\(\int e^{-3 i \arctan (a+b x)} x \, dx\) [224]

Optimal result

Integrand size = 14, antiderivative size = 163 \[ \int e^{-3 i \arctan (a+b x)} x \, dx=-\frac {(1+i a) (1-i a-i b x)^{5/2}}{b^2 \sqrt {1+i a+i b x}}-\frac {3 (3+2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3+2 i a) (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{2 b^2}-\frac {3 (3 i-2 a) \text {arcsinh}(a+b x)}{2 b^2} \] Output:

-(1+I*a)*(1-I*a-I*b*x)^(5/2)/b^2/(1+I*a+I*b*x)^(1/2)-3/2*(3+2*I*a)*(1-I*a- 
I*b*x)^(1/2)*(1+I*a+I*b*x)^(1/2)/b^2-1/2*(3+2*I*a)*(1-I*a-I*b*x)^(3/2)*(1+ 
I*a+I*b*x)^(1/2)/b^2-3/2*(3*I-2*a)*arcsinh(b*x+a)/b^2
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.96 \[ \int e^{-3 i \arctan (a+b x)} x \, dx=\frac {i \left (14 i-a^3+9 b x+6 i b^2 x^2+b^3 x^3+a^2 (14 i-b x)+a \left (-1+20 i b x+b^2 x^2\right )\right )}{2 b^2 \sqrt {1+a^2+2 a b x+b^2 x^2}}+\frac {3 \sqrt [4]{-1} (-3 i+2 a) \sqrt {-i b} \text {arcsinh}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (i+a+b x)}}{\sqrt {-i b}}\right )}{b^{5/2}} \] Input:

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

Output:

((I/2)*(14*I - a^3 + 9*b*x + (6*I)*b^2*x^2 + b^3*x^3 + a^2*(14*I - b*x) + 
a*(-1 + (20*I)*b*x + b^2*x^2)))/(b^2*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]) + 
(3*(-1)^(1/4)*(-3*I + 2*a)*Sqrt[(-I)*b]*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[ 
(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/b^(5/2)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5618, 87, 60, 60, 62, 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.

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

\(\Big \downarrow \) 5618

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 62

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

\(\Big \downarrow \) 1090

\(\displaystyle -\frac {(-2 a+3 i) \left (\frac {3}{2} \left (\frac {\int \frac {1}{\sqrt {\frac {\left (2 x b^2+2 a b\right )^2}{4 b^2}+1}}d\left (2 x b^2+2 a b\right )}{2 b^2}-\frac {i \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{b}\right )-\frac {i (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}{2 b}\right )}{b}-\frac {(1+i a) (-i a-i b x+1)^{5/2}}{b^2 \sqrt {i a+i b x+1}}\)

\(\Big \downarrow \) 222

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, 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 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

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 [A] (verified)

Time = 0.81 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.20

method	result	size
risch	\(-\frac {i \left (-b x +a -6 i\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{2}}+\frac {-\frac {9 i \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}-\frac {i \left (-8 i a -8\right ) \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{b^{2} \left (x -\frac {i-a}{b}\right )}+\frac {6 a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}}{2 b}\)	\(195\)
default	\(\frac {i \left (-\frac {i \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}}}{b \left (x -\frac {i-a}{b}\right )^{2}}+3 i b \left (\frac {\left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {3}{2}}}{3}+i b \left (\frac {\left (2 \left (x -\frac {i-a}{b}\right ) b^{2}+2 i b \right ) \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{4 b^{2}}+\frac {\ln \left (\frac {i b +\left (x -\frac {i-a}{b}\right ) b^{2}}{\sqrt {b^{2}}}+\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\right )}{2 \sqrt {b^{2}}}\right )\right )\right )}{b^{3}}+\frac {i \left (i-a \right ) \left (\frac {i \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}}}{b \left (x -\frac {i-a}{b}\right )^{3}}-2 i b \left (-\frac {i \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}}}{b \left (x -\frac {i-a}{b}\right )^{2}}+3 i b \left (\frac {\left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {3}{2}}}{3}+i b \left (\frac {\left (2 \left (x -\frac {i-a}{b}\right ) b^{2}+2 i b \right ) \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{4 b^{2}}+\frac {\ln \left (\frac {i b +\left (x -\frac {i-a}{b}\right ) b^{2}}{\sqrt {b^{2}}}+\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\right )}{2 \sqrt {b^{2}}}\right )\right )\right )\right )}{b^{4}}\)	\(595\)

