\(\int \frac {e^{i \arctan (a+b x)}}{x^4} \, dx\) [170]

Optimal result

Integrand size = 16, antiderivative size = 283 \[ \int \frac {e^{i \arctan (a+b x)}}{x^4} \, dx=-\frac {\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{3 (1-i a) x^3}-\frac {(3 i-2 a) b \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{6 (1-i a) \left (1+a^2\right ) x^2}+\frac {\left (4+9 i a-2 a^2\right ) b^2 \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{6 (1-i a) \left (1+a^2\right )^2 x}+\frac {\left (2 a-i \left (1-2 a^2\right )\right ) b^3 \text {arctanh}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{(i-a)^{5/2} (i+a)^{7/2}} \]

[Out]

(2*a-I*(-2*a^2+1))*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)^(5/2)/(I
+a)^(7/2)-1/3*(1-I*a-I*b*x)^(1/2)*(1+I*a+I*b*x)^(1/2)/(1-I*a)/x^3-1/6*(3*I-2*a)*b*(1-I*a-I*b*x)^(1/2)*(1+I*a+I
*b*x)^(1/2)/(1-I*a)/(a^2+1)/x^2+1/6*(4+9*I*a-2*a^2)*b^2*(1-I*a-I*b*x)^(1/2)*(1+I*a+I*b*x)^(1/2)/(1-I*a)/(a^2+1
)^2/x

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5203, 101, 156, 12, 95, 214} \[ \int \frac {e^{i \arctan (a+b x)}}{x^4} \, dx=\frac {\left (2 a-i \left (1-2 a^2\right )\right ) b^3 \text {arctanh}\left (\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-a+i} \sqrt {-i a-i b x+1}}\right )}{(-a+i)^{5/2} (a+i)^{7/2}}+\frac {\left (-2 a^2+9 i a+4\right ) b^2 \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{6 (1-i a) \left (a^2+1\right )^2 x}-\frac {(-2 a+3 i) b \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{6 (1-i a) \left (a^2+1\right ) x^2}-\frac {\sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{3 (1-i a) x^3} \]

[In]

Int[E^(I*ArcTan[a + b*x])/x^4,x]

[Out]

-1/3*(Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/((1 - I*a)*x^3) - ((3*I - 2*a)*b*Sqrt[1 - I*a - I*b*x]*Sqrt
[1 + I*a + I*b*x])/(6*(1 - I*a)*(1 + a^2)*x^2) + ((4 + (9*I)*a - 2*a^2)*b^2*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a
+ I*b*x])/(6*(1 - I*a)*(1 + a^2)^2*x) + ((2*a - I*(1 - 2*a^2))*b^3*ArcTanh[(Sqrt[I + a]*Sqrt[1 + I*a + I*b*x]
)/(Sqrt[I - a]*Sqrt[1 - I*a - I*b*x])])/((I - a)^(5/2)*(I + a)^(7/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[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] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 101

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> 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] + Dist[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 214

