\(\int \frac {e^{3 i \arctan (a+b x)}}{x^4} \, dx\) [202]

Optimal result

Integrand size = 16, antiderivative size = 338 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x^4} \, dx=\frac {\left (52+51 i a-2 a^2\right ) b^3 \sqrt {1+i a+i b x}}{6 (i-a) (i+a)^4 \sqrt {1-i a-i b x}}-\frac {(i-a) \sqrt {1+i a+i b x}}{3 (i+a) x^3 \sqrt {1-i a-i b x}}+\frac {7 i b \sqrt {1+i a+i b x}}{6 (i+a)^2 x^2 \sqrt {1-i a-i b x}}+\frac {(19+16 i a) b^2 \sqrt {1+i a+i b x}}{6 (i-a) (i+a)^3 x \sqrt {1-i a-i b x}}-\frac {\left (11 i-18 a-6 i a^2\right ) b^3 \text {arctanh}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{(i-a)^{3/2} (i+a)^{9/2}} \] Output:

1/6*(52+51*I*a-2*a^2)*b^3*(1+I*a+I*b*x)^(1/2)/(I-a)/(I+a)^4/(1-I*a-I*b*x)^ 
(1/2)-1/3*(I-a)*(1+I*a+I*b*x)^(1/2)/(I+a)/x^3/(1-I*a-I*b*x)^(1/2)+7/6*I*b* 
(1+I*a+I*b*x)^(1/2)/(I+a)^2/x^2/(1-I*a-I*b*x)^(1/2)+1/6*(19+16*I*a)*b^2*(1 
+I*a+I*b*x)^(1/2)/(I-a)/(I+a)^3/x/(1-I*a-I*b*x)^(1/2)-(11*I-18*a-6*I*a^2)* 
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)^(3/2)/(I+a)^(9/2)
 

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.83 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x^4} \, dx=-\frac {2 (-1+i a)^{3/2} (1+i a) (i+a)^2 (1+i a+i b x)^{5/2}+(3 i-4 a) (-1+i a)^{5/2} b x (1+i a+i b x)^{5/2}-i \left (-11-18 i a+6 a^2\right ) b^2 x^2 \left (i \sqrt {-1+i a} \sqrt {1+i a+i b x} \left (1+a^2-5 i b x+a b x\right )-6 \sqrt {-1-i a} b x \sqrt {-i (i+a+b x)} \text {arctanh}\left (\frac {\sqrt {-1-i a} \sqrt {-i (i+a+b x)}}{\sqrt {-1+i a} \sqrt {1+i a+i b x}}\right )\right )}{6 (-1+i a)^{5/2} \left (1+a^2\right )^2 x^3 \sqrt {-i (i+a+b x)}} \] Input:

Integrate[E^((3*I)*ArcTan[a + b*x])/x^4,x]
 

Output:

-1/6*(2*(-1 + I*a)^(3/2)*(1 + I*a)*(I + a)^2*(1 + I*a + I*b*x)^(5/2) + (3* 
I - 4*a)*(-1 + I*a)^(5/2)*b*x*(1 + I*a + I*b*x)^(5/2) - I*(-11 - (18*I)*a 
+ 6*a^2)*b^2*x^2*(I*Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x]*(1 + a^2 - (5*I)* 
b*x + a*b*x) - 6*Sqrt[-1 - I*a]*b*x*Sqrt[(-I)*(I + a + b*x)]*ArcTanh[(Sqrt 
[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)])/(Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x] 
)]))/((-1 + I*a)^(5/2)*(1 + a^2)^2*x^3*Sqrt[(-I)*(I + a + b*x)])
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {5618, 109, 25, 27, 168, 27, 168, 25, 27, 169, 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.