Input:

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

Output:

-1/2*I*(-b*x+a-6*I)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)/b^2+1/2/b*(-9*I*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)-I*(-8-8*I*a) 
/b^2/(x-(I-a)/b)*((x-(I-a)/b)^2*b^2+2*I*b*(x-(I-a)/b))^(1/2)+6*a*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))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.83 \[ \int e^{-3 i \arctan (a+b x)} x \, dx=\frac {-3 i \, a^{3} + {\left (-3 i \, a^{2} - 44 \, a + 32 i\right )} b x - 47 \, a^{2} - 12 \, {\left ({\left (2 \, a - 3 i\right )} b x + 2 \, a^{2} - 5 i \, a - 3\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 4 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (-i \, b^{2} x^{2} + i \, a^{2} + 5 \, b x + 15 \, a - 14 i\right )} + 76 i \, a + 32}{8 \, {\left (b^{3} x + {\left (a - i\right )} b^{2}\right )}} \] Input:

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

Output:

1/8*(-3*I*a^3 + (-3*I*a^2 - 44*a + 32*I)*b*x - 47*a^2 - 12*((2*a - 3*I)*b* 
x + 2*a^2 - 5*I*a - 3)*log(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 
 4*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(-I*b^2*x^2 + I*a^2 + 5*b*x + 15*a - 
14*I) + 76*I*a + 32)/(b^3*x + (a - I)*b^2)
 

Sympy [F]

\[ \int e^{-3 i \arctan (a+b x)} x \, dx=i \left (\int \frac {x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a^{3} + 3 a^{2} b x - 3 i a^{2} + 3 a b^{2} x^{2} - 6 i a b x - 3 a + b^{3} x^{3} - 3 i b^{2} x^{2} - 3 b x + i}\, dx + \int \frac {a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a^{3} + 3 a^{2} b x - 3 i a^{2} + 3 a b^{2} x^{2} - 6 i a b x - 3 a + b^{3} x^{3} - 3 i b^{2} x^{2} - 3 b x + i}\, dx + \int \frac {b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a^{3} + 3 a^{2} b x - 3 i a^{2} + 3 a b^{2} x^{2} - 6 i a b x - 3 a + b^{3} x^{3} - 3 i b^{2} x^{2} - 3 b x + i}\, dx + \int \frac {2 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a^{3} + 3 a^{2} b x - 3 i a^{2} + 3 a b^{2} x^{2} - 6 i a b x - 3 a + b^{3} x^{3} - 3 i b^{2} x^{2} - 3 b x + i}\, dx\right ) \] Input:

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

Output:

I*(Integral(x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/(a**3 + 3*a**2*b*x - 3* 
I*a**2 + 3*a*b**2*x**2 - 6*I*a*b*x - 3*a + b**3*x**3 - 3*I*b**2*x**2 - 3*b 
*x + I), x) + Integral(a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/(a**3 + 
 3*a**2*b*x - 3*I*a**2 + 3*a*b**2*x**2 - 6*I*a*b*x - 3*a + b**3*x**3 - 3*I 
*b**2*x**2 - 3*b*x + I), x) + Integral(b**2*x**3*sqrt(a**2 + 2*a*b*x + b** 
2*x**2 + 1)/(a**3 + 3*a**2*b*x - 3*I*a**2 + 3*a*b**2*x**2 - 6*I*a*b*x - 3* 
a + b**3*x**3 - 3*I*b**2*x**2 - 3*b*x + I), x) + Integral(2*a*b*x**2*sqrt( 
a**2 + 2*a*b*x + b**2*x**2 + 1)/(a**3 + 3*a**2*b*x - 3*I*a**2 + 3*a*b**2*x 
**2 - 6*I*a*b*x - 3*a + b**3*x**3 - 3*I*b**2*x**2 - 3*b*x + I), 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. 293 vs. \(2 (112) = 224\).

