\(\int \frac {e^{3 i \arctan (a+b x)}}{x^3} \, dx\) [201]

Optimal result

Integrand size = 16, antiderivative size = 264 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x^3} \, dx=\frac {3 (3 i-2 a) b^2 \sqrt {1+i a+i b x}}{(1+i a) (i+a)^3 \sqrt {1-i a-i b x}}+\frac {(3 i-2 a) b (1+i a+i b x)^{3/2}}{2 (1+i a) (i+a)^2 x \sqrt {1-i a-i b x}}-\frac {(1+i a+i b x)^{5/2}}{2 \left (1+a^2\right ) x^2 \sqrt {1-i a-i b x}}+\frac {3 (3+2 i a) b^2 \text {arctanh}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{\sqrt {i-a} (i+a)^{7/2}} \] Output:

3*(3*I-2*a)*b^2*(1+I*a+I*b*x)^(1/2)/(1+I*a)/(I+a)^3/(1-I*a-I*b*x)^(1/2)+1/ 
2*(3*I-2*a)*b*(1+I*a+I*b*x)^(3/2)/(1+I*a)/(I+a)^2/x/(1-I*a-I*b*x)^(1/2)-1/ 
2*(1+I*a+I*b*x)^(5/2)/(a^2+1)/x^2/(1-I*a-I*b*x)^(1/2)+3*(3+2*I*a)*b^2*arct 
anh((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) 
^(1/2)/(I+a)^(7/2)
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.73 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x^3} \, dx=\frac {\frac {\sqrt {1+i a+i b x} \left (i+a+i a^2+a^3-5 b x+5 i a b x+14 i b^2 x^2-a b^2 x^2\right )}{x^2 \sqrt {-i (i+a+b x)}}-\frac {6 i \sqrt {-1-i a} (-3 i+2 a) b^2 \text {arctanh}\left (\frac {\sqrt {-1-i a} \sqrt {-i (i+a+b x)}}{\sqrt {-1+i a} \sqrt {1+i a+i b x}}\right )}{\sqrt {-1+i a} (-i+a)}}{2 (i+a)^3} \] Input:

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

Output:

((Sqrt[1 + I*a + I*b*x]*(I + a + I*a^2 + a^3 - 5*b*x + (5*I)*a*b*x + (14*I 
)*b^2*x^2 - a*b^2*x^2))/(x^2*Sqrt[(-I)*(I + a + b*x)]) - ((6*I)*Sqrt[-1 - 
I*a]*(-3*I + 2*a)*b^2*ArcTanh[(Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)])/(S 
qrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x])])/(Sqrt[-1 + I*a]*(-I + a)))/(2*(I + 
a)^3)
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5618, 107, 105, 105, 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.

\(\displaystyle \int \frac {e^{3 i \arctan (a+b x)}}{x^3} \, dx\)

\(\Big \downarrow \) 5618

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

\(\Big \downarrow \) 107

\(\displaystyle \frac {(-2 a+3 i) b \int \frac {(i a+i b x+1)^{3/2}}{x^2 (-i a-i b x+1)^{3/2}}dx}{2 \left (a^2+1\right )}-\frac {(i a+i b x+1)^{5/2}}{2 \left (a^2+1\right ) x^2 \sqrt {-i a-i b x+1}}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {(-2 a+3 i) b \left (-\frac {3 b \int \frac {\sqrt {i a+i b x+1}}{x (-i a-i b x+1)^{3/2}}dx}{a+i}-\frac {(i a+i b x+1)^{3/2}}{(1-i a) x \sqrt {-i a-i b x+1}}\right )}{2 \left (a^2+1\right )}-\frac {(i a+i b x+1)^{5/2}}{2 \left (a^2+1\right ) x^2 \sqrt {-i a-i b x+1}}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {(-2 a+3 i) b \left (-\frac {3 b \left (\frac {(-a+i) \int \frac {1}{x \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}dx}{a+i}+\frac {2 \sqrt {i a+i b x+1}}{(1-i a) \sqrt {-i a-i b x+1}}\right )}{a+i}-\frac {(i a+i b x+1)^{3/2}}{(1-i a) x \sqrt {-i a-i b x+1}}\right )}{2 \left (a^2+1\right )}-\frac {(i a+i b x+1)^{5/2}}{2 \left (a^2+1\right ) x^2 \sqrt {-i a-i b x+1}}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {(-2 a+3 i) b \left (-\frac {3 b \left (\frac {2 (-a+i) \int \frac {1}{-i a+\frac {(1-i a) (i a+i b x+1)}{-i a-i b x+1}-1}d\frac {\sqrt {i a+i b x+1}}{\sqrt {-i a-i b x+1}}}{a+i}+\frac {2 \sqrt {i a+i b x+1}}{(1-i a) \sqrt {-i a-i b x+1}}\right )}{a+i}-\frac {(i a+i b x+1)^{3/2}}{(1-i a) x \sqrt {-i a-i b x+1}}\right )}{2 \left (a^2+1\right )}-\frac {(i a+i b x+1)^{5/2}}{2 \left (a^2+1\right ) x^2 \sqrt {-i a-i b x+1}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(-2 a+3 i) b \left (-\frac {3 b \left (\frac {2 \sqrt {i a+i b x+1}}{(1-i a) \sqrt {-i a-i b x+1}}-\frac {2 i \sqrt {-a+i} \text {arctanh}\left (\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-a+i} \sqrt {-i a-i b x+1}}\right )}{(a+i)^{3/2}}\right )}{a+i}-\frac {(i a+i b x+1)^{3/2}}{(1-i a) x \sqrt {-i a-i b x+1}}\right )}{2 \left (a^2+1\right )}-\frac {(i a+i b x+1)^{5/2}}{2 \left (a^2+1\right ) x^2 \sqrt {-i a-i b x+1}}\)