\(\displaystyle \int \frac {e^{3 i \arctan (a+b x)}}{x^4} \, dx\)

\(\Big \downarrow \) 5618

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

\(\Big \downarrow \) 109

\(\displaystyle -\frac {\int -\frac {b (7 (i-a)-6 b x)}{x^3 (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}dx}{3 (1-i a)}-\frac {(-a+i) \sqrt {i a+i b x+1}}{3 (a+i) x^3 \sqrt {-i a-i b x+1}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {b (7 (i-a)-6 b x)}{x^3 (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}dx}{3 (1-i a)}-\frac {(-a+i) \sqrt {i a+i b x+1}}{3 (a+i) x^3 \sqrt {-i a-i b x+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \int \frac {7 (i-a)-6 b x}{x^3 (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}dx}{3 (1-i a)}-\frac {(-a+i) \sqrt {i a+i b x+1}}{3 (a+i) x^3 \sqrt {-i a-i b x+1}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {b \left (\frac {7 \sqrt {i a+i b x+1}}{2 (a+i) x^2 \sqrt {-i a-i b x+1}}-\frac {\int \frac {b \left (-16 a^2+35 i a+14 (i-a) b x+19\right )}{x^2 (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}dx}{2 \left (a^2+1\right )}\right )}{3 (1-i a)}-\frac {(-a+i) \sqrt {i a+i b x+1}}{3 (a+i) x^3 \sqrt {-i a-i b x+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {7 \sqrt {i a+i b x+1}}{2 (a+i) x^2 \sqrt {-i a-i b x+1}}-\frac {b \int \frac {-16 a^2+35 i a+14 (i-a) b x+19}{x^2 (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}dx}{2 \left (a^2+1\right )}\right )}{3 (1-i a)}-\frac {(-a+i) \sqrt {i a+i b x+1}}{3 (a+i) x^3 \sqrt {-i a-i b x+1}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {b \left (\frac {7 \sqrt {i a+i b x+1}}{2 (a+i) x^2 \sqrt {-i a-i b x+1}}-\frac {b \left (-\frac {\int -\frac {b \left (3 (i-a) \left (-6 a^2+18 i a+11\right )-\left (-16 a^2+35 i a+19\right ) b x\right )}{x (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}dx}{a^2+1}-\frac {(-16 a+19 i) \sqrt {i a+i b x+1}}{(a+i) x \sqrt {-i a-i b x+1}}\right )}{2 \left (a^2+1\right )}\right )}{3 (1-i a)}-\frac {(-a+i) \sqrt {i a+i b x+1}}{3 (a+i) x^3 \sqrt {-i a-i b x+1}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {7 \sqrt {i a+i b x+1}}{2 (a+i) x^2 \sqrt {-i a-i b x+1}}-\frac {b \left (\frac {\int \frac {b \left (3 (i-a) \left (-6 a^2+18 i a+11\right )-\left (-16 a^2+35 i a+19\right ) b x\right )}{x (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}dx}{a^2+1}-\frac {(-16 a+19 i) \sqrt {i a+i b x+1}}{(a+i) x \sqrt {-i a-i b x+1}}\right )}{2 \left (a^2+1\right )}\right )}{3 (1-i a)}-\frac {(-a+i) \sqrt {i a+i b x+1}}{3 (a+i) x^3 \sqrt {-i a-i b x+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {7 \sqrt {i a+i b x+1}}{2 (a+i) x^2 \sqrt {-i a-i b x+1}}-\frac {b \left (\frac {b \int \frac {3 (i-a) \left (-6 a^2+18 i a+11\right )-\left (-16 a^2+35 i a+19\right ) b x}{x (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}dx}{a^2+1}-\frac {(-16 a+19 i) \sqrt {i a+i b x+1}}{(a+i) x \sqrt {-i a-i b x+1}}\right )}{2 \left (a^2+1\right )}\right )}{3 (1-i a)}-\frac {(-a+i) \sqrt {i a+i b x+1}}{3 (a+i) x^3 \sqrt {-i a-i b x+1}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {b \left (\frac {7 \sqrt {i a+i b x+1}}{2 (a+i) x^2 \sqrt {-i a-i b x+1}}-\frac {b \left (\frac {b \left (-\frac {\int \frac {3 \left (-6 i a^3-24 a^2+29 i a+11\right ) b}{x \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}dx}{(a+i) b}-\frac {\left (-2 i a^3-53 a^2+103 i a+52\right ) \sqrt {i a+i b x+1}}{(a+i) \sqrt {-i a-i b x+1}}\right )}{a^2+1}-\frac {(-16 a+19 i) \sqrt {i a+i b x+1}}{(a+i) x \sqrt {-i a-i b x+1}}\right )}{2 \left (a^2+1\right )}\right )}{3 (1-i a)}-\frac {(-a+i) \sqrt {i a+i b x+1}}{3 (a+i) x^3 \sqrt {-i a-i b x+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {7 \sqrt {i a+i b x+1}}{2 (a+i) x^2 \sqrt {-i a-i b x+1}}-\frac {b \left (\frac {b \left (-\frac {3 \left (-6 i a^3-24 a^2+29 i a+11\right ) \int \frac {1}{x \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}dx}{a+i}-\frac {\left (-2 i a^3-53 a^2+103 i a+52\right ) \sqrt {i a+i b x+1}}{(a+i) \sqrt {-i a-i b x+1}}\right )}{a^2+1}-\frac {(-16 a+19 i) \sqrt {i a+i b x+1}}{(a+i) x \sqrt {-i a-i b x+1}}\right )}{2 \left (a^2+1\right )}\right )}{3 (1-i a)}-\frac {(-a+i) \sqrt {i a+i b x+1}}{3 (a+i) x^3 \sqrt {-i a-i b x+1}}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {b \left (\frac {7 \sqrt {i a+i b x+1}}{2 (a+i) x^2 \sqrt {-i a-i b x+1}}-\frac {b \left (\frac {b \left (-\frac {6 \left (-6 i a^3-24 a^2+29 i a+11\right ) \int \frac {1}{-i a+\frac {(1-i a) (i a+i b x+1)}{-i a-i b x+1}-1}d\frac {\sqrt {i a+i b x+1}}{\sqrt {-i a-i b x+1}}}{a+i}-\frac {\left (-2 i a^3-53 a^2+103 i a+52\right ) \sqrt {i a+i b x+1}}{(a+i) \sqrt {-i a-i b x+1}}\right )}{a^2+1}-\frac {(-16 a+19 i) \sqrt {i a+i b x+1}}{(a+i) x \sqrt {-i a-i b x+1}}\right )}{2 \left (a^2+1\right )}\right )}{3 (1-i a)}-\frac {(-a+i) \sqrt {i a+i b x+1}}{3 (a+i) x^3 \sqrt {-i a-i b x+1}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {b \left (\frac {7 \sqrt {i a+i b x+1}}{2 (a+i) x^2 \sqrt {-i a-i b x+1}}-\frac {b \left (\frac {b \left (\frac {6 i \left (-6 i a^3-24 a^2+29 i a+11\right ) \text {arctanh}\left (\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-a+i} \sqrt {-i a-i b x+1}}\right )}{\sqrt {-a+i} (a+i)^{3/2}}-\frac {\left (-2 i a^3-53 a^2+103 i a+52\right ) \sqrt {i a+i b x+1}}{(a+i) \sqrt {-i a-i b x+1}}\right )}{a^2+1}-\frac {(-16 a+19 i) \sqrt {i a+i b x+1}}{(a+i) x \sqrt {-i a-i b x+1}}\right )}{2 \left (a^2+1\right )}\right )}{3 (1-i a)}-\frac {(-a+i) \sqrt {i a+i b x+1}}{3 (a+i) x^3 \sqrt {-i a-i b x+1}}\)