Time = 0.11 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.80 \[ \int e^{-3 i \arctan (a+b x)} x \, dx=-\frac {i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a}{b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2} - 2 i \, b^{3} x - 2 i \, a b^{2} - b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2} - 2 i \, b^{3} x - 2 i \, a b^{2} - b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{2 i \, b^{3} x + 2 i \, a b^{2} + 2 \, b^{2}} - \frac {6 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{i \, b^{3} x + i \, a b^{2} + b^{2}} + \frac {3 \, a \operatorname {arsinh}\left (b x + a\right )}{b^{2}} - \frac {6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{i \, b^{3} x + i \, a b^{2} + b^{2}} - \frac {9 i \, \operatorname {arsinh}\left (b x + a\right )}{2 \, b^{2}} - \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{2 \, b^{2}} \] Input:

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

Output:

-I*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*a/(b^4*x^2 + 2*a*b^3*x + a^2*b^2 - 
2*I*b^3*x - 2*I*a*b^2 - b^2) - (b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)/(b^4*x^ 
2 + 2*a*b^3*x + a^2*b^2 - 2*I*b^3*x - 2*I*a*b^2 - b^2) - (b^2*x^2 + 2*a*b* 
x + a^2 + 1)^(3/2)/(2*I*b^3*x + 2*I*a*b^2 + 2*b^2) - 6*I*sqrt(b^2*x^2 + 2* 
a*b*x + a^2 + 1)*a/(I*b^3*x + I*a*b^2 + b^2) + 3*a*arcsinh(b*x + a)/b^2 - 
6*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/(I*b^3*x + I*a*b^2 + b^2) - 9/2*I*arcs 
inh(b*x + a)/b^2 - 3/2*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/b^2
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.29 \[ \int e^{-3 i \arctan (a+b x)} x \, dx=-\frac {1}{2} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left (-\frac {i \, x}{b} - \frac {-i \, a b^{2} - 6 \, b^{2}}{b^{4}}\right )} - \frac {{\left (2 \, a - 3 i\right )} \log \left (3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b + a^{3} b + {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{3} {\left | b \right |} + 3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{2} {\left | b \right |} - 2 i \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b - 2 i \, a^{2} b + 4 \, {\left (-i \, x {\left | b \right |} + i \, \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a {\left | b \right |} - a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{2 \, b {\left | b \right |}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int e^{-3 i \arctan (a+b x)} x \, dx=-\left (\int \frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x^{3}}{b^{3} i \,x^{3}+3 a \,b^{2} i \,x^{2}+3 a^{2} b i x +a^{3} i +3 b^{2} x^{2}+6 a b x -3 b i x +3 a^{2}-3 a i -1}d x \right ) b^{2}-2 \left (\int \frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x^{2}}{b^{3} i \,x^{3}+3 a \,b^{2} i \,x^{2}+3 a^{2} b i x +a^{3} i +3 b^{2} x^{2}+6 a b x -3 b i x +3 a^{2}-3 a i -1}d x \right ) a b -\left (\int \frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x}{b^{3} i \,x^{3}+3 a \,b^{2} i \,x^{2}+3 a^{2} b i x +a^{3} i +3 b^{2} x^{2}+6 a b x -3 b i x +3 a^{2}-3 a i -1}d x \right ) a^{2}-\left (\int \frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x}{b^{3} i \,x^{3}+3 a \,b^{2} i \,x^{2}+3 a^{2} b i x +a^{3} i +3 b^{2} x^{2}+6 a b x -3 b i x +3 a^{2}-3 a i -1}d x \right ) \] Input:

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

Output:

 - int((sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*x**3)/(a**3*i + 3*a**2*b*i*x 
+ 3*a**2 + 3*a*b**2*i*x**2 + 6*a*b*x - 3*a*i + b**3*i*x**3 + 3*b**2*x**2 - 
 3*b*i*x - 1),x)*b**2 - 2*int((sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*x**2)/ 
(a**3*i + 3*a**2*b*i*x + 3*a**2 + 3*a*b**2*i*x**2 + 6*a*b*x - 3*a*i + b**3 
*i*x**3 + 3*b**2*x**2 - 3*b*i*x - 1),x)*a*b - int((sqrt(a**2 + 2*a*b*x + b 
**2*x**2 + 1)*x)/(a**3*i + 3*a**2*b*i*x + 3*a**2 + 3*a*b**2*i*x**2 + 6*a*b 
*x - 3*a*i + b**3*i*x**3 + 3*b**2*x**2 - 3*b*i*x - 1),x)*a**2 - int((sqrt( 
a**2 + 2*a*b*x + b**2*x**2 + 1)*x)/(a**3*i + 3*a**2*b*i*x + 3*a**2 + 3*a*b 
**2*i*x**2 + 6*a*b*x - 3*a*i + b**3*i*x**3 + 3*b**2*x**2 - 3*b*i*x - 1),x)