^3/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/2*b^3 
*(-2*I*a+2*a^2-1)/(a^2+1)^(5/2)/(I+a)*ln((2*a^2+2+2*a*b*x+2*(a^2+1)^(1/2)* 
(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/x)
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 690 vs. \(2 (198) = 396\).

Time = 0.09 (sec) , antiderivative size = 690, normalized size of antiderivative = 2.44 \[ \int \frac {e^{i \arctan (a+b x)}}{x^4} \, dx =\text {Too large to display} \] Input:

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

Output:

1/6*((-2*I*a^2 - 9*a + 4*I)*b^3*x^3 - 3*sqrt((4*a^4 - 8*I*a^3 - 8*a^2 + 4* 
I*a + 1)*b^6/(a^12 + 2*I*a^11 + 4*a^10 + 10*I*a^9 + 5*a^8 + 20*I*a^7 + 20* 
I*a^5 - 5*a^4 + 10*I*a^3 - 4*a^2 + 2*I*a - 1))*(a^5 + I*a^4 + 2*a^3 + 2*I* 
a^2 + a + I)*x^3*log(-((2*a^2 - 2*I*a - 1)*b^4*x - sqrt(b^2*x^2 + 2*a*b*x 
+ a^2 + 1)*(2*a^2 - 2*I*a - 1)*b^3 + (a^7 + I*a^6 + 3*a^5 + 3*I*a^4 + 3*a^ 
3 + 3*I*a^2 + a + I)*sqrt((4*a^4 - 8*I*a^3 - 8*a^2 + 4*I*a + 1)*b^6/(a^12 
+ 2*I*a^11 + 4*a^10 + 10*I*a^9 + 5*a^8 + 20*I*a^7 + 20*I*a^5 - 5*a^4 + 10* 
I*a^3 - 4*a^2 + 2*I*a - 1)))/((2*a^2 - 2*I*a - 1)*b^3)) + 3*sqrt((4*a^4 - 
8*I*a^3 - 8*a^2 + 4*I*a + 1)*b^6/(a^12 + 2*I*a^11 + 4*a^10 + 10*I*a^9 + 5* 
a^8 + 20*I*a^7 + 20*I*a^5 - 5*a^4 + 10*I*a^3 - 4*a^2 + 2*I*a - 1))*(a^5 + 
I*a^4 + 2*a^3 + 2*I*a^2 + a + I)*x^3*log(-((2*a^2 - 2*I*a - 1)*b^4*x - sqr 
t(b^2*x^2 + 2*a*b*x + a^2 + 1)*(2*a^2 - 2*I*a - 1)*b^3 - (a^7 + I*a^6 + 3* 
a^5 + 3*I*a^4 + 3*a^3 + 3*I*a^2 + a + I)*sqrt((4*a^4 - 8*I*a^3 - 8*a^2 + 4 
*I*a + 1)*b^6/(a^12 + 2*I*a^11 + 4*a^10 + 10*I*a^9 + 5*a^8 + 20*I*a^7 + 20 
*I*a^5 - 5*a^4 + 10*I*a^3 - 4*a^2 + 2*I*a - 1)))/((2*a^2 - 2*I*a - 1)*b^3) 
) + ((-2*I*a^2 - 9*a + 4*I)*b^2*x^2 - 2*I*a^4 + (2*I*a^3 + 3*a^2 + 2*I*a + 
 3)*b*x - 4*I*a^2 - 2*I)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))/((a^5 + I*a^4 
+ 2*a^3 + 2*I*a^2 + a + I)*x^3)
 

Sympy [F]

\[ \int \frac {e^{i \arctan (a+b x)}}{x^4} \, dx=i \left (\int \left (- \frac {i}{x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {a}{x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \frac {b}{x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx\right ) \] Input:

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

Output:

I*(Integral(-I/(x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral( 
a/(x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(b/(x**3*sqrt( 
a**2 + 2*a*b*x + b**2*x**2 + 1)), x))
 

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 644 vs. \(2 (198) = 396\).

