\(\int e^{-3 \text {arctanh}(a+b x)} x^3 \, dx\) [897]

Optimal result

Integrand size = 14, antiderivative size = 213 \[ \int e^{-3 \text {arctanh}(a+b x)} x^3 \, dx=-\frac {2 x^3 (1-a-b x)^{3/2}}{b \sqrt {1+a+b x}}+\frac {3 \left (17+44 a+36 a^2+8 a^3\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{8 b^4}+\frac {7 \left (1+8 a+6 a^2\right ) (1-a-b x)^{3/2} \sqrt {1+a+b x}}{8 b^4}+\frac {9 x^2 (1-a-b x)^{3/2} \sqrt {1+a+b x}}{4 b^2}+\frac {(11+10 a) (1-a-b x)^{5/2} \sqrt {1+a+b x}}{4 b^4}+\frac {3 \left (17+44 a+36 a^2+8 a^3\right ) \arcsin (a+b x)}{8 b^4} \] Output:

-2*x^3*(-b*x-a+1)^(3/2)/b/(b*x+a+1)^(1/2)+3/8*(8*a^3+36*a^2+44*a+17)*(-b*x 
-a+1)^(1/2)*(b*x+a+1)^(1/2)/b^4+7/8*(6*a^2+8*a+1)*(-b*x-a+1)^(3/2)*(b*x+a+ 
1)^(1/2)/b^4+9/4*x^2*(-b*x-a+1)^(3/2)*(b*x+a+1)^(1/2)/b^2+1/4*(11+10*a)*(- 
b*x-a+1)^(5/2)*(b*x+a+1)^(1/2)/b^4+3/8*(8*a^3+36*a^2+44*a+17)*arcsin(b*x+a 
)/b^4
 

Mathematica [A] (warning: unable to verify)

Time = 0.26 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.08 \[ \int e^{-3 \text {arctanh}(a+b x)} x^3 \, dx=\frac {\sqrt {-b} \left (80-2 a^5-51 b x-40 b^2 x^2+17 b^3 x^3-8 b^4 x^4+2 b^5 x^5-2 a^4 (38+b x)-5 a^3 (31+20 b x)-a^2 \left (4+265 b x+12 b^2 x^2\right )+a \left (157-212 b x-53 b^2 x^2+4 b^3 x^3+2 b^4 x^4\right )\right )+6 \left (17+44 a+36 a^2+8 a^3\right ) \sqrt {b} \sqrt {1-a^2-2 a b x-b^2 x^2} \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {1-a-b x}}{\sqrt {2} \sqrt {-b}}\right )}{8 (-b)^{9/2} \sqrt {-((-1+a+b x) (1+a+b x))}} \] Input:

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

Output:

(Sqrt[-b]*(80 - 2*a^5 - 51*b*x - 40*b^2*x^2 + 17*b^3*x^3 - 8*b^4*x^4 + 2*b 
^5*x^5 - 2*a^4*(38 + b*x) - 5*a^3*(31 + 20*b*x) - a^2*(4 + 265*b*x + 12*b^ 
2*x^2) + a*(157 - 212*b*x - 53*b^2*x^2 + 4*b^3*x^3 + 2*b^4*x^4)) + 6*(17 + 
 44*a + 36*a^2 + 8*a^3)*Sqrt[b]*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSinh[ 
(Sqrt[b]*Sqrt[1 - a - b*x])/(Sqrt[2]*Sqrt[-b])])/(8*(-b)^(9/2)*Sqrt[-((-1 
+ a + b*x)*(1 + a + b*x))])
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6713, 108, 27, 170, 27, 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.

