\(\int \frac {e^{3 i \arctan (a+b x)}}{x} \, dx\) [199]

Optimal result

Integrand size = 16, antiderivative size = 134 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x} \, dx=\frac {4 \sqrt {1+i a+i b x}}{(1-i a) \sqrt {1-i a-i b x}}-i \text {arcsinh}(a+b x)-\frac {2 (i-a)^{3/2} \text {arctanh}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{(i+a)^{3/2}} \] Output:

4*(1+I*a+I*b*x)^(1/2)/(1-I*a)/(1-I*a-I*b*x)^(1/2)-I*arcsinh(b*x+a)-2*(I-a) 
^(3/2)*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)^(3/2)
 

Mathematica [A] (verified)

Time = 1.23 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.46 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x} \, dx=\frac {2 \left (\frac {2 i \sqrt {1+i a+i b x}}{\sqrt {-i (i+a+b x)}}+\frac {\sqrt [4]{-1} (i+a) (-i b)^{3/2} \text {arcsinh}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (i+a+b x)}}{\sqrt {-i b}}\right )}{b^{3/2}}+\frac {\sqrt {-1-i a} (-i+a) \text {arctanh}\left (\frac {\sqrt {-1-i a} \sqrt {-i (i+a+b x)}}{\sqrt {-1+i a} \sqrt {1+i a+i b x}}\right )}{\sqrt {-1+i a}}\right )}{i+a} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.21, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {5618, 109, 27, 175, 62, 104, 221, 1090, 222}

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.

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

\(\Big \downarrow \) 5618

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 175

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

\(\Big \downarrow \) 62

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 1090

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

\(\Big \downarrow \) 222

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

Input:

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

Output:

(4*Sqrt[1 + I*a + I*b*x])/((1 - I*a)*Sqrt[1 - I*a - I*b*x]) - (-((1 - I*a) 
*ArcSinh[(2*a*b + 2*b^2*x)/(2*b)]) + (2*(I - a)^(3/2)*ArcTanh[(Sqrt[I + a] 
*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a - I*b*x])])/Sqrt[I + a]) 
/(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 62

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Int[ 
1/Sqrt[a*c - b*(a - c)*x - b^2*x^2], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b 
+ d, 0] && GtQ[a + c, 0]
 

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 109

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

rule 175

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ 
)))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b   Int[(c + d*x)^n*(e + f*x)^p, x] 
, x] + Simp[(b*g - a*h)/b   Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] 
 /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
 

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 222

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

rule 1090

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* 
(c/(b^2 - 4*a*c)))^p)   Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, 
b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
 

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 [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (104 ) = 208\).

Time = 0.52 (sec) , antiderivative size = 485, normalized size of antiderivative = 3.62

method	result	size
default	\(\left (-i a^{3}-3 a^{2}+3 i a +1\right ) \left (\frac {1}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a b \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\right ) \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )-i b^{3} \left (-\frac {x}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {a \left (-\frac {1}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a \left (2 b^{2} x +2 a b \right )}{b \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}\right )}{b}+\frac {\ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b^{2} \sqrt {b^{2}}}\right )+\frac {6 i \left (i a +1\right )^{2} b \left (2 b^{2} x +2 a b \right )}{\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-3 \left (i a +1\right ) b^{2} \left (-\frac {1}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a \left (2 b^{2} x +2 a b \right )}{b \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}\right )\)	\(485\)

Input:

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

Output:

(-I*a^3-3*a^2+3*I*a+1)*(1/(a^2+1)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a*b/(a^2 
+1)*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2 
)-1/(a^2+1)^(3/2)*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))-I*b^3*(-x/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-a/b*(-1/b^2/(b^ 
2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2) 
/(b^2*x^2+2*a*b*x+a^2+1)^(1/2))+1/b^2*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+ 
2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2))+6*I*(1+I*a)^2*b*(2*b^2*x+2*a*b)/(4*b^2* 
(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-3*(1+I*a)*b^2*(-1/b^2/(b^ 
2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2) 
/(b^2*x^2+2*a*b*x+a^2+1)^(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. 356 vs. \(2 (90) = 180\).

Time = 0.13 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.66 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x} \, dx=-\frac {{\left ({\left (a + i\right )} b x + a^{2} + 2 i \, a - 1\right )} \sqrt {-\frac {a^{3} - 3 i \, a^{2} - 3 \, a + i}{a^{3} + 3 i \, a^{2} - 3 \, a - i}} \log \left (-\frac {{\left (a - i\right )} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a - i\right )} - {\left (i \, a^{2} - 2 \, a - i\right )} \sqrt {-\frac {a^{3} - 3 i \, a^{2} - 3 \, a + i}{a^{3} + 3 i \, a^{2} - 3 \, a - i}}}{a - i}\right ) - {\left ({\left (a + i\right )} b x + a^{2} + 2 i \, a - 1\right )} \sqrt {-\frac {a^{3} - 3 i \, a^{2} - 3 \, a + i}{a^{3} + 3 i \, a^{2} - 3 \, a - i}} \log \left (-\frac {{\left (a - i\right )} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a - i\right )} - {\left (-i \, a^{2} + 2 \, a + i\right )} \sqrt {-\frac {a^{3} - 3 i \, a^{2} - 3 \, a + i}{a^{3} + 3 i \, a^{2} - 3 \, a - i}}}{a - i}\right ) + 4 \, b x + {\left ({\left (-i \, a + 1\right )} b x - i \, a^{2} + 2 \, a + i\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + 4 \, a + 4 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 4 i}{{\left (a + i\right )} b x + a^{2} + 2 i \, a - 1} \] Input:

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

Output:

-(((a + I)*b*x + a^2 + 2*I*a - 1)*sqrt(-(a^3 - 3*I*a^2 - 3*a + I)/(a^3 + 3 
*I*a^2 - 3*a - I))*log(-((a - I)*b*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*( 
a - I) - (I*a^2 - 2*a - I)*sqrt(-(a^3 - 3*I*a^2 - 3*a + I)/(a^3 + 3*I*a^2 
- 3*a - I)))/(a - I)) - ((a + I)*b*x + a^2 + 2*I*a - 1)*sqrt(-(a^3 - 3*I*a 
^2 - 3*a + I)/(a^3 + 3*I*a^2 - 3*a - I))*log(-((a - I)*b*x - sqrt(b^2*x^2 
+ 2*a*b*x + a^2 + 1)*(a - I) - (-I*a^2 + 2*a + I)*sqrt(-(a^3 - 3*I*a^2 - 3 
*a + I)/(a^3 + 3*I*a^2 - 3*a - I)))/(a - I)) + 4*b*x + ((-I*a + 1)*b*x - I 
*a^2 + 2*a + I)*log(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + 4*a + 
4*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + 4*I)/((a + I)*b*x + a^2 + 2*I*a - 1)
 

Sympy [F]

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

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

Output:

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

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. 733 vs. \(2 (90) = 180\).

Time = 0.04 (sec) , antiderivative size = 733, normalized size of antiderivative = 5.47 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x} \, dx =\text {Too large to display} \] Input:

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

Output:

2*I*a^2*b^3*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) 
) + (-I*a^2 - I)*b^3*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + 
 a^2 + 1)) + (-I*a^3 - 3*a^2 + 3*I*a + 1)*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)) + I*(a^2 + 1)*a*b^2/((a^2* 
b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + (-I*a^3 - 3*a^2 
+ 3*I*a + 1)*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)) - 3*(I*a*b^2 + b^2)*a*b*x/((a^2*b^2 - (a^2 + 1)*b^2)*sq 
rt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + 3*(I*a^2*b + 2*a*b - I*b)*b^2*x/((a^2*b 
^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 3*(I*a*b^2 + b^2) 
*a^2/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + 3*(I* 
a^2*b + 2*a*b - I*b)*a*b/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x 
 + a^2 + 1)) - (-I*a^3 - 3*a^2 + 3*I*a + 1)*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) + 
 (-I*a^3 - 3*a^2 + 3*I*a + 1)/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1) 
) + 3*(I*a*b^2 + b^2)/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - I*arcsinh( 
2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))
 

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. 252 vs. \(2 (90) = 180\).

Time = 0.25 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.88 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x} \, dx=\frac {i \, b \log \left (-3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b - a^{3} b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{3} {\left | b \right |} - 3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{2} {\left | b \right |} - 2 i \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b - 2 i \, a^{2} b - 4 \, {\left (i \, x {\left | b \right |} - i \, \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a {\left | b \right |} + a b + {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{3 \, {\left | b \right |}} - \frac {{\left (i \, a^{2} + 2 \, a - i\right )} \log \left (\frac {{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {{\left (b x + a\right )}^{2} + 1} - 2 \, \sqrt {a^{2} + 1} \right |}}{{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {{\left (b x + a\right )}^{2} + 1} + 2 \, \sqrt {a^{2} + 1} \right |}}\right )}{\sqrt {a^{2} + 1} {\left (a + i\right )}} \] Input:

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

Output:

1/3*I*b*log(-3*(x*abs(b) - sqrt((b*x + a)^2 + 1))^2*a*b - a^3*b - (x*abs(b 
) - sqrt((b*x + a)^2 + 1))^3*abs(b) - 3*(x*abs(b) - sqrt((b*x + a)^2 + 1)) 
*a^2*abs(b) - 2*I*(x*abs(b) - sqrt((b*x + a)^2 + 1))^2*b - 2*I*a^2*b - 4*( 
I*x*abs(b) - I*sqrt((b*x + a)^2 + 1))*a*abs(b) + a*b + (x*abs(b) - sqrt((b 
*x + a)^2 + 1))*abs(b))/abs(b) - (I*a^2 + 2*a - I)*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)))/(sqrt(a^2 + 1)*(a + I))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1336, normalized size of antiderivative = 9.97 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*sqrt(a**2 + 1)*atan((sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*i + b*i*x)/sq 
rt(a**2 + 1))*a**5 + 4*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*x - 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**4*i + 2*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**2*x**2 - 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**3*b*i*x - 4*sqrt(a**2 + 1)*atan((s 
qrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*i + b*i*x)/sqrt(a**2 + 1))*a**3 - 6*sq 
rt(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**2*i*x**2 - 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**2*b*x - 4*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**2*i - 
 6*sqrt(a**2 + 1)*atan((sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*i + b*i*x)/sq 
rt(a**2 + 1))*a*b**2*x**2 + 4*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*i*x - 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 + 2*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))*b**2*i*x**2 + 2*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))*i - 4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 
1)*a**4 - 4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a**3*b*x + 8*sqrt(a**2...