\(\int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^4} \, dx\) [904]

Optimal result

Integrand size = 14, antiderivative size = 257 \[ \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^4} \, dx=-\frac {\left (52-51 a+2 a^2\right ) b^3 \sqrt {1-a-b x}}{6 (1-a) (1+a)^4 \sqrt {1+a+b x}}-\frac {(1-a) \sqrt {1-a-b x}}{3 (1+a) x^3 \sqrt {1+a+b x}}+\frac {7 b \sqrt {1-a-b x}}{6 (1+a)^2 x^2 \sqrt {1+a+b x}}-\frac {(19-16 a) b^2 \sqrt {1-a-b x}}{6 (1-a) (1+a)^3 x \sqrt {1+a+b x}}+\frac {\left (11-18 a+6 a^2\right ) b^3 \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{(1-a) (1+a)^4 \sqrt {1-a^2}} \] Output:

-1/6*(2*a^2-51*a+52)*b^3*(-b*x-a+1)^(1/2)/(1-a)/(1+a)^4/(b*x+a+1)^(1/2)-1/ 
3*(1-a)*(-b*x-a+1)^(1/2)/(1+a)/x^3/(b*x+a+1)^(1/2)+7/6*b*(-b*x-a+1)^(1/2)/ 
(1+a)^2/x^2/(b*x+a+1)^(1/2)-1/6*(19-16*a)*b^2*(-b*x-a+1)^(1/2)/(1-a)/(1+a) 
^3/x/(b*x+a+1)^(1/2)+(6*a^2-18*a+11)*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)/(1+a)^4/(-a^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.75 \[ \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^4} \, dx=\frac {-2 (1-a) (1+a) (1-a-b x)^{5/2}+(3-4 a) b x (1-a-b x)^{5/2}+\frac {\left (11-18 a+6 a^2\right ) b^2 x^2 \left (\sqrt {-1-a} \sqrt {1-a-b x} \left (-1+a^2-5 b x+a b x\right )+6 \sqrt {-1+a} b x \sqrt {1+a+b x} \text {arctanh}\left (\frac {\sqrt {-1-a} \sqrt {1-a-b x}}{\sqrt {-1+a} \sqrt {1+a+b x}}\right )\right )}{(-1-a)^{5/2}}}{6 \left (-1+a^2\right )^2 x^3 \sqrt {1+a+b x}} \] Input:

Integrate[1/(E^(3*ArcTanh[a + b*x])*x^4),x]
 

Output:

(-2*(1 - a)*(1 + a)*(1 - a - b*x)^(5/2) + (3 - 4*a)*b*x*(1 - a - b*x)^(5/2 
) + ((11 - 18*a + 6*a^2)*b^2*x^2*(Sqrt[-1 - a]*Sqrt[1 - a - b*x]*(-1 + a^2 
 - 5*b*x + a*b*x) + 6*Sqrt[-1 + a]*b*x*Sqrt[1 + a + b*x]*ArcTanh[(Sqrt[-1 
- a]*Sqrt[1 - a - b*x])/(Sqrt[-1 + a]*Sqrt[1 + a + b*x])]))/(-1 - a)^(5/2) 
)/(6*(-1 + a^2)^2*x^3*Sqrt[1 + a + b*x])
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6713, 109, 27, 168, 27, 168, 27, 169, 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.

\(\displaystyle \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^4} \, dx\)