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 5203

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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {1+i a+i b x}}{x^4 \sqrt {1-i a-i b x}} \, dx \\ & = -\frac {\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{3 (1-i a) x^3}+\frac {\int \frac {(3 i-2 a) b-2 b^2 x}{x^3 \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{3 (1-i a)} \\ & = -\frac {\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{3 (1-i a) x^3}-\frac {(3 i-2 a) b \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{6 (1-i a) \left (1+a^2\right ) x^2}-\frac {\int \frac {\left (4+9 i a-2 a^2\right ) b^2+(3 i-2 a) b^3 x}{x^2 \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{6 (1-i a) \left (1+a^2\right )} \\ & = -\frac {\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{3 (1-i a) x^3}-\frac {(3 i-2 a) b \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{6 (1-i a) \left (1+a^2\right ) x^2}+\frac {\left (4+9 i a-2 a^2\right ) b^2 \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{6 (1-i a) \left (1+a^2\right )^2 x}+\frac {\int -\frac {3 \left (i-2 a-2 i a^2\right ) b^3}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{6 (1-i a) \left (1+a^2\right )^2} \\ & = -\frac {\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{3 (1-i a) x^3}-\frac {(3 i-2 a) b \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{6 (1-i a) \left (1+a^2\right ) x^2}+\frac {\left (4+9 i a-2 a^2\right ) b^2 \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{6 (1-i a) \left (1+a^2\right )^2 x}+\frac {\left (\left (1+2 i a-2 a^2\right ) b^3\right ) \int \frac {1}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{2 (i-a)^2 (i+a)^3} \\ & = -\frac {\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{3 (1-i a) x^3}-\frac {(3 i-2 a) b \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{6 (1-i a) \left (1+a^2\right ) x^2}+\frac {\left (4+9 i a-2 a^2\right ) b^2 \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{6 (1-i a) \left (1+a^2\right )^2 x}+\frac {\left (\left (1+2 i a-2 a^2\right ) b^3\right ) \text {Subst}\left (\int \frac {1}{-1-i a-(-1+i a) x^2} \, dx,x,\frac {\sqrt {1+i a+i b x}}{\sqrt {1-i a-i b x}}\right )}{(i-a)^2 (i+a)^3} \\ & = -\frac {\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{3 (1-i a) x^3}-\frac {(3 i-2 a) b \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{6 (1-i a) \left (1+a^2\right ) x^2}+\frac {\left (4+9 i a-2 a^2\right ) b^2 \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{6 (1-i a) \left (1+a^2\right )^2 x}+\frac {\left (2 a-i \left (1-2 a^2\right )\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)^{5/2} (i+a)^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.83 \[ \int \frac {e^{i \arctan (a+b x)}}{x^4} \, dx=\frac {\frac {2 (1-i a) (-i+a) (-i+a+b x) \sqrt {1+a^2+2 a b x+b^2 x^2}}{x^3}+\frac {(1+4 i a) b (-i+a+b x) \sqrt {1+a^2+2 a b x+b^2 x^2}}{x^2}+3 i \left (1+2 i a-2 a^2\right ) b^2 \left (\frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{i x+a x}+\frac {2 b \text {arctanh}\left (\frac {\sqrt {-1-i a} \sqrt {-i (i+a+b x)}}{\sqrt {-1+i a} \sqrt {1+i a+i b x}}\right )}{\sqrt {-1-i a} (-1+i a)^{3/2}}\right )}{6 \left (1+a^2\right )^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.99

method	result	size
risch	\(-\frac {i \left (2 a^{2} b^{4} x^{4}-9 i a \,b^{4} x^{4}+2 a^{3} b^{3} x^{3}-15 i a^{2} b^{3} x^{3}-3 i a^{3} b^{2} x^{2}-4 x^{4} b^{4}+2 a^{5} b x +3 i a^{4} b x -10 a \,b^{3} x^{3}+3 i b^{3} x^{3}+2 a^{6}-2 a^{2} b^{2} x^{2}-3 i a \,b^{2} x^{2}+4 a^{3} b x +6 i a^{2} b x +6 a^{4}-2 b^{2} x^{2}+2 a b x +3 b x i+6 a^{2}+2\right )}{6 x^{3} \left (a -i\right )^{2} \left (i+a \right )^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {b^{3} \left (2 a^{2}-2 i a -1\right ) \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{2 \left (a^{2}+1\right )^{\frac {5}{2}} \left (i+a \right )}\)	\(281\)
default	\(i b \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 \left (a^{2}+1\right ) x^{2}}-\frac {3 a b \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) x}+\frac {a b \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+1\right )}+\frac {b^{2} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{2 \left (a^{2}+1\right )^{\frac {3}{2}}}\right )+\left (i a +1\right ) \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{3 \left (a^{2}+1\right ) x^{3}}-\frac {5 a b \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 \left (a^{2}+1\right ) x^{2}}-\frac {3 a b \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) x}+\frac {a b \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+1\right )}+\frac {b^{2} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{2 \left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{3 \left (a^{2}+1\right )}-\frac {2 b^{2} \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) x}+\frac {a b \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{3 \left (a^{2}+1\right )}\right )\)	\(529\)

[In]

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

[Out]

-1/6*I*(3*I*a^4*b*x+2*a^2*b^4*x^4-3*I*a*b^2*x^2+2*a^3*b^3*x^3-9*I*a*b^4*x^4-4*x^4*b^4-15*I*a^2*b^3*x^3-3*I*a^3
*b^2*x^2+2*a^5*b*x-10*a*b^3*x^3+6*I*a^2*b*x+2*a^6-2*a^2*b^2*x^2+3*I*b^3*x^3+4*a^3*b*x+6*a^4-2*b^2*x^2+3*I*b*x+
2*a*b*x+6*a^2+2)/x^3/(a-I)^2/(I+a)^3/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/2*b^3*(-2*I*a+2*a^2-1)/(a^2+1)^(5/2)/(I+a
)*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)

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. 690 vs. \(2 (198) = 396\).