Input:

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

Output:

-1/2*(1 + I*a + I*b*x)^(5/2)/((1 + a^2)*x^2*Sqrt[1 - I*a - I*b*x]) + ((3*I 
 - 2*a)*b*(-((1 + I*a + I*b*x)^(3/2)/((1 - I*a)*x*Sqrt[1 - I*a - I*b*x])) 
- (3*b*((2*Sqrt[1 + I*a + I*b*x])/((1 - I*a)*Sqrt[1 - I*a - I*b*x]) - ((2* 
I)*Sqrt[I - a]*ArcTanh[(Sqrt[I + a]*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sq 
rt[1 - I*a - I*b*x])])/(I + a)^(3/2)))/(I + a)))/(2*(1 + a^2))
 

Defintions of rubi rules used

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 105

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[n*((d*e - c*f)/((m + 1)*(b*e - a*f)))   Int[(a 
+ b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, 
e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] 
 ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]
 

rule 107

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(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] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 
 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x 
] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[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 = 1.23 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.02

method	result	size
risch	\(\frac {i \left (-a \,b^{3} x^{3}+6 i b^{3} x^{3}-a^{2} b^{2} x^{2}+12 i a \,b^{2} x^{2}+a^{3} b x +6 i a^{2} b x +a^{4}+b^{2} x^{2}+a b x +6 i b x +2 a^{2}+1\right )}{2 x^{2} \left (i+a \right )^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {b^{2} \left (-\frac {3 \left (-2 a^{2}+i a -3\right ) \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (i+a \right ) \sqrt {a^{2}+1}}+\frac {8 i \left (i a -1\right ) \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}}{b \left (i+a \right ) \left (x +\frac {i+a}{b}\right )}\right )}{2 a^{3}+6 i a^{2}-6 a -2 i}\)	\(269\)
default	\(-\frac {2 i b^{3} \left (2 b^{2} x +2 a b \right )}{\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\left (-i a^{3}-3 a^{2}+3 i a +1\right ) \left (-\frac {1}{2 \left (a^{2}+1\right ) x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {5 a b \left (-\frac {1}{\left (a^{2}+1\right ) x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {3 a b \left (\frac {1}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a b \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\right ) \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {\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 )}{a^{2}+1}-\frac {4 b^{2} \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\right ) \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}\right )}{2 \left (a^{2}+1\right )}-\frac {3 b^{2} \left (\frac {1}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a b \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\right ) \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {\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 )}\right )-3 b \left (i a^{2}+2 a -i\right ) \left (-\frac {1}{\left (a^{2}+1\right ) x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {3 a b \left (\frac {1}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a b \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\right ) \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {\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 )}{a^{2}+1}-\frac {4 b^{2} \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\right ) \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}\right )-3 \left (i a +1\right ) b^{2} \left (\frac {1}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a b \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\right ) \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {\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 )\)	\(946\)

Input:

int((1+I*(b*x+a))^3/(1+(b*x+a)^2)^(3/2)/x^3,x,method=_RETURNVERBOSE)
 

Output:

1/2*I*(-a*b^3*x^3-a^2*b^2*x^2+a^3*b*x+6*I*b^3*x^3+a^4+b^2*x^2+12*I*a*b^2*x 
^2+a*b*x+6*I*a^2*b*x+2*a^2+6*I*b*x+1)/x^2/(I+a)^3/(b^2*x^2+2*a*b*x+a^2+1)^ 
(1/2)+1/2/(a^3-3*a+3*I*a^2-I)*b^2*(-3*(I*a-2*a^2-3)/(I+a)/(a^2+1)^(1/2)*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)+8*I*(I 
*a-1)/b/(I+a)/(x+(I+a)/b)*((x+(I+a)/b)^2*b^2-2*I*b*(x+(I+a)/b))^(1/2))
 

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. 574 vs. \(2 (182) = 364\).