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

\(\Big \downarrow \) 6713

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

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 170

\(\displaystyle \frac {3 \left (\frac {3 x^2 (-a-b x+1)^{3/2} \sqrt {a+b x+1}}{4 b}-\frac {\int \frac {b x \sqrt {-a-b x+1} (6 (1-a) (a+1)-(10 a+11) b x)}{\sqrt {a+b x+1}}dx}{4 b^2}\right )}{b}-\frac {2 x^3 (-a-b x+1)^{3/2}}{b \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \left (\frac {3 x^2 (-a-b x+1)^{3/2} \sqrt {a+b x+1}}{4 b}-\frac {\int \frac {x \sqrt {-a-b x+1} \left (6 \left (1-a^2\right )-(10 a+11) b x\right )}{\sqrt {a+b x+1}}dx}{4 b}\right )}{b}-\frac {2 x^3 (-a-b x+1)^{3/2}}{b \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {3 \left (\frac {3 x^2 (-a-b x+1)^{3/2} \sqrt {a+b x+1}}{4 b}-\frac {-\frac {\left (8 a^3+36 a^2+44 a+17\right ) \int \frac {\sqrt {-a-b x+1}}{\sqrt {a+b x+1}}dx}{2 b}-\frac {\sqrt {a+b x+1} \left (22 a^2-2 (10 a+11) b x+54 a+29\right ) (-a-b x+1)^{3/2}}{6 b^2}}{4 b}\right )}{b}-\frac {2 x^3 (-a-b x+1)^{3/2}}{b \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {3 \left (\frac {3 x^2 (-a-b x+1)^{3/2} \sqrt {a+b x+1}}{4 b}-\frac {-\frac {\left (8 a^3+36 a^2+44 a+17\right ) \left (\int \frac {1}{\sqrt {-a-b x+1} \sqrt {a+b x+1}}dx+\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b}\right )}{2 b}-\frac {\sqrt {a+b x+1} \left (22 a^2-2 (10 a+11) b x+54 a+29\right ) (-a-b x+1)^{3/2}}{6 b^2}}{4 b}\right )}{b}-\frac {2 x^3 (-a-b x+1)^{3/2}}{b \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 62

\(\displaystyle \frac {3 \left (\frac {3 x^2 (-a-b x+1)^{3/2} \sqrt {a+b x+1}}{4 b}-\frac {-\frac {\left (8 a^3+36 a^2+44 a+17\right ) \left (\int \frac {1}{\sqrt {-b^2 x^2-2 a b x+(1-a) (a+1)}}dx+\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b}\right )}{2 b}-\frac {\sqrt {a+b x+1} \left (22 a^2-2 (10 a+11) b x+54 a+29\right ) (-a-b x+1)^{3/2}}{6 b^2}}{4 b}\right )}{b}-\frac {2 x^3 (-a-b x+1)^{3/2}}{b \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {3 \left (\frac {3 x^2 (-a-b x+1)^{3/2} \sqrt {a+b x+1}}{4 b}-\frac {-\frac {\left (8 a^3+36 a^2+44 a+17\right ) \left (\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b}-\frac {\int \frac {1}{\sqrt {1-\frac {\left (-2 x b^2-2 a b\right )^2}{4 b^2}}}d\left (-2 x b^2-2 a b\right )}{2 b^2}\right )}{2 b}-\frac {\sqrt {a+b x+1} \left (22 a^2-2 (10 a+11) b x+54 a+29\right ) (-a-b x+1)^{3/2}}{6 b^2}}{4 b}\right )}{b}-\frac {2 x^3 (-a-b x+1)^{3/2}}{b \sqrt {a+b x+1}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {3 \left (\frac {3 x^2 (-a-b x+1)^{3/2} \sqrt {a+b x+1}}{4 b}-\frac {-\frac {\sqrt {a+b x+1} \left (22 a^2-2 (10 a+11) b x+54 a+29\right ) (-a-b x+1)^{3/2}}{6 b^2}-\frac {\left (8 a^3+36 a^2+44 a+17\right ) \left (\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{b}-\frac {\arcsin \left (\frac {-2 a b-2 b^2 x}{2 b}\right )}{b}\right )}{2 b}}{4 b}\right )}{b}-\frac {2 x^3 (-a-b x+1)^{3/2}}{b \sqrt {a+b x+1}}\)

Input:

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

Output:

(-2*x^3*(1 - a - b*x)^(3/2))/(b*Sqrt[1 + a + b*x]) + (3*((3*x^2*(1 - a - b 
*x)^(3/2)*Sqrt[1 + a + b*x])/(4*b) - (-1/6*((1 - a - b*x)^(3/2)*Sqrt[1 + a 
 + b*x]*(29 + 54*a + 22*a^2 - 2*(11 + 10*a)*b*x))/b^2 - ((17 + 44*a + 36*a 
^2 + 8*a^3)*((Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/b - ArcSin[(-2*a*b - 2* 
b^2*x)/(2*b)]/b))/(2*b))/(4*b)))/b
 

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 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 108