Input:

Int[E^((3*I)*ArcTan[a + b*x])/x^4,x]
 

Output:

-1/3*((I - a)*Sqrt[1 + I*a + I*b*x])/((I + a)*x^3*Sqrt[1 - I*a - I*b*x]) + 
 (b*((7*Sqrt[1 + I*a + I*b*x])/(2*(I + a)*x^2*Sqrt[1 - I*a - I*b*x]) - (b* 
(-(((19*I - 16*a)*Sqrt[1 + I*a + I*b*x])/((I + a)*x*Sqrt[1 - I*a - I*b*x]) 
) + (b*(-(((52 + (103*I)*a - 53*a^2 - (2*I)*a^3)*Sqrt[1 + I*a + I*b*x])/(( 
I + a)*Sqrt[1 - I*a - I*b*x])) + ((6*I)*(11 + (29*I)*a - 24*a^2 - (6*I)*a^ 
3)*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)^(3/2))))/(1 + a^2)))/(2*(1 + a^2))))/(3*(1 
 - I*a))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 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 169

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] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

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 = 1.48 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.12

method	result	size
risch	\(\frac {i \left (2 a^{2} b^{4} x^{4}-27 i a \,b^{4} x^{4}+2 a^{3} b^{3} x^{3}-45 i a^{2} b^{3} x^{3}-9 i a^{3} b^{2} x^{2}-28 b^{4} x^{4}+2 a^{5} b x +9 i a^{4} b x -58 a \,b^{3} x^{3}+9 i b^{3} x^{3}+2 a^{6}-26 a^{2} b^{2} x^{2}-9 i a \,b^{2} x^{2}+4 a^{3} b x +18 i a^{2} b x +6 a^{4}-26 b^{2} x^{2}+2 a b x +9 i b x +6 a^{2}+2\right )}{6 x^{3} \left (-i+a \right ) \left (i+a \right )^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {b^{3} \left (-\frac {\left (-6 a^{3}+12 i a^{2}-7 a +11 i\right ) \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (i+a \right ) \sqrt {a^{2}+1}}-\frac {8 \left (a^{2}+1\right ) \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}}{b \left (i+a \right ) \left (x +\frac {i+a}{b}\right )}\right )}{2 \left (-i+a \right ) \left (a^{4}+4 i a^{3}-6 a^{2}-4 i a +1\right )}\)	\(379\)
default	\(\text {Expression too large to display}\)	\(1649\)

Input:

int((1+I*(b*x+a))^3/(1+(b*x+a)^2)^(3/2)/x^4,x,method=_RETURNVERBOSE)
 

Output:

1/6*I*(9*I*a^4*b*x+2*a^2*b^4*x^4-9*I*a*b^2*x^2+2*a^3*b^3*x^3-45*I*a^2*b^3* 
x^3-28*b^4*x^4-9*I*a^3*b^2*x^2+9*I*b^3*x^3+2*a^5*b*x-58*a*b^3*x^3-27*I*a*b 
^4*x^4+2*a^6-26*a^2*b^2*x^2+9*I*b*x+4*a^3*b*x+6*a^4-26*b^2*x^2+18*I*a^2*b* 
x+2*a*b*x+6*a^2+2)/x^3/(-I+a)/(I+a)^4/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-1/2/(- 
I+a)/(a^4-6*a^2+4*I*a^3+1-4*I*a)*b^3*(-1/(I+a)*(12*I*a^2-6*a^3+11*I-7*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*(a^2+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. 839 vs. \(2 (224) = 448\).

Time = 0.13 (sec) , antiderivative size = 839, normalized size of antiderivative = 2.48 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x^4} \, dx =\text {Too large to display} \] Input:

integrate((1+I*(b*x+a))^3/(1+(b*x+a)^2)^(3/2)/x^4,x, algorithm="fricas")
 

Output:

1/6*((2*I*a^2 + 51*a - 52*I)*b^4*x^4 + (2*I*a^3 + 49*a^2 - I*a + 52)*b^3*x 
^3 + 3*sqrt((36*a^4 - 216*I*a^3 - 456*a^2 + 396*I*a + 121)*b^6/(a^12 + 6*I 
*a^11 - 12*a^10 - 2*I*a^9 - 27*a^8 - 36*I*a^7 - 36*I*a^5 + 27*a^4 - 2*I*a^ 
3 + 12*a^2 + 6*I*a - 1))*((a^5 + 3*I*a^4 - 2*a^3 + 2*I*a^2 - 3*a - I)*b*x^ 
4 + (a^6 + 4*I*a^5 - 5*a^4 - 5*a^2 - 4*I*a + 1)*x^3)*log(-((6*a^2 - 18*I*a 
 - 11)*b^4*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(6*a^2 - 18*I*a - 11)*b^3 
 + (a^7 + 3*I*a^6 - a^5 + 5*I*a^4 - 5*a^3 + I*a^2 - 3*a - I)*sqrt((36*a^4 
- 216*I*a^3 - 456*a^2 + 396*I*a + 121)*b^6/(a^12 + 6*I*a^11 - 12*a^10 - 2* 
I*a^9 - 27*a^8 - 36*I*a^7 - 36*I*a^5 + 27*a^4 - 2*I*a^3 + 12*a^2 + 6*I*a - 
 1)))/((6*a^2 - 18*I*a - 11)*b^3)) - 3*sqrt((36*a^4 - 216*I*a^3 - 456*a^2 
+ 396*I*a + 121)*b^6/(a^12 + 6*I*a^11 - 12*a^10 - 2*I*a^9 - 27*a^8 - 36*I* 
a^7 - 36*I*a^5 + 27*a^4 - 2*I*a^3 + 12*a^2 + 6*I*a - 1))*((a^5 + 3*I*a^4 - 
 2*a^3 + 2*I*a^2 - 3*a - I)*b*x^4 + (a^6 + 4*I*a^5 - 5*a^4 - 5*a^2 - 4*I*a 
 + 1)*x^3)*log(-((6*a^2 - 18*I*a - 11)*b^4*x - sqrt(b^2*x^2 + 2*a*b*x + a^ 
2 + 1)*(6*a^2 - 18*I*a - 11)*b^3 - (a^7 + 3*I*a^6 - a^5 + 5*I*a^4 - 5*a^3 
+ I*a^2 - 3*a - I)*sqrt((36*a^4 - 216*I*a^3 - 456*a^2 + 396*I*a + 121)*b^6 
/(a^12 + 6*I*a^11 - 12*a^10 - 2*I*a^9 - 27*a^8 - 36*I*a^7 - 36*I*a^5 + 27* 
a^4 - 2*I*a^3 + 12*a^2 + 6*I*a - 1)))/((6*a^2 - 18*I*a - 11)*b^3)) + ((2*I 
*a^2 + 51*a - 52*I)*b^3*x^3 + 2*I*a^5 + (16*a^2 - 3*I*a + 19)*b^2*x^2 - 2* 
a^4 + 4*I*a^3 - 7*(a^3 + I*a^2 + a + I)*b*x - 4*a^2 + 2*I*a - 2)*sqrt(b...
 

Sympy [F]

\[ \int \frac {e^{3 i \arctan (a+b x)}}{x^4} \, dx =\text {Too large to display} \] Input:

integrate((1+I*(b*x+a))**3/(1+(b*x+a)**2)**(3/2)/x**4,x)
 

Output:

-I*(Integral(I/(a**2*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x** 
5*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**6*sqrt(a**2 + 2*a*b*x + b 
**2*x**2 + 1) + x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral( 
-3*a/(a**2*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**5*sqrt(a** 
2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**6*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 
 1) + x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(a**3/(a**2 
*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**5*sqrt(a**2 + 2*a*b* 
x + b**2*x**2 + 1) + b**2*x**6*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x**4 
*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*I*a**2/(a**2*x**4 
*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**5*sqrt(a**2 + 2*a*b*x + b 
**2*x**2 + 1) + b**2*x**6*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x**4*sqrt 
(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*b*x/(a**2*x**4*sqrt(a* 
*2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2 
 + 1) + b**2*x**6*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x**4*sqrt(a**2 + 
2*a*b*x + b**2*x**2 + 1)), x) + Integral(b**3*x**3/(a**2*x**4*sqrt(a**2 + 
2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) 
 + b**2*x**6*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x**4*sqrt(a**2 + 2*a*b 
*x + b**2*x**2 + 1)), x) + Integral(-3*I*b**2*x**2/(a**2*x**4*sqrt(a**2 + 
2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) 
 + b**2*x**6*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x**4*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. 2313 vs. \(2 (224) = 448\).