Time = 0.29 (sec) , antiderivative size = 690, normalized size of antiderivative = 2.44 \[ \int \frac {e^{i \arctan (a+b x)}}{x^4} \, dx=\frac {{\left (-2 i \, a^{2} - 9 \, a + 4 i\right )} b^{3} x^{3} - 3 \, \sqrt {\frac {{\left (4 \, a^{4} - 8 i \, a^{3} - 8 \, a^{2} + 4 i \, a + 1\right )} b^{6}}{a^{12} + 2 i \, a^{11} + 4 \, a^{10} + 10 i \, a^{9} + 5 \, a^{8} + 20 i \, a^{7} + 20 i \, a^{5} - 5 \, a^{4} + 10 i \, a^{3} - 4 \, a^{2} + 2 i \, a - 1}} {\left (a^{5} + i \, a^{4} + 2 \, a^{3} + 2 i \, a^{2} + a + i\right )} x^{3} \log \left (-\frac {{\left (2 \, a^{2} - 2 i \, a - 1\right )} b^{4} x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (2 \, a^{2} - 2 i \, a - 1\right )} b^{3} + {\left (a^{7} + i \, a^{6} + 3 \, a^{5} + 3 i \, a^{4} + 3 \, a^{3} + 3 i \, a^{2} + a + i\right )} \sqrt {\frac {{\left (4 \, a^{4} - 8 i \, a^{3} - 8 \, a^{2} + 4 i \, a + 1\right )} b^{6}}{a^{12} + 2 i \, a^{11} + 4 \, a^{10} + 10 i \, a^{9} + 5 \, a^{8} + 20 i \, a^{7} + 20 i \, a^{5} - 5 \, a^{4} + 10 i \, a^{3} - 4 \, a^{2} + 2 i \, a - 1}}}{{\left (2 \, a^{2} - 2 i \, a - 1\right )} b^{3}}\right ) + 3 \, \sqrt {\frac {{\left (4 \, a^{4} - 8 i \, a^{3} - 8 \, a^{2} + 4 i \, a + 1\right )} b^{6}}{a^{12} + 2 i \, a^{11} + 4 \, a^{10} + 10 i \, a^{9} + 5 \, a^{8} + 20 i \, a^{7} + 20 i \, a^{5} - 5 \, a^{4} + 10 i \, a^{3} - 4 \, a^{2} + 2 i \, a - 1}} {\left (a^{5} + i \, a^{4} + 2 \, a^{3} + 2 i \, a^{2} + a + i\right )} x^{3} \log \left (-\frac {{\left (2 \, a^{2} - 2 i \, a - 1\right )} b^{4} x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (2 \, a^{2} - 2 i \, a - 1\right )} b^{3} - {\left (a^{7} + i \, a^{6} + 3 \, a^{5} + 3 i \, a^{4} + 3 \, a^{3} + 3 i \, a^{2} + a + i\right )} \sqrt {\frac {{\left (4 \, a^{4} - 8 i \, a^{3} - 8 \, a^{2} + 4 i \, a + 1\right )} b^{6}}{a^{12} + 2 i \, a^{11} + 4 \, a^{10} + 10 i \, a^{9} + 5 \, a^{8} + 20 i \, a^{7} + 20 i \, a^{5} - 5 \, a^{4} + 10 i \, a^{3} - 4 \, a^{2} + 2 i \, a - 1}}}{{\left (2 \, a^{2} - 2 i \, a - 1\right )} b^{3}}\right ) + {\left ({\left (-2 i \, a^{2} - 9 \, a + 4 i\right )} b^{2} x^{2} - 2 i \, a^{4} + {\left (2 i \, a^{3} + 3 \, a^{2} + 2 i \, a + 3\right )} b x - 4 i \, a^{2} - 2 i\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{6 \, {\left (a^{5} + i \, a^{4} + 2 \, a^{3} + 2 i \, a^{2} + a + i\right )} x^{3}} \]

[In]

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

[Out]