\(\Big \downarrow \) 6713

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

\(\Big \downarrow \) 109

\(\displaystyle -\frac {\int \frac {b (7 (1-a)-6 b x)}{x^3 \sqrt {-a-b x+1} (a+b x+1)^{3/2}}dx}{3 (a+1)}-\frac {(1-a) \sqrt {-a-b x+1}}{3 (a+1) x^3 \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b \int \frac {7 (1-a)-6 b x}{x^3 \sqrt {-a-b x+1} (a+b x+1)^{3/2}}dx}{3 (a+1)}-\frac {(1-a) \sqrt {-a-b x+1}}{3 (a+1) x^3 \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {b \left (-\frac {\int \frac {(1-a) b (-16 a-14 b x+19)}{x^2 \sqrt {-a-b x+1} (a+b x+1)^{3/2}}dx}{2 \left (1-a^2\right )}-\frac {7 \sqrt {-a-b x+1}}{2 (a+1) x^2 \sqrt {a+b x+1}}\right )}{3 (a+1)}-\frac {(1-a) \sqrt {-a-b x+1}}{3 (a+1) x^3 \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b \left (-\frac {(1-a) b \int \frac {-16 a-14 b x+19}{x^2 \sqrt {-a-b x+1} (a+b x+1)^{3/2}}dx}{2 \left (1-a^2\right )}-\frac {7 \sqrt {-a-b x+1}}{2 (a+1) x^2 \sqrt {a+b x+1}}\right )}{3 (a+1)}-\frac {(1-a) \sqrt {-a-b x+1}}{3 (a+1) x^3 \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {b \left (-\frac {(1-a) b \left (-\frac {\int \frac {b \left (3 \left (6 a^2-18 a+11\right )-(19-16 a) b x\right )}{x \sqrt {-a-b x+1} (a+b x+1)^{3/2}}dx}{1-a^2}-\frac {(19-16 a) \sqrt {-a-b x+1}}{\left (1-a^2\right ) x \sqrt {a+b x+1}}\right )}{2 \left (1-a^2\right )}-\frac {7 \sqrt {-a-b x+1}}{2 (a+1) x^2 \sqrt {a+b x+1}}\right )}{3 (a+1)}-\frac {(1-a) \sqrt {-a-b x+1}}{3 (a+1) x^3 \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b \left (-\frac {(1-a) b \left (-\frac {b \int \frac {3 \left (6 a^2-18 a+11\right )-(19-16 a) b x}{x \sqrt {-a-b x+1} (a+b x+1)^{3/2}}dx}{1-a^2}-\frac {(19-16 a) \sqrt {-a-b x+1}}{\left (1-a^2\right ) x \sqrt {a+b x+1}}\right )}{2 \left (1-a^2\right )}-\frac {7 \sqrt {-a-b x+1}}{2 (a+1) x^2 \sqrt {a+b x+1}}\right )}{3 (a+1)}-\frac {(1-a) \sqrt {-a-b x+1}}{3 (a+1) x^3 \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {b \left (-\frac {(1-a) b \left (-\frac {b \left (\frac {\int \frac {3 \left (6 a^2-18 a+11\right ) b}{x \sqrt {-a-b x+1} \sqrt {a+b x+1}}dx}{(a+1) b}+\frac {\left (2 a^2-51 a+52\right ) \sqrt {-a-b x+1}}{(a+1) \sqrt {a+b x+1}}\right )}{1-a^2}-\frac {(19-16 a) \sqrt {-a-b x+1}}{\left (1-a^2\right ) x \sqrt {a+b x+1}}\right )}{2 \left (1-a^2\right )}-\frac {7 \sqrt {-a-b x+1}}{2 (a+1) x^2 \sqrt {a+b x+1}}\right )}{3 (a+1)}-\frac {(1-a) \sqrt {-a-b x+1}}{3 (a+1) x^3 \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b \left (-\frac {(1-a) b \left (-\frac {b \left (\frac {3 \left (6 a^2-18 a+11\right ) \int \frac {1}{x \sqrt {-a-b x+1} \sqrt {a+b x+1}}dx}{a+1}+\frac {\left (2 a^2-51 a+52\right ) \sqrt {-a-b x+1}}{(a+1) \sqrt {a+b x+1}}\right )}{1-a^2}-\frac {(19-16 a) \sqrt {-a-b x+1}}{\left (1-a^2\right ) x \sqrt {a+b x+1}}\right )}{2 \left (1-a^2\right )}-\frac {7 \sqrt {-a-b x+1}}{2 (a+1) x^2 \sqrt {a+b x+1}}\right )}{3 (a+1)}-\frac {(1-a) \sqrt {-a-b x+1}}{3 (a+1) x^3 \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {b \left (-\frac {(1-a) b \left (-\frac {b \left (\frac {6 \left (6 a^2-18 a+11\right ) \int \frac {1}{-a+\frac {(1-a) (a+b x+1)}{-a-b x+1}-1}d\frac {\sqrt {a+b x+1}}{\sqrt {-a-b x+1}}}{a+1}+\frac {\left (2 a^2-51 a+52\right ) \sqrt {-a-b x+1}}{(a+1) \sqrt {a+b x+1}}\right )}{1-a^2}-\frac {(19-16 a) \sqrt {-a-b x+1}}{\left (1-a^2\right ) x \sqrt {a+b x+1}}\right )}{2 \left (1-a^2\right )}-\frac {7 \sqrt {-a-b x+1}}{2 (a+1) x^2 \sqrt {a+b x+1}}\right )}{3 (a+1)}-\frac {(1-a) \sqrt {-a-b x+1}}{3 (a+1) x^3 \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {b \left (-\frac {(1-a) b \left (-\frac {b \left (\frac {\left (2 a^2-51 a+52\right ) \sqrt {-a-b x+1}}{(a+1) \sqrt {a+b x+1}}-\frac {6 \left (6 a^2-18 a+11\right ) \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{(a+1) \sqrt {1-a^2}}\right )}{1-a^2}-\frac {(19-16 a) \sqrt {-a-b x+1}}{\left (1-a^2\right ) x \sqrt {a+b x+1}}\right )}{2 \left (1-a^2\right )}-\frac {7 \sqrt {-a-b x+1}}{2 (a+1) x^2 \sqrt {a+b x+1}}\right )}{3 (a+1)}-\frac {(1-a) \sqrt {-a-b x+1}}{3 (a+1) x^3 \sqrt {a+b x+1}}\)

