\(\int e^{\text {arctanh}(a+b x)} x^3 \, dx\) [855]

Optimal result

Integrand size = 12, antiderivative size = 182 \[ \int e^{\text {arctanh}(a+b x)} x^3 \, dx=-\frac {\left (3-12 a+12 a^2-8 a^3\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{8 b^4}-\frac {\left (3-8 a+10 a^2\right ) \sqrt {1-a-b x} (1+a+b x)^{3/2}}{8 b^4}-\frac {x^2 \sqrt {1-a-b x} (1+a+b x)^{3/2}}{4 b^2}+\frac {(1-6 a) (1-a-b x)^{3/2} (1+a+b x)^{3/2}}{12 b^4}+\frac {\left (3-12 a+12 a^2-8 a^3\right ) \arcsin (a+b x)}{8 b^4} \] Output:

-1/8*(-8*a^3+12*a^2-12*a+3)*(-b*x-a+1)^(1/2)*(b*x+a+1)^(1/2)/b^4-1/8*(10*a 
^2-8*a+3)*(-b*x-a+1)^(1/2)*(b*x+a+1)^(3/2)/b^4-1/4*x^2*(-b*x-a+1)^(1/2)*(b 
*x+a+1)^(3/2)/b^2+1/12*(1-6*a)*(-b*x-a+1)^(3/2)*(b*x+a+1)^(3/2)/b^4+1/8*(- 
8*a^3+12*a^2-12*a+3)*arcsin(b*x+a)/b^4
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.82 \[ \int e^{\text {arctanh}(a+b x)} x^3 \, dx=-\frac {\sqrt {1-a^2-2 a b x-b^2 x^2} \left (16-6 a^3+9 b x+8 b^2 x^2+6 b^3 x^3+a^2 (44+6 b x)-a \left (39+20 b x+6 b^2 x^2\right )\right )}{24 b^4}-\frac {\left (-3+12 a-12 a^2+8 a^3\right ) \sqrt {-b} \text {arcsinh}\left (\frac {\sqrt {-b} \sqrt {1-a-b x}}{\sqrt {2} \sqrt {b}}\right )}{4 b^{9/2}} \] Input:

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

Output:

-1/24*(Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(16 - 6*a^3 + 9*b*x + 8*b^2*x^2 + 
 6*b^3*x^3 + a^2*(44 + 6*b*x) - a*(39 + 20*b*x + 6*b^2*x^2)))/b^4 - ((-3 + 
 12*a - 12*a^2 + 8*a^3)*Sqrt[-b]*ArcSinh[(Sqrt[-b]*Sqrt[1 - a - b*x])/(Sqr 
t[2]*Sqrt[b])])/(4*b^(9/2))
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.91, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6713, 111, 25, 164, 60, 62, 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.

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

\(\Big \downarrow \) 6713

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

\(\Big \downarrow \) 111

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 164

\(\displaystyle \frac {\frac {\left (-8 a^3+12 a^2-12 a+3\right ) \int \frac {\sqrt {a+b x+1}}{\sqrt {-a-b x+1}}dx}{2 b}-\frac {\sqrt {-a-b x+1} (a+b x+1)^{3/2} \left (18 a^2+2 (1-6 a) b x-10 a+7\right )}{6 b^2}}{4 b^2}-\frac {x^2 \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{4 b^2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {\left (-8 a^3+12 a^2-12 a+3\right ) \left (\int \frac {1}{\sqrt {-a-b x+1} \sqrt {a+b x+1}}dx-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b}\right )}{2 b}-\frac {\sqrt {-a-b x+1} (a+b x+1)^{3/2} \left (18 a^2+2 (1-6 a) b x-10 a+7\right )}{6 b^2}}{4 b^2}-\frac {x^2 \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{4 b^2}\)

\(\Big \downarrow \) 62

\(\displaystyle \frac {\frac {\left (-8 a^3+12 a^2-12 a+3\right ) \left (\int \frac {1}{\sqrt {-b^2 x^2-2 a b x+(1-a) (a+1)}}dx-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b}\right )}{2 b}-\frac {\sqrt {-a-b x+1} (a+b x+1)^{3/2} \left (18 a^2+2 (1-6 a) b x-10 a+7\right )}{6 b^2}}{4 b^2}-\frac {x^2 \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{4 b^2}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {\frac {\left (-8 a^3+12 a^2-12 a+3\right ) \left (-\frac {\int \frac {1}{\sqrt {1-\frac {\left (-2 x b^2-2 a b\right )^2}{4 b^2}}}d\left (-2 x b^2-2 a b\right )}{2 b^2}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b}\right )}{2 b}-\frac {\sqrt {-a-b x+1} (a+b x+1)^{3/2} \left (18 a^2+2 (1-6 a) b x-10 a+7\right )}{6 b^2}}{4 b^2}-\frac {x^2 \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{4 b^2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {\frac {\left (-8 a^3+12 a^2-12 a+3\right ) \left (-\frac {\arcsin \left (\frac {-2 a b-2 b^2 x}{2 b}\right )}{b}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b}\right )}{2 b}-\frac {\sqrt {-a-b x+1} (a+b x+1)^{3/2} \left (18 a^2+2 (1-6 a) b x-10 a+7\right )}{6 b^2}}{4 b^2}-\frac {x^2 \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{4 b^2}\)

