\(\int \frac {e^{\text {arctanh}(a+b x)}}{x^4} \, dx\) [847]

Optimal result

Integrand size = 12, antiderivative size = 213 \[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^4} \, dx=-\frac {\sqrt {1-a-b x} \sqrt {1+a+b x}}{3 (1-a) x^3}-\frac {(3+2 a) b \sqrt {1-a-b x} \sqrt {1+a+b x}}{6 (1-a)^2 (1+a) x^2}-\frac {(4+a) (1+2 a) b^2 \sqrt {1-a-b x} \sqrt {1+a+b x}}{6 (1-a)^3 (1+a)^2 x}-\frac {\left (1+2 a+2 a^2\right ) b^3 \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{(1-a) \left (1-a^2\right )^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.86 \[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^4} \, dx=\frac {\frac {2 (-1+a) (1+a) \sqrt {1-a-b x} (1+a+b x)^{3/2}}{x^3}-\frac {(1+4 a) b \sqrt {1-a-b x} (1+a+b x)^{3/2}}{x^2}+3 \left (1+2 a+2 a^2\right ) b^2 \left (\frac {\sqrt {-((-1+a+b x) (1+a+b x))}}{(-1+a) x}-\frac {2 b \text {arctanh}\left (\frac {\sqrt {-1-a} \sqrt {1-a-b x}}{\sqrt {-1+a} \sqrt {1+a+b x}}\right )}{\sqrt {-1-a} (-1+a)^{3/2}}\right )}{6 \left (-1+a^2\right )^2} \] Input:

Integrate[E^ArcTanh[a + b*x]/x^4,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6713, 110, 27, 168, 25, 27, 168, 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.

\(\displaystyle \int \frac {e^{\text {arctanh}(a+b x)}}{x^4} \, dx\)

\(\Big \downarrow \) 6713

\(\displaystyle \int \frac {\sqrt {a+b x+1}}{x^4 \sqrt {-a-b x+1}}dx\)

\(\Big \downarrow \) 110

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

\(\displaystyle \frac {b \left (-\frac {\int -\frac {b ((a+4) (2 a+1)+(2 a+3) b x)}{x^2 \sqrt {-a-b x+1} \sqrt {a+b x+1}}dx}{2 \left (1-a^2\right )}-\frac {(2 a+3) \sqrt {-a-b x+1} \sqrt {a+b x+1}}{2 \left (1-a^2\right ) x^2}\right )}{3 (1-a)}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{3 (1-a) x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {\int \frac {b ((a+4) (2 a+1)+(2 a+3) b x)}{x^2 \sqrt {-a-b x+1} \sqrt {a+b x+1}}dx}{2 \left (1-a^2\right )}-\frac {(2 a+3) \sqrt {-a-b x+1} \sqrt {a+b x+1}}{2 \left (1-a^2\right ) x^2}\right )}{3 (1-a)}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{3 (1-a) x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {b \int \frac {(a+4) (2 a+1)+(2 a+3) b x}{x^2 \sqrt {-a-b x+1} \sqrt {a+b x+1}}dx}{2 \left (1-a^2\right )}-\frac {(2 a+3) \sqrt {-a-b x+1} \sqrt {a+b x+1}}{2 \left (1-a^2\right ) x^2}\right )}{3 (1-a)}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{3 (1-a) x^3}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {b \left (\frac {b \left (-\frac {\int -\frac {3 \left (2 a^2+2 a+1\right ) b}{x \sqrt {-a-b x+1} \sqrt {a+b x+1}}dx}{1-a^2}-\frac {(a+4) (2 a+1) \sqrt {-a-b x+1} \sqrt {a+b x+1}}{\left (1-a^2\right ) x}\right )}{2 \left (1-a^2\right )}-\frac {(2 a+3) \sqrt {-a-b x+1} \sqrt {a+b x+1}}{2 \left (1-a^2\right ) x^2}\right )}{3 (1-a)}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{3 (1-a) x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {b \left (\frac {3 \left (2 a^2+2 a+1\right ) b \int \frac {1}{x \sqrt {-a-b x+1} \sqrt {a+b x+1}}dx}{1-a^2}-\frac {(a+4) (2 a+1) \sqrt {-a-b x+1} \sqrt {a+b x+1}}{\left (1-a^2\right ) x}\right )}{2 \left (1-a^2\right )}-\frac {(2 a+3) \sqrt {-a-b x+1} \sqrt {a+b x+1}}{2 \left (1-a^2\right ) x^2}\right )}{3 (1-a)}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{3 (1-a) x^3}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {b \left (\frac {b \left (\frac {6 \left (2 a^2+2 a+1\right ) b \int \frac {1}{-a+\frac {(1-a) (a+b x+1)}{-a-b x+1}-1}d\frac {\sqrt {a+b x+1}}{\sqrt {-a-b x+1}}}{1-a^2}-\frac {(a+4) (2 a+1) \sqrt {-a-b x+1} \sqrt {a+b x+1}}{\left (1-a^2\right ) x}\right )}{2 \left (1-a^2\right )}-\frac {(2 a+3) \sqrt {-a-b x+1} \sqrt {a+b x+1}}{2 \left (1-a^2\right ) x^2}\right )}{3 (1-a)}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{3 (1-a) x^3}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {b \left (\frac {b \left (-\frac {6 \left (2 a^2+2 a+1\right ) b \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{\left (1-a^2\right )^{3/2}}-\frac {(a+4) (2 a+1) \sqrt {-a-b x+1} \sqrt {a+b x+1}}{\left (1-a^2\right ) x}\right )}{2 \left (1-a^2\right )}-\frac {(2 a+3) \sqrt {-a-b x+1} \sqrt {a+b x+1}}{2 \left (1-a^2\right ) x^2}\right )}{3 (1-a)}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{3 (1-a) x^3}\)