Input:

Int[1/(E^(3*ArcTanh[a + b*x])*x^4),x]
 

Output:

-1/3*((1 - a)*Sqrt[1 - a - b*x])/((1 + a)*x^3*Sqrt[1 + a + b*x]) - (b*((-7 
*Sqrt[1 - a - b*x])/(2*(1 + a)*x^2*Sqrt[1 + a + b*x]) - ((1 - a)*b*(-(((19 
 - 16*a)*Sqrt[1 - a - b*x])/((1 - a^2)*x*Sqrt[1 + a + b*x])) - (b*(((52 - 
51*a + 2*a^2)*Sqrt[1 - a - b*x])/((1 + a)*Sqrt[1 + a + b*x]) - (6*(11 - 18 
*a + 6*a^2)*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - 
a - b*x])])/((1 + a)*Sqrt[1 - a^2])))/(1 - a^2)))/(2*(1 - a^2))))/(3*(1 + 
a))
 

Defintions of rubi rules used

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 109

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f 
*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) 
 Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) 
+ c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) 
 + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, 
d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 169

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] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

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 = 1.10 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.01

method	result	size
risch	\(-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}-1\right ) \left (2 a^{2} b^{2} x^{2}-2 a^{3} b x -27 a \,b^{2} x^{2}+2 a^{4}+9 a^{2} b x +28 b^{2} x^{2}+2 a b x -4 a^{2}-9 b x +2\right )}{6 \left (a +1\right )^{3} x^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \left (a^{2}-1\right )}+\frac {b^{3} \left (-\frac {\left (6 a^{2}-18 a +11\right ) \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\sqrt {-a^{2}+1}}-\frac {8 \left (-1+a \right ) \sqrt {-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b}}{b \left (x +\frac {a +1}{b}\right )}\right )}{2 \left (a^{2}-1\right ) \left (a +1\right )^{3}}\)	\(260\)
default	\(\text {Expression too large to display}\)	\(3700\)

