\(\int \frac {e^{-\text {arctanh}(a+b x)}}{x^3} \, dx\) [886]

Optimal result

Integrand size = 14, antiderivative size = 162 \[ \int \frac {e^{-\text {arctanh}(a+b x)}}{x^3} \, dx=\frac {(1-2 a) b \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 (1-a) (1+a)^2 x}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 \left (1-a^2\right ) x^2}-\frac {(1-2 a) b^2 \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{(1-a) (1+a)^2 \sqrt {1-a^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.75 \[ \int \frac {e^{-\text {arctanh}(a+b x)}}{x^3} \, dx=-\frac {\left (-1+a^2+2 b x-a b x\right ) \sqrt {1-a^2-2 a b x-b^2 x^2}}{2 (-1+a) (1+a)^2 x^2}+\frac {(-1+2 a) b^2 \text {arctanh}\left (\frac {\sqrt {-1-a} \sqrt {1-a-b x}}{\sqrt {-1+a} \sqrt {1+a+b x}}\right )}{(-1-a)^{5/2} (-1+a)^{3/2}} \] Input:

Integrate[1/(E^ArcTanh[a + b*x]*x^3),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6713, 107, 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.

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

\(\Big \downarrow \) 6713

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

\(\Big \downarrow \) 107

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

Int[1/(E^ArcTanh[a + b*x]*x^3),x]
 

Output:

-1/2*((1 - a - b*x)^(3/2)*Sqrt[1 + a + b*x])/((1 - a^2)*x^2) - ((1 - 2*a)* 
b*(-((Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/((1 + a)*x)) + (2*b*ArcTanh[(Sq 
rt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 + a)*Sq 
rt[1 - a^2])))/(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 = 153, normalized size of antiderivative = 0.94

method	result	size
risch	\(\frac {\left (b^{2} x^{2}+2 a b x +a^{2}-1\right ) \left (-a b x +a^{2}+2 b x -1\right )}{2 \left (a +1\right ) x^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \left (a^{2}-1\right )}-\frac {b^{2} \left (-1+2 a \right ) \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{2 \left (a^{2}-1\right ) \left (a +1\right ) \sqrt {-a^{2}+1}}\)	\(153\)
default	\(\text {Expression too large to display}\)	\(1046\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.20 \[ \int \frac {e^{-\text {arctanh}(a+b x)}}{x^3} \, dx=\left [-\frac {\sqrt {-a^{2} + 1} {\left (2 \, a - 1\right )} b^{2} x^{2} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x + 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) + 2 \, {\left (a^{4} - {\left (a^{3} - 2 \, a^{2} - a + 2\right )} b x - 2 \, a^{2} + 1\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{4 \, {\left (a^{5} + a^{4} - 2 \, a^{3} - 2 \, a^{2} + a + 1\right )} x^{2}}, \frac {\sqrt {a^{2} - 1} {\left (2 \, a - 1\right )} b^{2} x^{2} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) - {\left (a^{4} - {\left (a^{3} - 2 \, a^{2} - a + 2\right )} b x - 2 \, a^{2} + 1\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, {\left (a^{5} + a^{4} - 2 \, a^{3} - 2 \, a^{2} + a + 1\right )} x^{2}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 750 vs. \(2 (134) = 268\).

Time = 0.14 (sec) , antiderivative size = 750, normalized size of antiderivative = 4.63 \[ \int \frac {e^{-\text {arctanh}(a+b x)}}{x^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*a*b^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) + a^2*abs(b) - a*abs(b) - a 
bs(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) - 2*(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)*abs(b) + b)^3*a^3*b^3/(b^2*x + a*b)^3 - 2*a^3*b^3 + 6*(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 + 2*(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*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a*b^3/(b^2*x + a*b) 
 - 4*(sqrt(-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*(sqrt(-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) + a^4*abs(b) - 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. \[ \int \frac {e^{-\text {arctanh}(a+b x)}}{x^3} \, dx=\int \frac {\sqrt {1-{\left (a+b\,x\right )}^2}}{x^3\,\left (a+b\,x+1\right )} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 533, normalized size of antiderivative = 3.29 \[ \int \frac {e^{-\text {arctanh}(a+b x)}}{x^3} \, dx=\frac {2 \sqrt {-a^{2}+1}\, \mathit {atan} \left (\frac {\sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2} i +\sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a b i x -\sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, i}{a^{2} b^{2} x^{2}+2 a^{3} b x +a^{4}-b^{2} x^{2}-2 a b x -2 a^{2}+1}\right ) a \,b^{2} i \,x^{2}-\sqrt {-a^{2}+1}\, \mathit {atan} \left (\frac {\sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2} i +\sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a b i x -\sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, i}{a^{2} b^{2} x^{2}+2 a^{3} b x +a^{4}-b^{2} x^{2}-2 a b x -2 a^{2}+1}\right ) b^{2} i \,x^{2}-\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{4}+\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{3} b x -2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2} b x +2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2}-\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a b x +2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x -\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 x^{2} \left (a^{5}+a^{4}-2 a^{3}-2 a^{2}+a +1\right )} \] Input:

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

Output:

(2*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 - 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))*b**2*i*x**2 - sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a**4 + 
sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a**3*b*x - 2*sqrt( - a**2 - 2*a*b* 
x - b**2*x**2 + 1)*a**2*b*x + 2*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a* 
*2 - sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a*b*x + 2*sqrt( - a**2 - 2*a* 
b*x - b**2*x**2 + 1)*b*x - sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1))/(2*x** 
2*(a**5 + a**4 - 2*a**3 - 2*a**2 + a + 1))