1/6*((-2*I*a^2 - 9*a + 4*I)*b^3*x^3 - 3*sqrt((4*a^4 - 8*I*a^3 - 8*a^2 + 4*I*a + 1)*b^6/(a^12 + 2*I*a^11 + 4*a^
10 + 10*I*a^9 + 5*a^8 + 20*I*a^7 + 20*I*a^5 - 5*a^4 + 10*I*a^3 - 4*a^2 + 2*I*a - 1))*(a^5 + I*a^4 + 2*a^3 + 2*
I*a^2 + a + I)*x^3*log(-((2*a^2 - 2*I*a - 1)*b^4*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(2*a^2 - 2*I*a - 1)*b^3
+ (a^7 + I*a^6 + 3*a^5 + 3*I*a^4 + 3*a^3 + 3*I*a^2 + a + I)*sqrt((4*a^4 - 8*I*a^3 - 8*a^2 + 4*I*a + 1)*b^6/(a
^12 + 2*I*a^11 + 4*a^10 + 10*I*a^9 + 5*a^8 + 20*I*a^7 + 20*I*a^5 - 5*a^4 + 10*I*a^3 - 4*a^2 + 2*I*a - 1)))/((2
*a^2 - 2*I*a - 1)*b^3)) + 3*sqrt((4*a^4 - 8*I*a^3 - 8*a^2 + 4*I*a + 1)*b^6/(a^12 + 2*I*a^11 + 4*a^10 + 10*I*a^
9 + 5*a^8 + 20*I*a^7 + 20*I*a^5 - 5*a^4 + 10*I*a^3 - 4*a^2 + 2*I*a - 1))*(a^5 + I*a^4 + 2*a^3 + 2*I*a^2 + a +
I)*x^3*log(-((2*a^2 - 2*I*a - 1)*b^4*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(2*a^2 - 2*I*a - 1)*b^3 - (a^7 + I*
a^6 + 3*a^5 + 3*I*a^4 + 3*a^3 + 3*I*a^2 + a + I)*sqrt((4*a^4 - 8*I*a^3 - 8*a^2 + 4*I*a + 1)*b^6/(a^12 + 2*I*a^
11 + 4*a^10 + 10*I*a^9 + 5*a^8 + 20*I*a^7 + 20*I*a^5 - 5*a^4 + 10*I*a^3 - 4*a^2 + 2*I*a - 1)))/((2*a^2 - 2*I*a
- 1)*b^3)) + ((-2*I*a^2 - 9*a + 4*I)*b^2*x^2 - 2*I*a^4 + (2*I*a^3 + 3*a^2 + 2*I*a + 3)*b*x - 4*I*a^2 - 2*I)*s
qrt(b^2*x^2 + 2*a*b*x + a^2 + 1))/((a^5 + I*a^4 + 2*a^3 + 2*I*a^2 + a + I)*x^3)

Sympy [F]

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

[In]

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

[Out]

I*(Integral(-I/(x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(a/(x**4*sqrt(a**2 + 2*a*b*x + b**2*x
**2 + 1)), x) + Integral(b/(x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), 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. 644 vs. \(2 (198) = 396\).

Time = 0.21 (sec) , antiderivative size = 644, normalized size of antiderivative = 2.28 \[ \int \frac {e^{i \arctan (a+b x)}}{x^4} \, dx=\frac {5 \, a^{3} {\left (i \, a + 1\right )} b^{3} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{2 \, {\left (a^{2} + 1\right )}^{\frac {7}{2}}} - \frac {3 i \, a^{2} b^{3} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{2 \, {\left (a^{2} + 1\right )}^{\frac {5}{2}}} - \frac {3 \, a {\left (i \, a + 1\right )} b^{3} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{2 \, {\left (a^{2} + 1\right )}^{\frac {5}{2}}} + \frac {i \, b^{3} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{2 \, {\left (a^{2} + 1\right )}^{\frac {3}{2}}} - \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2} {\left (i \, a + 1\right )} b^{2}}{2 \, {\left (a^{2} + 1\right )}^{3} x} + \frac {3 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a b^{2}}{2 \, {\left (a^{2} + 1\right )}^{2} x} - \frac {2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (-i \, a - 1\right )} b^{2}}{3 \, {\left (a^{2} + 1\right )}^{2} x} + \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a {\left (i \, a + 1\right )} b}{6 \, {\left (a^{2} + 1\right )}^{2} x^{2}} - \frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b}{2 \, {\left (a^{2} + 1\right )} x^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (i \, a + 1\right )}}{3 \, {\left (a^{2} + 1\right )} x^{3}} \]

[In]

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

[Out]

