\(\int \frac {e^{3 i \arctan (a+b x)}}{x^2} \, dx\) [186]

Optimal result

Integrand size = 16, antiderivative size = 176 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x^2} \, dx=-\frac {6 i b \sqrt {1+i a+i b x}}{(i+a)^2 \sqrt {1-i a-i b x}}-\frac {(1+i a+i b x)^{3/2}}{(1-i a) x \sqrt {1-i a-i b x}}+\frac {6 i \sqrt {i-a} b \text {arctanh}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{(i+a)^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5203, 96, 95, 214} \[ \int \frac {e^{3 i \arctan (a+b x)}}{x^2} \, dx=\frac {6 i \sqrt {-a+i} b \text {arctanh}\left (\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-a+i} \sqrt {-i a-i b x+1}}\right )}{(a+i)^{5/2}}-\frac {(i a+i b x+1)^{3/2}}{(1-i a) x \sqrt {-i a-i b x+1}}-\frac {6 i b \sqrt {i a+i b x+1}}{(a+i)^2 \sqrt {-i a-i b x+1}} \]

[In]

Int[E^((3*I)*ArcTan[a + b*x])/x^2,x]

[Out]

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

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 96

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[n*((d*e - c*f)/((m + 1)*(b*e - a*f
))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[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 {(1+i a+i b x)^{3/2}}{x^2 (1-i a-i b x)^{3/2}} \, dx \\ & = -\frac {(1+i a+i b x)^{3/2}}{(1-i a) x \sqrt {1-i a-i b x}}-\frac {(3 b) \int \frac {\sqrt {1+i a+i b x}}{x (1-i a-i b x)^{3/2}} \, dx}{i+a} \\ & = -\frac {6 i b \sqrt {1+i a+i b x}}{(i+a)^2 \sqrt {1-i a-i b x}}-\frac {(1+i a+i b x)^{3/2}}{(1-i a) x \sqrt {1-i a-i b x}}-\frac {(3 (i-a) b) \int \frac {1}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{(i+a)^2} \\ & = -\frac {6 i b \sqrt {1+i a+i b x}}{(i+a)^2 \sqrt {1-i a-i b x}}-\frac {(1+i a+i b x)^{3/2}}{(1-i a) x \sqrt {1-i a-i b x}}-\frac {(6 (i-a) b) \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} \\ & = -\frac {6 i b \sqrt {1+i a+i b x}}{(i+a)^2 \sqrt {1-i a-i b x}}-\frac {(1+i a+i b x)^{3/2}}{(1-i a) x \sqrt {1-i a-i b x}}+\frac {6 i \sqrt {i-a} b \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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.82 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x^2} \, dx=\frac {\frac {\sqrt {1+i a+i b x} \left (1+a^2-5 i b x+a b x\right )}{x \sqrt {-i (i+a+b x)}}+\frac {6 i \sqrt {-1-i a} b \text {arctanh}\left (\frac {\sqrt {-1-i a} \sqrt {-i (i+a+b x)}}{\sqrt {-1+i a} \sqrt {1+i a+i b x}}\right )}{\sqrt {-1+i a}}}{(i+a)^2} \]

[In]

Integrate[E^((3*I)*ArcTan[a + b*x])/x^2,x]

[Out]

((Sqrt[1 + I*a + I*b*x]*(1 + a^2 - (5*I)*b*x + a*b*x))/(x*Sqrt[(-I)*(I + a + b*x)]) + ((6*I)*Sqrt[-1 - I*a]*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])/(I
+ a)^2

Maple [A] (verified)

Time = 1.58 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.03

method	result	size
risch	\(\frac {i \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \left (a -i\right )}{\left (i+a \right )^{2} x}+\frac {b \left (-\frac {\left (3 a^{2}+3\right ) \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (i+a \right ) \sqrt {a^{2}+1}}-\frac {4 i \left (i a -1\right ) \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}}{b \left (i+a \right ) \left (x +\frac {i+a}{b}\right )}\right )}{a^{2}+2 i a -1}\)	\(181\)
default	\(-\frac {6 b^{2} \left (2 b^{2} x +2 a b \right )}{\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-i b^{3} \left (-\frac {1}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a \left (2 b^{2} x +2 a b \right )}{b \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}\right )-\frac {6 i a \,b^{2} \left (2 b^{2} x +2 a b \right )}{\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-3 b \left (i a^{2}+2 a -i\right ) \left (\frac {1}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a b \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\right ) \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )+\left (-i a^{3}-3 a^{2}+3 i a +1\right ) \left (-\frac {1}{\left (a^{2}+1\right ) x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {3 a b \left (\frac {1}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a b \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\right ) \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{a^{2}+1}-\frac {4 b^{2} \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\right ) \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}\right )\)	\(631\)

[In]

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

[Out]

I*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*(a-I)/(I+a)^2/x+1/(2*I*a+a^2-1)*b*(-(3*a^2+3)/(I+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)-4*I*(I*a-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. 389 vs. \(2 (116) = 232\).