Time = 0.04 (sec) , antiderivative size = 644, normalized size of antiderivative = 2.28 \[ \int \frac {e^{i \arctan (a+b x)}}{x^4} \, dx =\text {Too large to display} \] Input:

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

Output:

5/2*a^3*(I*a + 1)*b^3*arcsinh(2*a*b*x/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)* 
abs(x)) + 2*a^2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2/(sqrt(-4*a 
^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)))/(a^2 + 1)^(7/2) - 3/2*I*a^2*b^3*arcsinh 
(2*a*b*x/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2*a^2/(sqrt(-4*a^2* 
b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs 
(x)))/(a^2 + 1)^(5/2) - 3/2*a*(I*a + 1)*b^3*arcsinh(2*a*b*x/(sqrt(-4*a^2*b 
^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2*a^2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)* 
abs(x)) + 2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)))/(a^2 + 1)^(5/2) + 
 1/2*I*b^3*arcsinh(2*a*b*x/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2 
*a^2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2/(sqrt(-4*a^2*b^2 + 4* 
(a^2 + 1)*b^2)*abs(x)))/(a^2 + 1)^(3/2) - 5/2*sqrt(b^2*x^2 + 2*a*b*x + a^2 
 + 1)*a^2*(I*a + 1)*b^2/((a^2 + 1)^3*x) + 3/2*I*sqrt(b^2*x^2 + 2*a*b*x + a 
^2 + 1)*a*b^2/((a^2 + 1)^2*x) - 2/3*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(-I* 
a - 1)*b^2/((a^2 + 1)^2*x) + 5/6*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a*(I*a 
+ 1)*b/((a^2 + 1)^2*x^2) - 1/2*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b/((a^2 
 + 1)*x^2) - 1/3*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(I*a + 1)/((a^2 + 1)*x^ 
3)
 

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 884 vs. \(2 (198) = 396\).

Time = 0.17 (sec) , antiderivative size = 884, normalized size of antiderivative = 3.12 \[ \int \frac {e^{i \arctan (a+b x)}}{x^4} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*(2*a^2*b^3 - 2*I*a*b^3 - b^3)*log(abs(2*x*abs(b) - 2*sqrt((b*x + a)^2 
+ 1) - 2*sqrt(a^2 + 1))/abs(2*x*abs(b) - 2*sqrt((b*x + a)^2 + 1) + 2*sqrt( 
a^2 + 1)))/((a^5 + I*a^4 + 2*a^3 + 2*I*a^2 + a + I)*sqrt(a^2 + 1)) + 1/3*( 
8*I*(x*abs(b) - sqrt((b*x + a)^2 + 1))^3*a^5*b^3 + 24*(I*x*abs(b) - I*sqrt 
((b*x + a)^2 + 1))*a^7*b^3 + 24*I*(x*abs(b) - sqrt((b*x + a)^2 + 1))^2*a^6 
*b^2*abs(b) + 8*I*a^8*b^2*abs(b) + 6*(x*abs(b) - sqrt((b*x + a)^2 + 1))^5* 
a^2*b^3 - 24*(x*abs(b) - sqrt((b*x + a)^2 + 1))^3*a^4*b^3 + 18*(x*abs(b) - 
 sqrt((b*x + a)^2 + 1))*a^6*b^3 - 12*(x*abs(b) - sqrt((b*x + a)^2 + 1))^2* 
a^5*b^2*abs(b) + 12*a^7*b^2*abs(b) - 6*I*(x*abs(b) - sqrt((b*x + a)^2 + 1) 
)^5*a*b^3 + 32*I*(x*abs(b) - sqrt((b*x + a)^2 + 1))^3*a^3*b^3 + 54*(I*x*ab 
s(b) - I*sqrt((b*x + a)^2 + 1))*a^5*b^3 + 60*I*(x*abs(b) - sqrt((b*x + a)^ 
2 + 1))^2*a^4*b^2*abs(b) + 20*I*a^6*b^2*abs(b) - 3*(x*abs(b) - sqrt((b*x + 
 a)^2 + 1))^5*b^3 - 24*(x*abs(b) - sqrt((b*x + a)^2 + 1))^3*a^2*b^3 + 39*( 
x*abs(b) - sqrt((b*x + a)^2 + 1))*a^4*b^3 - 24*(x*abs(b) - sqrt((b*x + a)^ 
2 + 1))^2*a^3*b^2*abs(b) + 36*a^5*b^2*abs(b) + 24*I*(x*abs(b) - sqrt((b*x 
+ a)^2 + 1))^3*a*b^3 + 36*(I*x*abs(b) - I*sqrt((b*x + a)^2 + 1))*a^3*b^3 + 
 48*I*(x*abs(b) - sqrt((b*x + a)^2 + 1))^2*a^2*b^2*abs(b) + 12*I*a^4*b^2*a 
bs(b) + 24*(x*abs(b) - sqrt((b*x + a)^2 + 1))*a^2*b^3 - 12*(x*abs(b) - sqr 
t((b*x + a)^2 + 1))^2*a*b^2*abs(b) + 36*a^3*b^2*abs(b) + 6*(I*x*abs(b) - I 
*sqrt((b*x + a)^2 + 1))*a*b^3 + 12*I*(x*abs(b) - sqrt((b*x + a)^2 + 1))...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{i \arctan (a+b x)}}{x^4} \, dx=\int \frac {1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}{x^4\,\sqrt {{\left (a+b\,x\right )}^2+1}} \,d x \] Input:

int((a*1i + b*x*1i + 1)/(x^4*((a + b*x)^2 + 1)^(1/2)),x)
                                                                                    