5/2*a^3*(I*a + 1)*b^3*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) - 3/2*I*a^2*b^3*arcsi
nh(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)^(5/2) - 3/2*a*(I*a + 1)*b^3*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)^(5/2) + 1/2*I*b^3*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) - 5/2*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^2*(I*a + 1)*b^2/((a^2 + 1)^3*x) + 3/2*I*sqrt(b^2*x^2 + 2*a
*b*x + a^2 + 1)*a*b^2/((a^2 + 1)^2*x) - 2/3*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(-I*a - 1)*b^2/((a^2 + 1)^2*x) +
5/6*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a*(I*a + 1)*b/((a^2 + 1)^2*x^2) - 1/2*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 +
1)*b/((a^2 + 1)*x^2) - 1/3*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(I*a + 1)/((a^2 + 1)*x^3)

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. 884 vs. \(2 (198) = 396\).

Time = 0.34 (sec) , antiderivative size = 884, normalized size of antiderivative = 3.12 \[ \int \frac {e^{i \arctan (a+b x)}}{x^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*(2*a^2*b^3 - 2*I*a*b^3 - b^3)*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)))/((a^5 + I*a^4 + 2*a^3 + 2*I*a^2 + a + I)*sqrt(a^2 + 1)) + 1/3
*(8*I*(x*abs(b) - sqrt((b*x + a)^2 + 1))^3*a^5*b^3 + 24*(I*x*abs(b) - I*sqrt((b*x + a)^2 + 1))*a^7*b^3 + 24*I*
(x*abs(b) - sqrt((b*x + a)^2 + 1))^2*a^6*b^2*abs(b) + 8*I*a^8*b^2*abs(b) + 6*(x*abs(b) - sqrt((b*x + a)^2 + 1)
)^5*a^2*b^3 - 24*(x*abs(b) - sqrt((b*x + a)^2 + 1))^3*a^4*b^3 + 18*(x*abs(b) - sqrt((b*x + a)^2 + 1))*a^6*b^3
- 12*(x*abs(b) - sqrt((b*x + a)^2 + 1))^2*a^5*b^2*abs(b) + 12*a^7*b^2*abs(b) - 6*I*(x*abs(b) - sqrt((b*x + a)^
2 + 1))^5*a*b^3 + 32*I*(x*abs(b) - sqrt((b*x + a)^2 + 1))^3*a^3*b^3 + 54*(I*x*abs(b) - I*sqrt((b*x + a)^2 + 1)
)*a^5*b^3 + 60*I*(x*abs(b) - sqrt((b*x + a)^2 + 1))^2*a^4*b^2*abs(b) + 20*I*a^6*b^2*abs(b) - 3*(x*abs(b) - sqr
t((b*x + a)^2 + 1))^5*b^3 - 24*(x*abs(b) - sqrt((b*x + a)^2 + 1))^3*a^2*b^3 + 39*(x*abs(b) - sqrt((b*x + a)^2
+ 1))*a^4*b^3 - 24*(x*abs(b) - sqrt((b*x + a)^2 + 1))^2*a^3*b^2*abs(b) + 36*a^5*b^2*abs(b) + 24*I*(x*abs(b) -
sqrt((b*x + a)^2 + 1))^3*a*b^3 + 36*(I*x*abs(b) - I*sqrt((b*x + a)^2 + 1))*a^3*b^3 + 48*I*(x*abs(b) - sqrt((b*
x + a)^2 + 1))^2*a^2*b^2*abs(b) + 12*I*a^4*b^2*abs(b) + 24*(x*abs(b) - sqrt((b*x + a)^2 + 1))*a^2*b^3 - 12*(x*
abs(b) - sqrt((b*x + a)^2 + 1))^2*a*b^2*abs(b) + 36*a^3*b^2*abs(b) + 6*(I*x*abs(b) - I*sqrt((b*x + a)^2 + 1))*
a*b^3 + 12*I*(x*abs(b) - sqrt((b*x + a)^2 + 1))^2*b^2*abs(b) - 4*I*a^2*b^2*abs(b) + 3*(x*abs(b) - sqrt((b*x +
a)^2 + 1))*b^3 + 12*a*b^2*abs(b) - 4*I*b^2*abs(b))/((a^5 + I*a^4 + 2*a^3 + 2*I*a^2 + a + I)*((x*abs(b) - sqrt(
(b*x + a)^2 + 1))^2 - a^2 - 1)^3)

Mupad [F(-1)]

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

[In]

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

[Out]

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