Time = 0.06 (sec) , antiderivative size = 2313, normalized size of antiderivative = 6.84 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x^4} \, dx=\text {Too large to display} \] Input:

integrate((1+I*(b*x+a))^3/(1+(b*x+a)^2)^(3/2)/x^4,x, algorithm="maxima")
 

Output:

-35/2*(-I*a^3 - 3*a^2 + 3*I*a + 1)*a^4*b^6*x/((a^2*b^2 - (a^2 + 1)*b^2)*sq 
rt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^4) - 35/2*(-I*a^3 - 3*a^2 + 3*I* 
a + 1)*a^5*b^5/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1 
)*(a^2 + 1)^4) - 45/2*(I*a^2*b + 2*a*b - I*b)*a^3*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)^3) - I*a*b^6*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)) + 115/6* 
(-I*a^3 - 3*a^2 + 3*I*a + 1)*a^2*b^6*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) - 45/2*(I*a^2*b + 2*a*b - I*b)*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) - I*a^2*b^5/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 
1)*(a^2 + 1)) + 115/6*(-I*a^3 - 3*a^2 + 3*I*a + 1)*a^3*b^5/((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*b^2 + 
b^2)*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) + 39/2*(I*a^2*b + 2*a*b - I*b)*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) - 8/3*(-I*a^3 - 3*a^ 
2 + 3*I*a + 1)*b^6*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*b^2 + b^2)*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) + 39/2*(I*a^2*b + 2*a*b - 
 I*b)*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) - 8/3*(-I*a^3 - 3*a^2 + 3*I*a + 1)*a*b^5/((a^2*b^2 - (a^2...
 

Giac [F]

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

integrate((1+I*(b*x+a))^3/(1+(b*x+a)^2)^(3/2)/x^4,x, algorithm="giac")
 

Output:

undef
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{3 i \arctan (a+b x)}}{x^4} \, dx=\int \frac {{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3}{x^4\,{\left ({\left (a+b\,x\right )}^2+1\right )}^{3/2}} \,d x \] Input:

int((a*1i + b*x*1i + 1)^3/(x^4*((a + b*x)^2 + 1)^(3/2)),x)
 

Output:

int((a*1i + b*x*1i + 1)^3/(x^4*((a + b*x)^2 + 1)^(3/2)), x)
 

Reduce [B] (verification not implemented)

Time = 5.75 (sec) , antiderivative size = 2831, normalized size of antiderivative = 8.38 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x^4} \, dx =\text {Too large to display} \] Input:

int((1+I*(b*x+a))^3/(1+(b*x+a)^2)^(3/2)/x^4,x)
 

Output:

(36*sqrt(a**2 + 1)*atan((sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*i + b*i*x)/s 
qrt(a**2 + 1))*a**8*b**3*i*x**3 + 72*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**7*b**4*i*x**4 + 216*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**7*b**3*x**3 + 36*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**6*b**5*i*x**5 + 432*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 
**6*b**4*x**4 - 462*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**6*b**3*i*x**3 + 216*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**5*b**5 
*x**5 - 996*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**5*b**4*i*x**4 - 342*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**5*b**3*x**3 - 
498*sqrt(a**2 + 1)*atan((sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*i + b*i*x)/s 
qrt(a**2 + 1))*a**4*b**5*i*x**5 - 1116*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**4*x**4 - 192*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**4*b**3*i*x**3 - 558*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**5*x**5 + 612*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...