                                                                                    
 

Output:

int((a*1i + b*x*1i + 1)/(x^4*((a + b*x)^2 + 1)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 852, normalized size of antiderivative = 3.01 \[ \int \frac {e^{i \arctan (a+b x)}}{x^4} \, dx=\frac {-2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{2} i -a \,b^{3} x^{3}-b^{3} i \,x^{3}+5 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{6} b x -11 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{5} b^{2} x^{2}-6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{5}-6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{3}-7 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{3} b^{2} x^{2}+4 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a \,b^{2} x^{2}+2 a^{6} b^{3} i \,x^{3}-7 a^{4} b^{3} i \,x^{3}-10 a^{2} b^{3} i \,x^{3}-2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{8} i +9 a^{5} b^{3} x^{3}+8 a^{3} b^{3} x^{3}+10 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{4} b x +5 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{2} b x -6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{6} i -6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{4} i -24 \sqrt {a^{2}+1}\, \mathit {atan} \left (\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, i +b i x}{\sqrt {a^{2}+1}}\right ) a^{3} b^{3} x^{3}+6 \sqrt {a^{2}+1}\, \mathit {atan} \left (\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, i +b i x}{\sqrt {a^{2}+1}}\right ) a \,b^{3} x^{3}+2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{7} b i x -2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{6} b^{2} i \,x^{2}+11 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{4} b^{2} i \,x^{2}+13 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{2} b^{2} i \,x^{2}-2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{7}-3 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a b i x -2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a +\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{5} b i x -4 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{3} b i x -12 \sqrt {a^{2}+1}\, \mathit {atan} \left (\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, i +b i x}{\sqrt {a^{2}+1}}\right ) a^{4} b^{3} i \,x^{3}+18 \sqrt {a^{2}+1}\, \mathit {atan} \left (\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, i +b i x}{\sqrt {a^{2}+1}}\right ) a^{2} b^{3} i \,x^{3}}{6 a \,x^{3} \left (a^{8}+4 a^{6}+6 a^{4}+4 a^{2}+1\right )} \] Input:

int((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2)/x^4,x)
 

Output:

( - 12*sqrt(a**2 + 1)*atan((sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*i + b*i*x 
)/sqrt(a**2 + 1))*a**4*b**3*i*x**3 - 24*sqrt(a**2 + 1)*atan((sqrt(a**2 + 2 
*a*b*x + b**2*x**2 + 1)*i + b*i*x)/sqrt(a**2 + 1))*a**3*b**3*x**3 + 18*sqr 
t(a**2 + 1)*atan((sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*i + b*i*x)/sqrt(a** 
2 + 1))*a**2*b**3*i*x**3 + 6*sqrt(a**2 + 1)*atan((sqrt(a**2 + 2*a*b*x + b* 
*2*x**2 + 1)*i + b*i*x)/sqrt(a**2 + 1))*a*b**3*x**3 - 2*sqrt(a**2 + 2*a*b* 
x + b**2*x**2 + 1)*a**8*i + 2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a**7*b* 
i*x - 2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a**7 - 2*sqrt(a**2 + 2*a*b*x 
+ b**2*x**2 + 1)*a**6*b**2*i*x**2 + 5*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) 
*a**6*b*x - 6*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a**6*i - 11*sqrt(a**2 + 
 2*a*b*x + b**2*x**2 + 1)*a**5*b**2*x**2 + sqrt(a**2 + 2*a*b*x + b**2*x**2 
 + 1)*a**5*b*i*x - 6*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a**5 + 11*sqrt(a 
**2 + 2*a*b*x + b**2*x**2 + 1)*a**4*b**2*i*x**2 + 10*sqrt(a**2 + 2*a*b*x + 
 b**2*x**2 + 1)*a**4*b*x - 6*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a**4*i - 
 7*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a**3*b**2*x**2 - 4*sqrt(a**2 + 2*a 
*b*x + b**2*x**2 + 1)*a**3*b*i*x - 6*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)* 
a**3 + 13*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a**2*b**2*i*x**2 + 5*sqrt(a 
**2 + 2*a*b*x + b**2*x**2 + 1)*a**2*b*x - 2*sqrt(a**2 + 2*a*b*x + b**2*x** 
2 + 1)*a**2*i + 4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a*b**2*x**2 - 3*sqr 
t(a**2 + 2*a*b*x + b**2*x**2 + 1)*a*b*i*x - 2*sqrt(a**2 + 2*a*b*x + b**...