Input:

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

Output:

-1/4*(x^2*Sqrt[1 - a - b*x]*(1 + a + b*x)^(3/2))/b^2 + (-1/6*(Sqrt[1 - a - 
 b*x]*(1 + a + b*x)^(3/2)*(7 - 10*a + 18*a^2 + 2*(1 - 6*a)*b*x))/b^2 + ((3 
 - 12*a + 12*a^2 - 8*a^3)*(-((Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/b) - Ar 
cSin[(-2*a*b - 2*b^2*x)/(2*b)]/b))/(2*b))/(4*b^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, 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 111

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 
)/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1))   Int[(a + b*x) 
^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m 
 - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m 
 + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & 
& GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
 

rule 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
 

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 [A] (verified)

Time = 0.40 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.90

method	result	size
risch	\(-\frac {\left (-6 b^{3} x^{3}+6 a \,b^{2} x^{2}-6 a^{2} b x -8 b^{2} x^{2}+6 a^{3}+20 a b x -44 a^{2}-9 b x +39 a -16\right ) \left (b^{2} x^{2}+2 a b x +a^{2}-1\right )}{24 b^{4} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {\left (8 a^{3}-12 a^{2}+12 a -3\right ) \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{8 b^{3} \sqrt {b^{2}}}\)	\(163\)
default	\(b \left (-\frac {x^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{4 b^{2}}-\frac {7 a \left (-\frac {x^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{3 b^{2}}-\frac {5 a \left (-\frac {x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{2}}-\frac {3 a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{2}}-\frac {a \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 \sqrt {b^{2}}}\right )}{2 b}+\frac {\left (-a^{2}+1\right ) \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 )}{2 b^{2} \sqrt {b^{2}}}\right )}{3 b}+\frac {2 \left (-a^{2}+1\right ) \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{2}}-\frac {a \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 \sqrt {b^{2}}}\right )}{3 b^{2}}\right )}{4 b}+\frac {3 \left (-a^{2}+1\right ) \left (-\frac {x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{2}}-\frac {3 a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{2}}-\frac {a \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 \sqrt {b^{2}}}\right )}{2 b}+\frac {\left (-a^{2}+1\right ) \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 )}{2 b^{2} \sqrt {b^{2}}}\right )}{4 b^{2}}\right )+\left (a +1\right ) \left (-\frac {x^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{3 b^{2}}-\frac {5 a \left (-\frac {x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{2}}-\frac {3 a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{2}}-\frac {a \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 \sqrt {b^{2}}}\right )}{2 b}+\frac {\left (-a^{2}+1\right ) \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 )}{2 b^{2} \sqrt {b^{2}}}\right )}{3 b}+\frac {2 \left (-a^{2}+1\right ) \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{2}}-\frac {a \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 \sqrt {b^{2}}}\right )}{3 b^{2}}\right )\)	\(794\)

Input:

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

Output:

-1/24*(-6*b^3*x^3+6*a*b^2*x^2-6*a^2*b*x-8*b^2*x^2+6*a^3+20*a*b*x-44*a^2-9* 
b*x+39*a-16)*(b^2*x^2+2*a*b*x+a^2-1)/b^4/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-1/ 
8/b^3*(8*a^3-12*a^2+12*a-3)/(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))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.79 \[ \int e^{\text {arctanh}(a+b x)} x^3 \, dx=\frac {3 \, {\left (8 \, a^{3} - 12 \, a^{2} + 12 \, a - 3\right )} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - {\left (6 \, b^{3} x^{3} - 2 \, {\left (3 \, a - 4\right )} b^{2} x^{2} - 6 \, a^{3} + {\left (6 \, a^{2} - 20 \, a + 9\right )} b x + 44 \, a^{2} - 39 \, a + 16\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{24 \, b^{4}} \] Input:

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

Output:

1/24*(3*(8*a^3 - 12*a^2 + 12*a - 3)*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)) - (6*b^3*x^3 - 2*(3*a - 4)*b^ 
2*x^2 - 6*a^3 + (6*a^2 - 20*a + 9)*b*x + 44*a^2 - 39*a + 16)*sqrt(-b^2*x^2 
 - 2*a*b*x - a^2 + 1))/b^4
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 818 vs. \(2 (165) = 330\).

