\(\int \frac {e^{3 \text {arctanh}(a+b x)}}{x^3} \, dx\) [878]

Optimal result

Integrand size = 14, antiderivative size = 202 \[ \int \frac {e^{3 \text {arctanh}(a+b x)}}{x^3} \, dx=\frac {3 (3+2 a) b^2 \sqrt {1+a+b x}}{(1-a)^3 (1+a) \sqrt {1-a-b x}}-\frac {(3+2 a) b (1+a+b x)^{3/2}}{2 (1-a)^2 (1+a) x \sqrt {1-a-b x}}-\frac {(1+a+b x)^{5/2}}{2 \left (1-a^2\right ) x^2 \sqrt {1-a-b x}}-\frac {3 (3+2 a) b^2 \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{(1-a)^3 \sqrt {1-a^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.70 \[ \int \frac {e^{3 \text {arctanh}(a+b x)}}{x^3} \, dx=\frac {\sqrt {1+a+b x} \left (1-a^2+a^3+5 b x-14 b^2 x^2-a \left (1+5 b x+b^2 x^2\right )\right )}{2 (-1+a)^3 x^2 \sqrt {1-a-b x}}-\frac {3 (3+2 a) b^2 \text {arctanh}\left (\frac {\sqrt {-1-a} \sqrt {1-a-b x}}{\sqrt {-1+a} \sqrt {1+a+b x}}\right )}{\sqrt {-1-a} (-1+a)^{7/2}} \] Input:

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

Output:

(Sqrt[1 + a + b*x]*(1 - a^2 + a^3 + 5*b*x - 14*b^2*x^2 - a*(1 + 5*b*x + b^ 
2*x^2)))/(2*(-1 + a)^3*x^2*Sqrt[1 - a - b*x]) - (3*(3 + 2*a)*b^2*ArcTanh[( 
Sqrt[-1 - a]*Sqrt[1 - a - b*x])/(Sqrt[-1 + a]*Sqrt[1 + a + b*x])])/(Sqrt[- 
1 - a]*(-1 + a)^(7/2))
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6713, 107, 105, 105, 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.

\(\displaystyle \int \frac {e^{3 \text {arctanh}(a+b x)}}{x^3} \, dx\)

\(\Big \downarrow \) 6713

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

\(\Big \downarrow \) 107

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/2*(1 + a + b*x)^(5/2)/((1 - a^2)*x^2*Sqrt[1 - a - b*x]) + ((3 + 2*a)*b* 
(-((1 + a + b*x)^(3/2)/((1 - a)*x*Sqrt[1 - a - b*x])) + (3*b*((2*Sqrt[1 + 
a + b*x])/((1 - a)*Sqrt[1 - a - b*x]) - (2*(1 + a)*ArcTanh[(Sqrt[1 - a]*Sq 
rt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 - a)*Sqrt[1 - a^2]) 
))/(1 - a)))/(2*(1 - a^2))
 

Defintions of rubi rules used

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 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[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 107

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(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] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 
 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x 
] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
 

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 6713

Int[E^(ArcTanh[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.) 
, x_Symbol] :> Int[(d + e*x)^m*((1 + a*c + b*c*x)^(n/2)/(1 - a*c - b*c*x)^( 
n/2)), x] /; FreeQ[{a, b, c, d, e, m, n}, x]
 

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.94

method	result	size
risch	\(\frac {\left (b^{2} x^{2}+2 a b x +a^{2}-1\right ) \left (-a b x +a^{2}-6 b x -1\right )}{2 \left (-1+a \right )^{3} x^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {b^{2} \left (-\frac {\left (6 a +9\right ) \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\sqrt {-a^{2}+1}}-\frac {8 \sqrt {-\left (x +\frac {-1+a}{b}\right )^{2} b^{2}-2 \left (x +\frac {-1+a}{b}\right ) b}}{b \left (x +\frac {-1+a}{b}\right )}\right )}{2 \left (-1+a \right )^{3}}\)	\(189\)
default	\(\text {Expression too large to display}\)	\(1052\)

Input:

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

Output:

1/2*(b^2*x^2+2*a*b*x+a^2-1)*(-a*b*x+a^2-6*b*x-1)/(-1+a)^3/x^2/(-b^2*x^2-2* 
a*b*x-a^2+1)^(1/2)-1/2*b^2/(-1+a)^3*(-(6*a+9)/(-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/b/(x+(-1+a)/ 
b)*(-(x+(-1+a)/b)^2*b^2-2*(x+(-1+a)/b)*b)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 523, normalized size of antiderivative = 2.59 \[ \int \frac {e^{3 \text {arctanh}(a+b x)}}{x^3} \, dx=\left [-\frac {3 \, {\left ({\left (2 \, a + 3\right )} b^{3} x^{3} + {\left (2 \, a^{2} + a - 3\right )} b^{2} x^{2}\right )} \sqrt {-a^{2} + 1} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) + 2 \, {\left (a^{5} - {\left (a^{3} + 14 \, a^{2} - a - 14\right )} b^{2} x^{2} - a^{4} - 2 \, a^{3} - 5 \, {\left (a^{3} - a^{2} - a + 1\right )} b x + 2 \, a^{2} + a - 1\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{4 \, {\left ({\left (a^{5} - 3 \, a^{4} + 2 \, a^{3} + 2 \, a^{2} - 3 \, a + 1\right )} b x^{3} + {\left (a^{6} - 4 \, a^{5} + 5 \, a^{4} - 5 \, a^{2} + 4 \, a - 1\right )} x^{2}\right )}}, -\frac {3 \, {\left ({\left (2 \, a + 3\right )} b^{3} x^{3} + {\left (2 \, a^{2} + a - 3\right )} b^{2} x^{2}\right )} \sqrt {a^{2} - 1} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) + {\left (a^{5} - {\left (a^{3} + 14 \, a^{2} - a - 14\right )} b^{2} x^{2} - a^{4} - 2 \, a^{3} - 5 \, {\left (a^{3} - a^{2} - a + 1\right )} b x + 2 \, a^{2} + a - 1\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, {\left ({\left (a^{5} - 3 \, a^{4} + 2 \, a^{3} + 2 \, a^{2} - 3 \, a + 1\right )} b x^{3} + {\left (a^{6} - 4 \, a^{5} + 5 \, a^{4} - 5 \, a^{2} + 4 \, a - 1\right )} x^{2}\right )}}\right ] \] Input:

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

Output:

[-1/4*(3*((2*a + 3)*b^3*x^3 + (2*a^2 + a - 3)*b^2*x^2)*sqrt(-a^2 + 1)*log( 
((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x - 2*sqrt(-b^2*x^2 - 2*a*b*x 
 - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(-a^2 + 1) - 4*a^2 + 2)/x^2) + 2*(a^5 - 
(a^3 + 14*a^2 - a - 14)*b^2*x^2 - a^4 - 2*a^3 - 5*(a^3 - a^2 - a + 1)*b*x 
+ 2*a^2 + a - 1)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^5 - 3*a^4 + 2*a^3 
 + 2*a^2 - 3*a + 1)*b*x^3 + (a^6 - 4*a^5 + 5*a^4 - 5*a^2 + 4*a - 1)*x^2), 
-1/2*(3*((2*a + 3)*b^3*x^3 + (2*a^2 + a - 3)*b^2*x^2)*sqrt(a^2 - 1)*arctan 
(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(a^2 - 1)/((a^2 
- 1)*b^2*x^2 + a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1)) + (a^5 - (a^3 + 14*a^2 
- a - 14)*b^2*x^2 - a^4 - 2*a^3 - 5*(a^3 - a^2 - a + 1)*b*x + 2*a^2 + a - 
1)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^5 - 3*a^4 + 2*a^3 + 2*a^2 - 3*a 
 + 1)*b*x^3 + (a^6 - 4*a^5 + 5*a^4 - 5*a^2 + 4*a - 1)*x^2)]
 

Sympy [F]

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

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

Output:

Integral((a + b*x + 1)**3/(x**3*(-(a + b*x - 1)*(a + b*x + 1))**(3/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {e^{3 \text {arctanh}(a+b x)}}{x^3} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a-1>0)', see `assume?` for more 
details)Is
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 826 vs. \(2 (168) = 336\).

