\(\int \frac {e^{3 \text {arctanh}(a+b x)}}{x} \, dx\) [861]

Optimal result

Integrand size = 14, antiderivative size = 107 \[ \int \frac {e^{3 \text {arctanh}(a+b x)}}{x} \, dx=\frac {4 \sqrt {1+a+b x}}{(1-a) \sqrt {1-a-b x}}-\arcsin (a+b x)-\frac {2 (1+a)^2 \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{(1-a) \sqrt {1-a^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.43 \[ \int \frac {e^{3 \text {arctanh}(a+b x)}}{x} \, dx=\frac {2 \sqrt {b} \text {arcsinh}\left (\frac {\sqrt {-b} \sqrt {1-a-b x}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {-b}}-\frac {2 \left (2 \sqrt {-1+a} \sqrt {1+a+b x}+(-1-a)^{3/2} \sqrt {1-a-b x} \text {arctanh}\left (\frac {\sqrt {-1-a} \sqrt {1-a-b x}}{\sqrt {-1+a} \sqrt {1+a+b x}}\right )\right )}{(-1+a)^{3/2} \sqrt {1-a-b x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.18, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6713, 109, 27, 175, 62, 104, 221, 1090, 223}

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.

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

\(\Big \downarrow \) 6713

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 175

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

\(\Big \downarrow \) 62

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 1090

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

\(\Big \downarrow \) 223

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

Input:

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

Output:

(4*Sqrt[1 + a + b*x])/((1 - a)*Sqrt[1 - a - b*x]) + ((1 - a)*ArcSin[(-2*a* 
b - 2*b^2*x)/(2*b)] - (2*(1 + a)^2*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x]) 
/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/Sqrt[1 - a^2])/(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 62

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Int[ 
1/Sqrt[a*c - b*(a - c)*x - b^2*x^2], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b 
+ d, 0] && GtQ[a + c, 0]
 

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 175

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ 
)))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b   Int[(c + d*x)^n*(e + f*x)^p, x] 
, x] + Simp[(b*g - a*h)/b   Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] 
 /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
 

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 1090

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* 
(c/(b^2 - 4*a*c)))^p)   Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, 
b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(721\) vs. \(2(93)=186\).

Time = 0.51 (sec) , antiderivative size = 722, normalized size of antiderivative = 6.75

method	result	size
default	\(b^{3} \left (\frac {x}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {a \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 a b \right )}{b \left (-4 b^{2} \left (-a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b}-\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b^{2} \sqrt {b^{2}}}\right )+\left (a^{3}+3 a^{2}+3 a +1\right ) \left (\frac {1}{\left (-a^{2}+1\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {2 a b \left (-2 b^{2} x -2 a b \right )}{\left (-a^{2}+1\right ) \left (-4 b^{2} \left (-a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {\ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}\right )+\frac {6 b \left (-2 b^{2} x -2 a b \right )}{\left (-4 b^{2} \left (-a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {12 a b \left (-2 b^{2} x -2 a b \right )}{\left (-4 b^{2} \left (-a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {6 a^{2} b \left (-2 b^{2} x -2 a b \right )}{\left (-4 b^{2} \left (-a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+3 b^{2} \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 a b \right )}{b \left (-4 b^{2} \left (-a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )+3 a \,b^{2} \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 a b \right )}{b \left (-4 b^{2} \left (-a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )\)	\(722\)

Input:

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

Output:

b^3*(x/b^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-a/b*(1/b^2/(-b^2*x^2-2*a*b*x-a^2 
+1)^(1/2)-2*a/b*(-2*b^2*x-2*a*b)/(-4*b^2*(-a^2+1)-4*a^2*b^2)/(-b^2*x^2-2*a 
*b*x-a^2+1)^(1/2))-1/b^2/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+1/b*a)/(-b^2*x^ 
2-2*a*b*x-a^2+1)^(1/2)))+(a^3+3*a^2+3*a+1)*(1/(-a^2+1)/(-b^2*x^2-2*a*b*x-a 
^2+1)^(1/2)+2*a*b/(-a^2+1)*(-2*b^2*x-2*a*b)/(-4*b^2*(-a^2+1)-4*a^2*b^2)/(- 
b^2*x^2-2*a*b*x-a^2+1)^(1/2)-1/(-a^2+1)^(3/2)*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))+6*b*(-2*b^2*x-2*a*b)/(-4*b^2 
*(-a^2+1)-4*a^2*b^2)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+12*a*b*(-2*b^2*x-2*a*b 
)/(-4*b^2*(-a^2+1)-4*a^2*b^2)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+6*a^2*b*(-2*b 
^2*x-2*a*b)/(-4*b^2*(-a^2+1)-4*a^2*b^2)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+3*b 
^2*(1/b^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-2*a/b*(-2*b^2*x-2*a*b)/(-4*b^2*(- 
a^2+1)-4*a^2*b^2)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))+3*a*b^2*(1/b^2/(-b^2*x^2 
-2*a*b*x-a^2+1)^(1/2)-2*a/b*(-2*b^2*x-2*a*b)/(-4*b^2*(-a^2+1)-4*a^2*b^2)/( 
-b^2*x^2-2*a*b*x-a^2+1)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (89) = 178\).

Time = 0.15 (sec) , antiderivative size = 439, normalized size of antiderivative = 4.10 \[ \int \frac {e^{3 \text {arctanh}(a+b x)}}{x} \, dx=\left [\frac {{\left ({\left (a + 1\right )} b x + a^{2} - 1\right )} \sqrt {-\frac {a + 1}{a - 1}} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x - 4 \, a^{2} + 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{3} + {\left (a^{2} - a\right )} b x - a^{2} - a + 1\right )} \sqrt {-\frac {a + 1}{a - 1}} + 2}{x^{2}}\right ) + 2 \, {\left ({\left (a - 1\right )} b x + a^{2} - 2 \, a + 1\right )} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + 8 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, {\left ({\left (a - 1\right )} b x + a^{2} - 2 \, a + 1\right )}}, -\frac {{\left ({\left (a + 1\right )} b x + a^{2} - 1\right )} \sqrt {\frac {a + 1}{a - 1}} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {\frac {a + 1}{a - 1}}}{{\left (a + 1\right )} b^{2} x^{2} + a^{3} + 2 \, {\left (a^{2} + a\right )} b x + a^{2} - a - 1}\right ) - {\left ({\left (a - 1\right )} b x + a^{2} - 2 \, a + 1\right )} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - 4 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{{\left (a - 1\right )} b x + a^{2} - 2 \, a + 1}\right ] \] Input:

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

Output:

[1/2*(((a + 1)*b*x + a^2 - 1)*sqrt(-(a + 1)/(a - 1))*log(((2*a^2 - 1)*b^2* 
x^2 + 2*a^4 + 4*(a^3 - a)*b*x - 4*a^2 + 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 
1)*(a^3 + (a^2 - a)*b*x - a^2 - a + 1)*sqrt(-(a + 1)/(a - 1)) + 2)/x^2) + 
2*((a - 1)*b*x + a^2 - 2*a + 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)* 
(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) + 8*sqrt(-b^2*x^2 - 2*a*b*x - a^2 
 + 1))/((a - 1)*b*x + a^2 - 2*a + 1), -(((a + 1)*b*x + a^2 - 1)*sqrt((a + 
1)/(a - 1))*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sq 
rt((a + 1)/(a - 1))/((a + 1)*b^2*x^2 + a^3 + 2*(a^2 + a)*b*x + a^2 - a - 1 
)) - ((a - 1)*b*x + a^2 - 2*a + 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 
1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) - 4*sqrt(-b^2*x^2 - 2*a*b*x - 
a^2 + 1))/((a - 1)*b*x + a^2 - 2*a + 1)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((b*x+a+1)^3/(1-(b*x+a)^2)^(3/2)/x,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 [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.47 \[ \int \frac {e^{3 \text {arctanh}(a+b x)}}{x} \, dx=\frac {b \arcsin \left (-b x - a\right ) \mathrm {sgn}\left (b\right )}{{\left | b \right |}} - \frac {2 \, {\left (a^{2} b + 2 \, a b + b\right )} \arctan \left (\frac {\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a}{b^{2} x + a b} - 1}{\sqrt {a^{2} - 1}}\right )}{\sqrt {a^{2} - 1} {\left (a {\left | b \right |} - {\left | b \right |}\right )}} - \frac {8 \, b}{{\left (a {\left | b \right |} - {\left | b \right |}\right )} {\left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b}{b^{2} x + a b} - 1\right )}} \] Input:

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

Output:

b*arcsin(-b*x - a)*sgn(b)/abs(b) - 2*(a^2*b + 2*a*b + b)*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)) 
/(sqrt(a^2 - 1)*(a*abs(b) - abs(b))) - 8*b/((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))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 295, normalized size of antiderivative = 2.76 \[ \int \frac {e^{3 \text {arctanh}(a+b x)}}{x} \, dx=\frac {-\mathit {asin} \left (b x +a \right ) \tan \left (\frac {\mathit {asin} \left (b x +a \right )}{2}\right ) a^{2}+2 \mathit {asin} \left (b x +a \right ) \tan \left (\frac {\mathit {asin} \left (b x +a \right )}{2}\right ) a -\mathit {asin} \left (b x +a \right ) \tan \left (\frac {\mathit {asin} \left (b x +a \right )}{2}\right )+\mathit {asin} \left (b x +a \right ) a^{2}-2 \mathit {asin} \left (b x +a \right ) a +\mathit {asin} \left (b x +a \right )+2 \sqrt {a^{2}-1}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (b x +a \right )}{2}\right ) a -1}{\sqrt {a^{2}-1}}\right ) \tan \left (\frac {\mathit {asin} \left (b x +a \right )}{2}\right ) a +2 \sqrt {a^{2}-1}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (b x +a \right )}{2}\right ) a -1}{\sqrt {a^{2}-1}}\right ) \tan \left (\frac {\mathit {asin} \left (b x +a \right )}{2}\right )-2 \sqrt {a^{2}-1}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (b x +a \right )}{2}\right ) a -1}{\sqrt {a^{2}-1}}\right ) a -2 \sqrt {a^{2}-1}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (b x +a \right )}{2}\right ) a -1}{\sqrt {a^{2}-1}}\right )+8 \tan \left (\frac {\mathit {asin} \left (b x +a \right )}{2}\right ) a -8 \tan \left (\frac {\mathit {asin} \left (b x +a \right )}{2}\right )}{\tan \left (\frac {\mathit {asin} \left (b x +a \right )}{2}\right ) a^{2}-2 \tan \left (\frac {\mathit {asin} \left (b x +a \right )}{2}\right ) a +\tan \left (\frac {\mathit {asin} \left (b x +a \right )}{2}\right )-a^{2}+2 a -1} \] Input:

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

Output:

( - asin(a + b*x)*tan(asin(a + b*x)/2)*a**2 + 2*asin(a + b*x)*tan(asin(a + 
 b*x)/2)*a - asin(a + b*x)*tan(asin(a + b*x)/2) + asin(a + b*x)*a**2 - 2*a 
sin(a + b*x)*a + asin(a + b*x) + 2*sqrt(a**2 - 1)*atan((tan(asin(a + b*x)/ 
2)*a - 1)/sqrt(a**2 - 1))*tan(asin(a + b*x)/2)*a + 2*sqrt(a**2 - 1)*atan(( 
tan(asin(a + b*x)/2)*a - 1)/sqrt(a**2 - 1))*tan(asin(a + b*x)/2) - 2*sqrt( 
a**2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/sqrt(a**2 - 1))*a - 2*sqrt(a** 
2 - 1)*atan((tan(asin(a + b*x)/2)*a - 1)/sqrt(a**2 - 1)) + 8*tan(asin(a + 
b*x)/2)*a - 8*tan(asin(a + b*x)/2))/(tan(asin(a + b*x)/2)*a**2 - 2*tan(asi 
n(a + b*x)/2)*a + tan(asin(a + b*x)/2) - a**2 + 2*a - 1)