Time = 0.16 (sec) , antiderivative size = 574, normalized size of antiderivative = 2.17 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x^3} \, dx=\frac {{\left (-i \, a - 14\right )} b^{3} x^{3} + {\left (-i \, a^{2} - 13 \, a - 14 i\right )} b^{2} x^{2} - 3 \, {\left ({\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} b x^{3} + {\left (a^{4} + 4 i \, a^{3} - 6 \, a^{2} - 4 i \, a + 1\right )} x^{2}\right )} \sqrt {\frac {{\left (4 \, a^{2} - 12 i \, a - 9\right )} b^{4}}{a^{8} + 6 i \, a^{7} - 14 \, a^{6} - 14 i \, a^{5} - 14 i \, a^{3} + 14 \, a^{2} + 6 i \, a - 1}} \log \left (-\frac {{\left (2 \, a - 3 i\right )} b^{3} x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (2 \, a - 3 i\right )} b^{2} + {\left (a^{5} + 3 i \, a^{4} - 2 \, a^{3} + 2 i \, a^{2} - 3 \, a - i\right )} \sqrt {\frac {{\left (4 \, a^{2} - 12 i \, a - 9\right )} b^{4}}{a^{8} + 6 i \, a^{7} - 14 \, a^{6} - 14 i \, a^{5} - 14 i \, a^{3} + 14 \, a^{2} + 6 i \, a - 1}}}{{\left (2 \, a - 3 i\right )} b^{2}}\right ) + 3 \, {\left ({\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} b x^{3} + {\left (a^{4} + 4 i \, a^{3} - 6 \, a^{2} - 4 i \, a + 1\right )} x^{2}\right )} \sqrt {\frac {{\left (4 \, a^{2} - 12 i \, a - 9\right )} b^{4}}{a^{8} + 6 i \, a^{7} - 14 \, a^{6} - 14 i \, a^{5} - 14 i \, a^{3} + 14 \, a^{2} + 6 i \, a - 1}} \log \left (-\frac {{\left (2 \, a - 3 i\right )} b^{3} x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (2 \, a - 3 i\right )} b^{2} - {\left (a^{5} + 3 i \, a^{4} - 2 \, a^{3} + 2 i \, a^{2} - 3 \, a - i\right )} \sqrt {\frac {{\left (4 \, a^{2} - 12 i \, a - 9\right )} b^{4}}{a^{8} + 6 i \, a^{7} - 14 \, a^{6} - 14 i \, a^{5} - 14 i \, a^{3} + 14 \, a^{2} + 6 i \, a - 1}}}{{\left (2 \, a - 3 i\right )} b^{2}}\right ) + {\left ({\left (-i \, a - 14\right )} b^{2} x^{2} + i \, a^{3} - 5 \, {\left (a + i\right )} b x - a^{2} + i \, a - 1\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{2 \, {\left ({\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} b x^{3} + {\left (a^{4} + 4 i \, a^{3} - 6 \, a^{2} - 4 i \, a + 1\right )} x^{2}\right )}} \] Input:

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

Output:

1/2*((-I*a - 14)*b^3*x^3 + (-I*a^2 - 13*a - 14*I)*b^2*x^2 - 3*((a^3 + 3*I* 
a^2 - 3*a - I)*b*x^3 + (a^4 + 4*I*a^3 - 6*a^2 - 4*I*a + 1)*x^2)*sqrt((4*a^ 
2 - 12*I*a - 9)*b^4/(a^8 + 6*I*a^7 - 14*a^6 - 14*I*a^5 - 14*I*a^3 + 14*a^2 
 + 6*I*a - 1))*log(-((2*a - 3*I)*b^3*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) 
*(2*a - 3*I)*b^2 + (a^5 + 3*I*a^4 - 2*a^3 + 2*I*a^2 - 3*a - I)*sqrt((4*a^2 
 - 12*I*a - 9)*b^4/(a^8 + 6*I*a^7 - 14*a^6 - 14*I*a^5 - 14*I*a^3 + 14*a^2 
+ 6*I*a - 1)))/((2*a - 3*I)*b^2)) + 3*((a^3 + 3*I*a^2 - 3*a - I)*b*x^3 + ( 
a^4 + 4*I*a^3 - 6*a^2 - 4*I*a + 1)*x^2)*sqrt((4*a^2 - 12*I*a - 9)*b^4/(a^8 
 + 6*I*a^7 - 14*a^6 - 14*I*a^5 - 14*I*a^3 + 14*a^2 + 6*I*a - 1))*log(-((2* 
a - 3*I)*b^3*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(2*a - 3*I)*b^2 - (a^5 
+ 3*I*a^4 - 2*a^3 + 2*I*a^2 - 3*a - I)*sqrt((4*a^2 - 12*I*a - 9)*b^4/(a^8 
+ 6*I*a^7 - 14*a^6 - 14*I*a^5 - 14*I*a^3 + 14*a^2 + 6*I*a - 1)))/((2*a - 3 
*I)*b^2)) + ((-I*a - 14)*b^2*x^2 + I*a^3 - 5*(a + I)*b*x - a^2 + I*a - 1)* 
sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))/((a^3 + 3*I*a^2 - 3*a - I)*b*x^3 + (a^4 
 + 4*I*a^3 - 6*a^2 - 4*I*a + 1)*x^2)
 

Sympy [F]

\[ \int \frac {e^{3 i \arctan (a+b x)}}{x^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-I*(Integral(I/(a**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x** 
4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**5*sqrt(a**2 + 2*a*b*x + b 
**2*x**2 + 1) + x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral( 
-3*a/(a**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**4*sqrt(a** 
2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 
 1) + x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(a**3/(a**2 
*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**4*sqrt(a**2 + 2*a*b* 
x + b**2*x**2 + 1) + b**2*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x**3 
*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*I*a**2/(a**2*x**3 
*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**4*sqrt(a**2 + 2*a*b*x + b 
**2*x**2 + 1) + b**2*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x**3*sqrt 
(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*b*x/(a**2*x**3*sqrt(a* 
*2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 
 + 1) + b**2*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x**3*sqrt(a**2 + 
2*a*b*x + b**2*x**2 + 1)), x) + Integral(b**3*x**3/(a**2*x**3*sqrt(a**2 + 
2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) 
 + b**2*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x**3*sqrt(a**2 + 2*a*b 
*x + b**2*x**2 + 1)), x) + Integral(-3*I*b**2*x**2/(a**2*x**3*sqrt(a**2 + 
2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) 
 + b**2*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x**3*sqrt(a**2 + 2*...
 

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. 1536 vs. \(2 (182) = 364\).

Time = 0.05 (sec) , antiderivative size = 1536, normalized size of antiderivative = 5.82 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x^3} \, dx=\text {Too large to display} \] Input:

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

Output:

15/2*(-I*a^3 - 3*a^2 + 3*I*a + 1)*a^3*b^5*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqr 
t(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^3) + 15/2*(-I*a^3 - 3*a^2 + 3*I*a 
 + 1)*a^4*b^4/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) 
*(a^2 + 1)^3) + 9*(I*a^2*b + 2*a*b - I*b)*a^2*b^4*x/((a^2*b^2 - (a^2 + 1)* 
b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^2) + I*b^5*x/((a^2*b^2 - 
(a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 13/2*(-I*a^3 - 3*a^2 + 
 3*I*a + 1)*a*b^5*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^ 
2 + 1)*(a^2 + 1)^2) + 9*(I*a^2*b + 2*a*b - I*b)*a^3*b^3/((a^2*b^2 - (a^2 + 
 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^2) + I*a*b^4/((a^2*b^ 
2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 13/2*(-I*a^3 - 3*a 
^2 + 3*I*a + 1)*a^2*b^4/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x 
+ a^2 + 1)*(a^2 + 1)^2) - 3*(I*a*b^2 + b^2)*a*b^3*x/((a^2*b^2 - (a^2 + 1)* 
b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)) - 6*(I*a^2*b + 2*a*b - I 
*b)*b^4*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^ 
2 + 1)) - 3*(I*a*b^2 + b^2)*a^2*b^2/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^ 
2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)) - 6*(I*a^2*b + 2*a*b - I*b)*a*b^3/((a^2* 
b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)) - 15/2*( 
-I*a^3 - 3*a^2 + 3*I*a + 1)*a^2*b^2*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) + 15/2...
 

Giac [F]

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

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

Output:

undef
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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