Input:

Int[E^ArcTanh[a + b*x]/x^4,x]
 

Output:

-1/3*(Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/((1 - a)*x^3) + (b*(-1/2*((3 + 
2*a)*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/((1 - a^2)*x^2) + (b*(-(((4 + a) 
*(1 + 2*a)*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/((1 - a^2)*x)) - (6*(1 + 2 
*a + 2*a^2)*b*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 
- a - b*x])])/(1 - a^2)^(3/2)))/(2*(1 - a^2))))/(3*(1 - 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 110

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[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] && Gt 
Q[n, 0] && (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 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 = 0.54 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.97

method	result	size
risch	\(-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}-1\right ) \left (2 a^{2} b^{2} x^{2}-2 a^{3} b x +9 a \,b^{2} x^{2}+2 a^{4}-3 a^{2} b x +4 b^{2} x^{2}+2 a b x -4 a^{2}+3 b x +2\right )}{6 \left (-1+a \right ) x^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \left (a^{2}-1\right )^{2}}+\frac {b^{3} \left (2 a^{2}+2 a +1\right ) \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{2 \left (a^{2}-1\right )^{2} \left (-1+a \right ) \sqrt {-a^{2}+1}}\)	\(207\)
default	\(b \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 \left (-a^{2}+1\right ) x^{2}}+\frac {3 a b \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (-a^{2}+1\right ) x}-\frac {a b \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (-a^{2}+1\right )}-\frac {b^{2} \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{2 \left (-a^{2}+1\right )^{\frac {3}{2}}}\right )+\left (a +1\right ) \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{3 \left (-a^{2}+1\right ) x^{3}}+\frac {5 a b \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 \left (-a^{2}+1\right ) x^{2}}+\frac {3 a b \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (-a^{2}+1\right ) x}-\frac {a b \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (-a^{2}+1\right )}-\frac {b^{2} \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{2 \left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{3 \left (-a^{2}+1\right )}+\frac {2 b^{2} \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (-a^{2}+1\right ) x}-\frac {a b \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{3 \left (-a^{2}+1\right )}\right )\)	\(600\)

Input:

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

Output:

-1/6*(b^2*x^2+2*a*b*x+a^2-1)*(2*a^2*b^2*x^2-2*a^3*b*x+9*a*b^2*x^2+2*a^4-3* 
a^2*b*x+4*b^2*x^2+2*a*b*x-4*a^2+3*b*x+2)/(-1+a)/x^3/(-b^2*x^2-2*a*b*x-a^2+ 
1)^(1/2)/(a^2-1)^2+1/2*b^3*(2*a^2+2*a+1)/(a^2-1)^2/(-1+a)/(-a^2+1)^(1/2)*l 
n((-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 = 488, normalized size of antiderivative = 2.29 \[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^4} \, dx=\left [-\frac {3 \, {\left (2 \, a^{2} + 2 \, a + 1\right )} \sqrt {-a^{2} + 1} b^{3} x^{3} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) - 2 \, {\left (2 \, a^{6} + {\left (2 \, a^{4} + 9 \, a^{3} + 2 \, a^{2} - 9 \, a - 4\right )} b^{2} x^{2} - 6 \, a^{4} - {\left (2 \, a^{5} + 3 \, a^{4} - 4 \, a^{3} - 6 \, a^{2} + 2 \, a + 3\right )} b x + 6 \, a^{2} - 2\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{12 \, {\left (a^{7} - a^{6} - 3 \, a^{5} + 3 \, a^{4} + 3 \, a^{3} - 3 \, a^{2} - a + 1\right )} x^{3}}, -\frac {3 \, {\left (2 \, a^{2} + 2 \, a + 1\right )} \sqrt {a^{2} - 1} b^{3} x^{3} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) - {\left (2 \, a^{6} + {\left (2 \, a^{4} + 9 \, a^{3} + 2 \, a^{2} - 9 \, a - 4\right )} b^{2} x^{2} - 6 \, a^{4} - {\left (2 \, a^{5} + 3 \, a^{4} - 4 \, a^{3} - 6 \, a^{2} + 2 \, a + 3\right )} b x + 6 \, a^{2} - 2\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{6 \, {\left (a^{7} - a^{6} - 3 \, a^{5} + 3 \, a^{4} + 3 \, a^{3} - 3 \, a^{2} - a + 1\right )} x^{3}}\right ] \] Input:

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

Output:

[-1/12*(3*(2*a^2 + 2*a + 1)*sqrt(-a^2 + 1)*b^3*x^3*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*(2*a^6 + (2*a^4 + 9*a^3 + 
2*a^2 - 9*a - 4)*b^2*x^2 - 6*a^4 - (2*a^5 + 3*a^4 - 4*a^3 - 6*a^2 + 2*a + 
3)*b*x + 6*a^2 - 2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^7 - a^6 - 3*a^ 
5 + 3*a^4 + 3*a^3 - 3*a^2 - a + 1)*x^3), -1/6*(3*(2*a^2 + 2*a + 1)*sqrt(a^ 
2 - 1)*b^3*x^3*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)) - 
(2*a^6 + (2*a^4 + 9*a^3 + 2*a^2 - 9*a - 4)*b^2*x^2 - 6*a^4 - (2*a^5 + 3*a^ 
4 - 4*a^3 - 6*a^2 + 2*a + 3)*b*x + 6*a^2 - 2)*sqrt(-b^2*x^2 - 2*a*b*x - a^ 
2 + 1))/((a^7 - a^6 - 3*a^5 + 3*a^4 + 3*a^3 - 3*a^2 - a + 1)*x^3)]
 

Sympy [F]

\[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^4} \, dx=\int \frac {a + b x + 1}{x^{4} \sqrt {- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}\, dx \] Input:

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

Output:

Integral((a + b*x + 1)/(x**4*sqrt(-(a + b*x - 1)*(a + b*x + 1))), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^4} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/x^4,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. 1664 vs. \(2 (176) = 352\).

Time = 0.15 (sec) , antiderivative size = 1664, normalized size of antiderivative = 7.81 \[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^4} \, dx=\text {Too large to display} \] Input:

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

Output:

-(2*a^2*b^4 + 2*a*b^4 + b^4)*arctan(((sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a 
bs(b) + b)*a/(b^2*x + a*b) - 1)/sqrt(a^2 - 1))/((a^5*abs(b) - a^4*abs(b) - 
 2*a^3*abs(b) + 2*a^2*abs(b) + a*abs(b) - abs(b))*sqrt(a^2 - 1)) + 1/3*(12 
*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a^7*b^4/(b^2*x + a*b)^2 
 + 6*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^4*a^7*b^4/(b^2*x + a* 
b)^4 + 6*a^7*b^4 - 24*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a^6* 
b^4/(b^2*x + a*b) + 24*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a 
^6*b^4/(b^2*x + a*b)^2 - 36*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b 
)^3*a^6*b^4/(b^2*x + a*b)^3 + 12*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b 
) + b)^4*a^6*b^4/(b^2*x + a*b)^4 - 12*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)* 
abs(b) + b)^5*a^6*b^4/(b^2*x + a*b)^5 + 12*a^6*b^4 - 57*(sqrt(-b^2*x^2 - 2 
*a*b*x - a^2 + 1)*abs(b) + b)*a^5*b^4/(b^2*x + a*b) + 36*(sqrt(-b^2*x^2 - 
2*a*b*x - a^2 + 1)*abs(b) + b)^2*a^5*b^4/(b^2*x + a*b)^2 - 72*(sqrt(-b^2*x 
^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*a^5*b^4/(b^2*x + a*b)^3 + 30*(sqrt(- 
b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^4*a^5*b^4/(b^2*x + a*b)^4 - 15*(s 
qrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^5*a^5*b^4/(b^2*x + a*b)^5 - 
2*a^5*b^4 + 84*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a^4*b^4/( 
b^2*x + a*b)^2 - 12*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*a^4* 
b^4/(b^2*x + a*b)^3 + 51*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^4 
*a^4*b^4/(b^2*x + a*b)^4 + 12*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^4} \, dx=\int \frac {a+b\,x+1}{x^4\,\sqrt {1-{\left (a+b\,x\right )}^2}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 950, normalized size of antiderivative = 4.46 \[ \int \frac {e^{\text {arctanh}(a+b x)}}{x^4} \, dx=\frac {-6 \sqrt {-a^{2}+1}\, \mathit {atan} \left (\frac {\sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2} i +\sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a b i x -\sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, i}{a^{2} b^{2} x^{2}+2 a^{3} b x +a^{4}-b^{2} x^{2}-2 a b x -2 a^{2}+1}\right ) a^{2} b^{3} i \,x^{3}-6 \sqrt {-a^{2}+1}\, \mathit {atan} \left (\frac {\sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2} i +\sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a b i x -\sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, i}{a^{2} b^{2} x^{2}+2 a^{3} b x +a^{4}-b^{2} x^{2}-2 a b x -2 a^{2}+1}\right ) a \,b^{3} i \,x^{3}-3 \sqrt {-a^{2}+1}\, \mathit {atan} \left (\frac {\sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2} i +\sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a b i x -\sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, i}{a^{2} b^{2} x^{2}+2 a^{3} b x +a^{4}-b^{2} x^{2}-2 a b x -2 a^{2}+1}\right ) b^{3} i \,x^{3}+2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{6}-2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{5} b x +2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{4} b^{2} x^{2}-3 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{4} b x -6 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{4}+9 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{3} b^{2} x^{2}+4 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{3} b x +2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2} b^{2} x^{2}+6 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2} b x +6 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2}-9 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a \,b^{2} x^{2}-2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a b x -4 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b^{2} x^{2}-3 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x -2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{6 x^{3} \left (a^{7}-a^{6}-3 a^{5}+3 a^{4}+3 a^{3}-3 a^{2}-a +1\right )} \] Input:

int((b*x+a+1)/(1-(b*x+a)^2)^(1/2)/x^4,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**3*i*x**3 - 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))*b**3*i*x**3 + 2*sqrt( - a**2 - 2*a*b*x 
 - b**2*x**2 + 1)*a**6 - 2*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a**5*b* 
x + 2*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a**4*b**2*x**2 - 3*sqrt( - a 
**2 - 2*a*b*x - b**2*x**2 + 1)*a**4*b*x - 6*sqrt( - a**2 - 2*a*b*x - b**2* 
x**2 + 1)*a**4 + 9*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a**3*b**2*x**2 
+ 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**2*x**2 + 6*sqrt( - a**2 - 2*a*b*x - b**2*x* 
*2 + 1)*a**2*b*x + 6*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a**2 - 9*s...