Time = 0.15 (sec) , antiderivative size = 826, normalized size of antiderivative = 4.09 \[ \int \frac {e^{3 \text {arctanh}(a+b x)}}{x^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-8*b^3/((a^3*abs(b) - 3*a^2*abs(b) + 3*a*abs(b) - abs(b))*((sqrt(-b^2*x^2 
- 2*a*b*x - a^2 + 1)*abs(b) + b)/(b^2*x + a*b) - 1)) - 3*(2*a*b^3 + 3*b^3) 
*arctan(((sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a/(b^2*x + a*b) - 
 1)/sqrt(a^2 - 1))/((a^3*abs(b) - 3*a^2*abs(b) + 3*a*abs(b) - abs(b))*sqrt 
(a^2 - 1)) + (2*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a^4*b^3/ 
(b^2*x + a*b)^2 + 2*a^4*b^3 - 5*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) 
 + b)*a^3*b^3/(b^2*x + a*b) + 6*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) 
 + b)^2*a^3*b^3/(b^2*x + a*b)^2 - 3*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*ab 
s(b) + b)^3*a^3*b^3/(b^2*x + a*b)^3 + 6*a^3*b^3 - 18*(sqrt(-b^2*x^2 - 2*a* 
b*x - a^2 + 1)*abs(b) + b)*a^2*b^3/(b^2*x + a*b) + 3*(sqrt(-b^2*x^2 - 2*a* 
b*x - a^2 + 1)*abs(b) + b)^2*a^2*b^3/(b^2*x + a*b)^2 - 6*(sqrt(-b^2*x^2 - 
2*a*b*x - a^2 + 1)*abs(b) + b)^3*a^2*b^3/(b^2*x + a*b)^3 - a^2*b^3 + 2*(sq 
rt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a*b^3/(b^2*x + a*b) + 12*(sqr 
t(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a*b^3/(b^2*x + a*b)^2 + 2*(s 
qrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*a*b^3/(b^2*x + a*b)^3 - 2* 
(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*b^3/(b^2*x + a*b)^2)/((a 
^5*abs(b) - 3*a^4*abs(b) + 3*a^3*abs(b) - a^2*abs(b))*((sqrt(-b^2*x^2 - 2* 
a*b*x - a^2 + 1)*abs(b) + b)^2*a/(b^2*x + a*b)^2 + a - 2*(sqrt(-b^2*x^2 - 
2*a*b*x - a^2 + 1)*abs(b) + b)/(b^2*x + a*b))^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 1258, normalized size of antiderivative = 6.23 \[ \int \frac {e^{3 \text {arctanh}(a+b x)}}{x^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 6*sqrt( - a**2 + 1)*atan((sqrt( - a**2 + 1)*sqrt( - a**2 - 2*a*b*x - b 
**2*x**2 + 1)*a**2*i + sqrt( - a**2 + 1)*sqrt( - a**2 - 2*a*b*x - b**2*x** 
2 + 1)*a*b*i*x - sqrt( - a**2 + 1)*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1) 
*i)/(a**4 + 2*a**3*b*x + a**2*b**2*x**2 - 2*a**2 - 2*a*b*x - b**2*x**2 + 1 
))*a**2*b**2*i*x**2 - 6*sqrt( - a**2 + 1)*atan((sqrt( - a**2 + 1)*sqrt( - 
a**2 - 2*a*b*x - b**2*x**2 + 1)*a**2*i + sqrt( - a**2 + 1)*sqrt( - a**2 - 
2*a*b*x - b**2*x**2 + 1)*a*b*i*x - sqrt( - a**2 + 1)*sqrt( - a**2 - 2*a*b* 
x - b**2*x**2 + 1)*i)/(a**4 + 2*a**3*b*x + a**2*b**2*x**2 - 2*a**2 - 2*a*b 
*x - b**2*x**2 + 1))*a*b**3*i*x**3 - 3*sqrt( - a**2 + 1)*atan((sqrt( - a** 
2 + 1)*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a**2*i + sqrt( - a**2 + 1)* 
sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a*b*i*x - sqrt( - a**2 + 1)*sqrt( 
- a**2 - 2*a*b*x - b**2*x**2 + 1)*i)/(a**4 + 2*a**3*b*x + a**2*b**2*x**2 - 
 2*a**2 - 2*a*b*x - b**2*x**2 + 1))*a*b**2*i*x**2 - 9*sqrt( - a**2 + 1)*at 
an((sqrt( - a**2 + 1)*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a**2*i + sqr 
t( - a**2 + 1)*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a*b*i*x - sqrt( - a 
**2 + 1)*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*i)/(a**4 + 2*a**3*b*x + a 
**2*b**2*x**2 - 2*a**2 - 2*a*b*x - b**2*x**2 + 1))*b**3*i*x**3 + 9*sqrt( - 
 a**2 + 1)*atan((sqrt( - a**2 + 1)*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1) 
*a**2*i + sqrt( - a**2 + 1)*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a*b*i* 
x - sqrt( - a**2 + 1)*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*i)/(a**4 ...