Time = 0.30 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.21 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x^2} \, dx=-\frac {{\left (-i \, a - 5\right )} b^{2} x^{2} + {\left (-i \, a^{2} - 4 \, a - 5 i\right )} b x - 3 \, {\left ({\left (a^{2} + 2 i \, a - 1\right )} b x^{2} + {\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} x\right )} \sqrt {\frac {{\left (a - i\right )} b^{2}}{a^{5} + 5 i \, a^{4} - 10 \, a^{3} - 10 i \, a^{2} + 5 \, a + i}} \log \left (-\frac {b^{2} x + {\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} \sqrt {\frac {{\left (a - i\right )} b^{2}}{a^{5} + 5 i \, a^{4} - 10 \, a^{3} - 10 i \, a^{2} + 5 \, a + i}} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b}{b}\right ) + 3 \, {\left ({\left (a^{2} + 2 i \, a - 1\right )} b x^{2} + {\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} x\right )} \sqrt {\frac {{\left (a - i\right )} b^{2}}{a^{5} + 5 i \, a^{4} - 10 \, a^{3} - 10 i \, a^{2} + 5 \, a + i}} \log \left (-\frac {b^{2} x - {\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} \sqrt {\frac {{\left (a - i\right )} b^{2}}{a^{5} + 5 i \, a^{4} - 10 \, a^{3} - 10 i \, a^{2} + 5 \, a + i}} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b}{b}\right ) + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left ({\left (-i \, a - 5\right )} b x - i \, a^{2} - i\right )}}{{\left (a^{2} + 2 i \, a - 1\right )} b x^{2} + {\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} x} \]

[In]

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

[Out]

-((-I*a - 5)*b^2*x^2 + (-I*a^2 - 4*a - 5*I)*b*x - 3*((a^2 + 2*I*a - 1)*b*x^2 + (a^3 + 3*I*a^2 - 3*a - I)*x)*sq
rt((a - I)*b^2/(a^5 + 5*I*a^4 - 10*a^3 - 10*I*a^2 + 5*a + I))*log(-(b^2*x + (a^3 + 3*I*a^2 - 3*a - I)*sqrt((a
- I)*b^2/(a^5 + 5*I*a^4 - 10*a^3 - 10*I*a^2 + 5*a + I)) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b)/b) + 3*((a^2 +
2*I*a - 1)*b*x^2 + (a^3 + 3*I*a^2 - 3*a - I)*x)*sqrt((a - I)*b^2/(a^5 + 5*I*a^4 - 10*a^3 - 10*I*a^2 + 5*a + I)
)*log(-(b^2*x - (a^3 + 3*I*a^2 - 3*a - I)*sqrt((a - I)*b^2/(a^5 + 5*I*a^4 - 10*a^3 - 10*I*a^2 + 5*a + I)) - sq
rt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b)/b) + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*((-I*a - 5)*b*x - I*a^2 - I))/((a^2
+ 2*I*a - 1)*b*x^2 + (a^3 + 3*I*a^2 - 3*a - I)*x)

Sympy [F]

\[ \int \frac {e^{3 i \arctan (a+b x)}}{x^2} \, dx=- i \left (\int \frac {i}{a^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a}{a^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3}}{a^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i a^{2}}{a^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \left (- \frac {3 b x}{a^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {b^{3} x^{3}}{a^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i b^{2} x^{2}}{a^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {3 a b^{2} x^{2}}{a^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} b x}{a^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {6 i a b x}{a^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx\right ) \]

[In]

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

[Out]

-I*(Integral(I/(a**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 +
1) + b**2*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integra
l(-3*a/(a**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**
2*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(a**3/(
a**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**4*s
qrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*I*a**2/(a**
2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**4*sqrt
(a**2 + 2*a*b*x + b**2*x**2 + 1) + x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*b*x/(a**2*x**2
*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**4*sqrt(a**2
+ 2*a*b*x + b**2*x**2 + 1) + x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(b**3*x**3/(a**2*x**2*sq
rt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**4*sqrt(a**2 + 2
*a*b*x + b**2*x**2 + 1) + x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*I*b**2*x**2/(a**2*x**2*
sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**4*sqrt(a**2 +
2*a*b*x + b**2*x**2 + 1) + x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(3*a*b**2*x**2/(a**2*x**2
*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**4*sqrt(a**2
+ 2*a*b*x + b**2*x**2 + 1) + x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(3*a**2*b*x/(a**2*x**2*s
qrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**4*sqrt(a**2 +
2*a*b*x + b**2*x**2 + 1) + x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-6*I*a*b*x/(a**2*x**2*sqr
t(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**4*sqrt(a**2 + 2*
a*b*x + b**2*x**2 + 1) + x**2*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. 992 vs. \(2 (116) = 232\).

Time = 0.21 (sec) , antiderivative size = 992, normalized size of antiderivative = 5.64 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

undef

Mupad [F(-1)]

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

[In]

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

[Out]

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