Input:

int(1/(b*x+a+1)^3*(1-(b*x+a)^2)^(3/2)/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-27*a*b^2*x^2+2*a^4+9 
*a^2*b*x+28*b^2*x^2+2*a*b*x-4*a^2-9*b*x+2)/(a+1)^3/x^3/(-b^2*x^2-2*a*b*x-a 
^2+1)^(1/2)/(a^2-1)+1/2*b^3/(a^2-1)/(a+1)^3*(-(6*a^2-18*a+11)/(-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*(-1+a)/b/(x+(a+1)/b)*(-(x+(a+1)/b)^2*b^2+2*(x+(a+1)/b)*b)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 697, normalized size of antiderivative = 2.71 \[ \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^4} \, dx =\text {Too large to display} \] Input:

integrate(1/(b*x+a+1)^3*(1-(b*x+a)^2)^(3/2)/x^4,x, algorithm="fricas")
 

Output:

[-1/12*(3*((6*a^2 - 18*a + 11)*b^4*x^4 + (6*a^3 - 12*a^2 - 7*a + 11)*b^3*x 
^3)*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*(2*a^7 + (2*a^4 - 51*a^3 + 50*a^2 + 51*a - 52)*b^3*x^3 + 2 
*a^6 - 6*a^5 - (16*a^4 - 3*a^3 - 35*a^2 + 3*a + 19)*b^2*x^2 - 6*a^4 + 6*a^ 
3 + 7*(a^5 + a^4 - 2*a^3 - 2*a^2 + a + 1)*b*x + 6*a^2 - 2*a - 2)*sqrt(-b^2 
*x^2 - 2*a*b*x - a^2 + 1))/((a^7 + 3*a^6 + a^5 - 5*a^4 - 5*a^3 + a^2 + 3*a 
 + 1)*b*x^4 + (a^8 + 4*a^7 + 4*a^6 - 4*a^5 - 10*a^4 - 4*a^3 + 4*a^2 + 4*a 
+ 1)*x^3), 1/6*(3*((6*a^2 - 18*a + 11)*b^4*x^4 + (6*a^3 - 12*a^2 - 7*a + 1 
1)*b^3*x^3)*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)) + (2*a^7 + (2*a^4 - 51*a^3 + 50*a^2 + 51*a - 52)*b^3*x^3 + 2*a^6 
- 6*a^5 - (16*a^4 - 3*a^3 - 35*a^2 + 3*a + 19)*b^2*x^2 - 6*a^4 + 6*a^3 + 7 
*(a^5 + a^4 - 2*a^3 - 2*a^2 + a + 1)*b*x + 6*a^2 - 2*a - 2)*sqrt(-b^2*x^2 
- 2*a*b*x - a^2 + 1))/((a^7 + 3*a^6 + a^5 - 5*a^4 - 5*a^3 + a^2 + 3*a + 1) 
*b*x^4 + (a^8 + 4*a^7 + 4*a^6 - 4*a^5 - 10*a^4 - 4*a^3 + 4*a^2 + 4*a + 1)* 
x^3)]
 

Sympy [F]

\[ \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^4} \, dx=\int \frac {\left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac {3}{2}}}{x^{4} \left (a + b x + 1\right )^{3}}\, dx \] Input:

integrate(1/(b*x+a+1)**3*(1-(b*x+a)**2)**(3/2)/x**4,x)
 

Output:

Integral((-(a + b*x - 1)*(a + b*x + 1))**(3/2)/(x**4*(a + b*x + 1)**3), x)
 

Maxima [F]

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

integrate(1/(b*x+a+1)^3*(1-(b*x+a)^2)^(3/2)/x^4,x, algorithm="maxima")
 

Output:

integrate((-(b*x + a)^2 + 1)^(3/2)/((b*x + a + 1)^3*x^4), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1839 vs. \(2 (216) = 432\).