Time = 2.42 (sec) , antiderivative size = 818, normalized size of antiderivative = 4.49 \[ \int e^{\text {arctanh}(a+b x)} x^3 \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*(-x**3/(4*b) - x**2*(1 - 
3*a/4)/(3*b**2) - x*(-5*a*(1 - 3*a/4)/(3*b) + (3 - 3*a**2)/(4*b))/(2*b**2) 
 - (-3*a*(-5*a*(1 - 3*a/4)/(3*b) + (3 - 3*a**2)/(4*b))/(2*b) + (1 - 3*a/4) 
*(2 - 2*a**2)/(3*b**2))/b**2) + (-a*(-3*a*(-5*a*(1 - 3*a/4)/(3*b) + (3 - 3 
*a**2)/(4*b))/(2*b) + (1 - 3*a/4)*(2 - 2*a**2)/(3*b**2))/b + (1 - a**2)*(- 
5*a*(1 - 3*a/4)/(3*b) + (3 - 3*a**2)/(4*b))/(2*b**2))*log(-2*a*b - 2*b**2* 
x + 2*sqrt(-b**2)*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1))/sqrt(-b**2), Ne(b 
**2, 0)), (-(-(a**6*sqrt(-a**2 - 2*a*b*x + 1) - 3*a**4*sqrt(-a**2 - 2*a*b* 
x + 1) + 3*a**2*sqrt(-a**2 - 2*a*b*x + 1) + (3*a**2 - 3)*(-a**2 - 2*a*b*x 
+ 1)**(5/2)/5 + (-a**2 - 2*a*b*x + 1)**(7/2)/7 + (-a**2 - 2*a*b*x + 1)**(3 
/2)*(3*a**4 - 6*a**2 + 3)/3 - sqrt(-a**2 - 2*a*b*x + 1))/(4*a**2*b**3) - ( 
a**6*sqrt(-a**2 - 2*a*b*x + 1) - 3*a**4*sqrt(-a**2 - 2*a*b*x + 1) + 3*a**2 
*sqrt(-a**2 - 2*a*b*x + 1) + (3*a**2 - 3)*(-a**2 - 2*a*b*x + 1)**(5/2)/5 + 
 (-a**2 - 2*a*b*x + 1)**(7/2)/7 + (-a**2 - 2*a*b*x + 1)**(3/2)*(3*a**4 - 6 
*a**2 + 3)/3 - sqrt(-a**2 - 2*a*b*x + 1))/(4*a**3*b**3) + (a**8*sqrt(-a**2 
 - 2*a*b*x + 1) - 4*a**6*sqrt(-a**2 - 2*a*b*x + 1) + 6*a**4*sqrt(-a**2 - 2 
*a*b*x + 1) - 4*a**2*sqrt(-a**2 - 2*a*b*x + 1) + (4*a**2 - 4)*(-a**2 - 2*a 
*b*x + 1)**(7/2)/7 + (-a**2 - 2*a*b*x + 1)**(9/2)/9 + (-a**2 - 2*a*b*x + 1 
)**(5/2)*(6*a**4 - 12*a**2 + 6)/5 + (-a**2 - 2*a*b*x + 1)**(3/2)*(4*a**6 - 
 12*a**4 + 12*a**2 - 4)/3 + sqrt(-a**2 - 2*a*b*x + 1))/(8*a**4*b**3))/(...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 540 vs. \(2 (156) = 312\).

Time = 0.32 (sec) , antiderivative size = 540, normalized size of antiderivative = 2.97 \[ \int e^{\text {arctanh}(a+b x)} x^3 \, dx=-\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x^{3}}{4 \, b} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a + 1\right )} x^{2}}{3 \, b^{2}} + \frac {7 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a x^{2}}{12 \, b^{2}} + \frac {5 \, {\left (a + 1\right )} a^{3} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{2 \, b^{4}} - \frac {35 \, a^{4} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{8 \, b^{4}} + \frac {5 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a + 1\right )} a x}{6 \, b^{3}} - \frac {35 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2} x}{24 \, b^{3}} - \frac {3 \, {\left (a^{2} - 1\right )} {\left (a + 1\right )} a \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{2 \, b^{4}} + \frac {15 \, {\left (a^{2} - 1\right )} a^{2} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{4 \, b^{4}} - \frac {5 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a + 1\right )} a^{2}}{2 \, b^{4}} + \frac {35 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{3}}{8 \, b^{4}} + \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )} x}{8 \, b^{3}} - \frac {3 \, {\left (a^{2} - 1\right )}^{2} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{8 \, b^{4}} + \frac {2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )} {\left (a + 1\right )}}{3 \, b^{4}} - \frac {55 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )} a}{24 \, b^{4}} \] Input:

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

Output:

-1/4*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*x^3/b - 1/3*sqrt(-b^2*x^2 - 2*a*b* 
x - a^2 + 1)*(a + 1)*x^2/b^2 + 7/12*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a*x 
^2/b^2 + 5/2*(a + 1)*a^3*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^ 
2))/b^4 - 35/8*a^4*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^ 
4 + 5/6*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a + 1)*a*x/b^3 - 35/24*sqrt(-b 
^2*x^2 - 2*a*b*x - a^2 + 1)*a^2*x/b^3 - 3/2*(a^2 - 1)*(a + 1)*a*arcsin(-(b 
^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^4 + 15/4*(a^2 - 1)*a^2*arcsin 
(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^4 - 5/2*sqrt(-b^2*x^2 - 2 
*a*b*x - a^2 + 1)*(a + 1)*a^2/b^4 + 35/8*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1 
)*a^3/b^4 + 3/8*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^2 - 1)*x/b^3 - 3/8*( 
a^2 - 1)^2*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^4 + 2/3* 
sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^2 - 1)*(a + 1)/b^4 - 55/24*sqrt(-b^2 
*x^2 - 2*a*b*x - a^2 + 1)*(a^2 - 1)*a/b^4
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.81 \[ \int e^{\text {arctanh}(a+b x)} x^3 \, dx=-\frac {1}{24} \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left ({\left (2 \, x {\left (\frac {3 \, x}{b} - \frac {3 \, a b^{5} - 4 \, b^{5}}{b^{7}}\right )} + \frac {6 \, a^{2} b^{4} - 20 \, a b^{4} + 9 \, b^{4}}{b^{7}}\right )} x - \frac {6 \, a^{3} b^{3} - 44 \, a^{2} b^{3} + 39 \, a b^{3} - 16 \, b^{3}}{b^{7}}\right )} + \frac {{\left (8 \, a^{3} - 12 \, a^{2} + 12 \, a - 3\right )} \arcsin \left (-b x - a\right ) \mathrm {sgn}\left (b\right )}{8 \, b^{3} {\left | b \right |}} \] Input:

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

Output:

-1/24*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*((2*x*(3*x/b - (3*a*b^5 - 4*b^5)/ 
b^7) + (6*a^2*b^4 - 20*a*b^4 + 9*b^4)/b^7)*x - (6*a^3*b^3 - 44*a^2*b^3 + 3 
9*a*b^3 - 16*b^3)/b^7) + 1/8*(8*a^3 - 12*a^2 + 12*a - 3)*arcsin(-b*x - a)* 
sgn(b)/(b^3*abs(b))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.79 \[ \int e^{\text {arctanh}(a+b x)} x^3 \, dx=\frac {-24 \mathit {asin} \left (b x +a \right ) a^{3}+36 \mathit {asin} \left (b x +a \right ) a^{2}-36 \mathit {asin} \left (b x +a \right ) a +9 \mathit {asin} \left (b x +a \right )+6 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{3}-6 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2} b x -44 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2}+6 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a \,b^{2} x^{2}+20 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a b x +39 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a -6 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b^{3} x^{3}-8 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b^{2} x^{2}-9 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x -16 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}-24 a^{3}+72 a^{2}-48 a +16}{24 b^{4}} \] Input:

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

Output:

( - 24*asin(a + b*x)*a**3 + 36*asin(a + b*x)*a**2 - 36*asin(a + b*x)*a + 9 
*asin(a + b*x) + 6*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a**3 - 6*sqrt( 
- a**2 - 2*a*b*x - b**2*x**2 + 1)*a**2*b*x - 44*sqrt( - a**2 - 2*a*b*x - b 
**2*x**2 + 1)*a**2 + 6*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a*b**2*x**2 
 + 20*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a*b*x + 39*sqrt( - a**2 - 2* 
a*b*x - b**2*x**2 + 1)*a - 6*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*b**3* 
x**3 - 8*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*b**2*x**2 - 9*sqrt( - a** 
2 - 2*a*b*x - b**2*x**2 + 1)*b*x - 16*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 
 1) - 24*a**3 + 72*a**2 - 48*a + 16)/(24*b**4)