\(\int e^{3 i \arctan (a+b x)} x \, dx\) [197]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.81 \[ \int e^{3 i \arctan (a+b x)} x \, dx=\frac {\sqrt {1+i a+i b x} \left (-14+15 i a+a^2+5 i b x-b^2 x^2\right )}{2 b^2 \sqrt {-i (i+a+b x)}}+\frac {3 \sqrt [4]{-1} (3 i+2 a) \sqrt {-i b} \text {arcsinh}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (i+a+b x)}}{\sqrt {-i b}}\right )}{b^{5/2}} \] Input:

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

Output:

(Sqrt[1 + I*a + I*b*x]*(-14 + (15*I)*a + a^2 + (5*I)*b*x - b^2*x^2))/(2*b^ 
2*Sqrt[(-I)*(I + a + b*x)]) + (3*(-1)^(1/4)*(3*I + 2*a)*Sqrt[(-I)*b]*ArcSi 
nh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/b^(5/2)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.05, 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

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 62

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

\(\Big \downarrow \) 1090

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

\(\Big \downarrow \) 222

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

Input:

Int[E^((3*I)*ArcTan[a + b*x])*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)*(((I/2)*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2))/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 = 0.82 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.14

method	result	size
risch	\(\frac {i \left (-b x +a +6 i\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{2}}+\frac {\frac {9 i \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}-\frac {i \left (-8 i a +8\right ) \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}}{b^{2} \left (x +\frac {i+a}{b}\right )}+\frac {6 a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}}{2 b}\)	\(186\)
default	\(\text {Expression too large to display}\)	\(1052\)

Input:

int((1+I*(b*x+a))^3/(1+(b*x+a)^2)^(3/2)*x,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 = 0.12 (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((1+I*(b*x+a))^3/(1+(b*x+a)^2)^(3/2)*x,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]

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

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

Output:

-I*(Integral(I*x/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt 
(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x* 
*2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*a*x/(a** 
2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b** 
2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 
 2*a*b*x + b**2*x**2 + 1)), x) + Integral(a**3*x/(a**2*sqrt(a**2 + 2*a*b*x 
 + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x* 
*2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 
+ 1)), x) + Integral(-3*b*x**2/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) 
+ 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a 
*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integr 
al(b**3*x**4/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a** 
2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 
 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*I*a**2*x/(a* 
*2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b* 
*2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 
+ 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*I*b**2*x**3/(a**2*sqrt(a**2 
+ 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) 
+ b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b 
**2*x**2 + 1)), x) + Integral(3*a*b**2*x**3/(a**2*sqrt(a**2 + 2*a*b*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. 1108 vs. \(2 (113) = 226\).

Time = 0.04 (sec) , antiderivative size = 1108, normalized size of antiderivative = 6.80 \[ \int e^{3 i \arctan (a+b x)} x \, dx=\text {Too large to display} \] Input:

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

Output:

15*I*a^4*b*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) 
 - 31/2*I*(a^2 + 1)*a^2*b*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a* 
b*x + a^2 + 1)) - 1/2*I*b*x^3/sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + 5/2*I*(a 
^2 + 1)*a^3/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) 
+ 6*(I*a^2*b + 2*a*b - I*b)*a^2*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 
+ 2*a*b*x + a^2 + 1)) - 18*(I*a*b^2 + b^2)*a^3*x/((a^2*b^2 - (a^2 + 1)*b^2 
)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b) + 3/2*I*(a^2 + 1)^2*b*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*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) 
) + 5/2*I*a*x^2/sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) - 3/2*I*(a^2 + 1)^2*a/(( 
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/((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^2 + 1)*x/((a^2*b^2 - (a^2 + 1)*b^2 
)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + 15*(I*a*b^2 + b^2)*(a^2 + 1)*a*x/(( 
a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b) - 15/2*I*a^2 
*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^2 - 3*(I*a* 
b^2 + b^2)*(a^2 + 1)*a^2/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x 
 + a^2 + 1)*b^2) + 3*(I*a^2*b + 2*a*b - I*b)*(a^2 + 1)*a/((a^2*b^2 - (a^2 
+ 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b) - 3*(I*a*b^2 + b^2)*x^2/(sq 
rt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - 3/2*(-I*a^2 - I)*arcsinh(2*(b^2*...
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.28 \[ \int e^{3 i \arctan (a+b x)} x \, dx=-\frac {1}{2} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left (\frac {i \, x}{b} + \frac {-i \, a b^{2} + 6 \, b^{2}}{b^{4}}\right )} - \frac {{\left (2 \, a + 3 i\right )} \log \left (3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b + a^{3} b + {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{3} {\left | b \right |} + 3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{2} {\left | b \right |} + 2 i \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b + 2 i \, a^{2} b + 4 \, {\left (i \, x {\left | b \right |} - i \, \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a {\left | b \right |} - a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{2 \, b {\left | b \right |}} \] Input:

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 555, normalized size of antiderivative = 3.40 \[ \int e^{3 i \arctan (a+b x)} x \, dx=\frac {-33 i -32 a -33 b^{2} i \,x^{2}-80 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a b x -4 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b^{3} i \,x^{3}-36 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b i x +72 \,\mathrm {log}\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+a +b x \right ) a b i x -64 a^{2} b x -32 a \,b^{2} x^{2}-66 a b i x +4 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{3} i +4 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a i -24 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b^{2} x^{2}+36 \,\mathrm {log}\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+a +b x \right ) a^{2} i -33 a^{2} i -56 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+48 \,\mathrm {log}\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+a +b x \right ) a^{2} b x +24 \,\mathrm {log}\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+a +b x \right ) a \,b^{2} x^{2}+36 \,\mathrm {log}\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+a +b x \right ) b^{2} i \,x^{2}+4 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{2} b i x -4 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a \,b^{2} i \,x^{2}-32 a^{3}-56 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{2}+24 \,\mathrm {log}\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+a +b x \right ) a^{3}+24 \,\mathrm {log}\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+a +b x \right ) a +36 \,\mathrm {log}\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+a +b x \right ) i}{8 b^{2} \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )} \] Input:

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

Output:

(4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a**3*i + 4*sqrt(a**2 + 2*a*b*x + b 
**2*x**2 + 1)*a**2*b*i*x - 56*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a**2 - 
4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a*b**2*i*x**2 - 80*sqrt(a**2 + 2*a* 
b*x + b**2*x**2 + 1)*a*b*x + 4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a*i - 
4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*b**3*i*x**3 - 24*sqrt(a**2 + 2*a*b* 
x + b**2*x**2 + 1)*b**2*x**2 - 36*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*b*i 
*x - 56*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 24*log(sqrt(a**2 + 2*a*b*x 
+ b**2*x**2 + 1) + a + b*x)*a**3 + 48*log(sqrt(a**2 + 2*a*b*x + b**2*x**2 
+ 1) + a + b*x)*a**2*b*x + 36*log(sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + a 
 + b*x)*a**2*i + 24*log(sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + a + b*x)*a* 
b**2*x**2 + 72*log(sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + a + b*x)*a*b*i*x 
 + 24*log(sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + a + b*x)*a + 36*log(sqrt( 
a**2 + 2*a*b*x + b**2*x**2 + 1) + a + b*x)*b**2*i*x**2 + 36*log(sqrt(a**2 
+ 2*a*b*x + b**2*x**2 + 1) + a + b*x)*i - 32*a**3 - 64*a**2*b*x - 33*a**2* 
i - 32*a*b**2*x**2 - 66*a*b*i*x - 32*a - 33*b**2*i*x**2 - 33*i)/(8*b**2*(a 
**2 + 2*a*b*x + b**2*x**2 + 1))