Time = 0.16 (sec) , antiderivative size = 1839, normalized size of antiderivative = 7.16 \[ \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^4} \, dx=\text {Too large to display} \] Input:

integrate(1/(b*x+a+1)^3*(1-(b*x+a)^2)^(3/2)/x^4,x, algorithm="giac")
 

Output:

8*b^4/((a^4*abs(b) + 4*a^3*abs(b) + 6*a^2*abs(b) + 4*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)) + (6*a^ 
2*b^4 - 18*a*b^4 + 11*b^4)*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^5*abs(b) + 3*a^4*abs(b) + 
 2*a^3*abs(b) - 2*a^2*abs(b) - 3*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) - 72*(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 - 36*(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 - 36*a^6*b^4 + 171*(sqrt(-b^2*x^2 
- 2*a*b*x - a^2 + 1)*abs(b) + b)*a^5*b^4/(b^2*x + a*b) + 84*(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 + 216*(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 + 54*(sq 
rt(-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 + 4 
5*(sqrt(-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 + 22*a^5*b^4 - 120*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a^4*b 
^4/(b^2*x + a*b) - 252*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^...
 

Mupad [F(-1)]

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

int((1 - (a + b*x)^2)^(3/2)/(x^4*(a + b*x + 1)^3),x)
 

Output:

int((1 - (a + b*x)^2)^(3/2)/(x^4*(a + b*x + 1)^3), x)
 

Reduce [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 4014, normalized size of antiderivative = 15.62 \[ \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^4} \, dx =\text {Too large to display} \] Input:

int(1/(b*x+a+1)^3*(1-(b*x+a)^2)^(3/2)/x^4,x)
 

Output:

(b**3*( - 18*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/sqrt(a**2 - 
1))*tan(asin(a + b*x)/2)**7*a**8 + 162*sqrt(a**2 - 1)*atan((tan(asin(a + b 
*x)/2)*a - 1)/sqrt(a**2 - 1))*tan(asin(a + b*x)/2)**7*a**7 - 357*sqrt(a**2 
 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/sqrt(a**2 - 1))*tan(asin(a + b*x)/ 
2)**7*a**6 + 198*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/sqrt(a** 
2 - 1))*tan(asin(a + b*x)/2)**7*a**5 - 18*sqrt(a**2 - 1)*atan((tan(asin(a 
+ b*x)/2)*a - 1)/sqrt(a**2 - 1))*tan(asin(a + b*x)/2)**6*a**8 + 270*sqrt(a 
**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/sqrt(a**2 - 1))*tan(asin(a + b* 
x)/2)**6*a**7 - 1329*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/sqrt 
(a**2 - 1))*tan(asin(a + b*x)/2)**6*a**6 + 2340*sqrt(a**2 - 1)*atan((tan(a 
sin(a + b*x)/2)*a - 1)/sqrt(a**2 - 1))*tan(asin(a + b*x)/2)**6*a**5 - 1188 
*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/sqrt(a**2 - 1))*tan(asin 
(a + b*x)/2)**6*a**4 - 54*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1) 
/sqrt(a**2 - 1))*tan(asin(a + b*x)/2)**5*a**8 + 594*sqrt(a**2 - 1)*atan((t 
an(asin(a + b*x)/2)*a - 1)/sqrt(a**2 - 1))*tan(asin(a + b*x)/2)**5*a**7 - 
2259*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/sqrt(a**2 - 1))*tan( 
asin(a + b*x)/2)**5*a**6 + 4680*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/2)* 
a - 1)/sqrt(a**2 - 1))*tan(asin(a + b*x)/2)**5*a**5 - 5472*sqrt(a**2 - 1)* 
atan((tan(asin(a + b*x)/2)*a - 1)/sqrt(a**2 - 1))*tan(asin(a + b*x)/2)**5* 
a**4 + 2376*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/sqrt(a**2 ...