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

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 170

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

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 = 0.46 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.64

method	result	size
risch	\(-\frac {\left (-2 b^{3} x^{3}+2 a \,b^{2} x^{2}-2 a^{2} b x +8 b^{2} x^{2}+2 a^{3}-20 a b x +44 a^{2}-19 b x +93 a +48\right ) \left (b^{2} x^{2}+2 a b x +a^{2}-1\right )}{8 b^{4} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {-\frac {\left (-32 a^{3}-96 a^{2}-96 a -32\right ) \sqrt {-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b}}{b^{2} \left (x +\frac {a +1}{b}\right )}+\frac {51 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\sqrt {b^{2}}}+\frac {132 a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\sqrt {b^{2}}}+\frac {108 a^{2} \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\sqrt {b^{2}}}+\frac {24 a^{3} \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\sqrt {b^{2}}}}{8 b^{3}}\)	\(349\)
default	\(\frac {-\frac {\left (-2 b^{2} x -2 a b \right ) \left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}}}{8 b^{2}}-\frac {3 \left (-4 b^{2} \left (-a^{2}+1\right )-4 a^{2} b^{2}\right ) \left (-\frac {\left (-2 b^{2} x -2 a b \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{4 b^{2}}-\frac {\left (-4 b^{2} \left (-a^{2}+1\right )-4 a^{2} b^{2}\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^{2} \sqrt {b^{2}}}\right )}{16 b^{2}}}{b^{3}}-\frac {3 \left (a +1\right ) \left (\frac {\left (-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b \right )^{\frac {3}{2}}}{3}+b \left (-\frac {\left (-2 b^{2} \left (x +\frac {a +1}{b}\right )+2 b \right ) \sqrt {-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b}}{4 b^{2}}+\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a +1}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b}}\right )}{2 \sqrt {b^{2}}}\right )\right )}{b^{4}}+\frac {3 \left (a^{2}+2 a +1\right ) \left (\frac {\left (-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b \right )^{\frac {5}{2}}}{b \left (x +\frac {a +1}{b}\right )^{2}}+3 b \left (\frac {\left (-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b \right )^{\frac {3}{2}}}{3}+b \left (-\frac {\left (-2 b^{2} \left (x +\frac {a +1}{b}\right )+2 b \right ) \sqrt {-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b}}{4 b^{2}}+\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a +1}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b}}\right )}{2 \sqrt {b^{2}}}\right )\right )\right )}{b^{5}}-\frac {\left (a^{3}+3 a^{2}+3 a +1\right ) \left (-\frac {\left (-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b \right )^{\frac {5}{2}}}{b \left (x +\frac {a +1}{b}\right )^{3}}-2 b \left (\frac {\left (-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b \right )^{\frac {5}{2}}}{b \left (x +\frac {a +1}{b}\right )^{2}}+3 b \left (\frac {\left (-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b \right )^{\frac {3}{2}}}{3}+b \left (-\frac {\left (-2 b^{2} \left (x +\frac {a +1}{b}\right )+2 b \right ) \sqrt {-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b}}{4 b^{2}}+\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a +1}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {a +1}{b}\right )^{2} b^{2}+2 \left (x +\frac {a +1}{b}\right ) b}}\right )}{2 \sqrt {b^{2}}}\right )\right )\right )\right )}{b^{6}}\)	\(817\)

Input:

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

Output:

-1/8*(-2*b^3*x^3+2*a*b^2*x^2-2*a^2*b*x+8*b^2*x^2+2*a^3-20*a*b*x+44*a^2-19* 
b*x+93*a+48)*(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*(-(-32*a^3-96*a^2-96*a-32)/b^2/(x+(a+1)/b)*(-(x+(a+1)/b)^2*b^2+2*(x+ 
(a+1)/b)*b)^(1/2)+51/(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))+132*a/(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))+108*a^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))+24*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 = 191, normalized size of antiderivative = 0.90 \[ \int e^{-3 \text {arctanh}(a+b x)} x^3 \, dx=-\frac {3 \, {\left (8 \, a^{4} + 44 \, a^{3} + {\left (8 \, a^{3} + 36 \, a^{2} + 44 \, a + 17\right )} b x + 80 \, a^{2} + 61 \, a + 17\right )} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + {\left (2 \, b^{4} x^{4} - 6 \, b^{3} x^{3} + {\left (10 \, a + 11\right )} b^{2} x^{2} - 2 \, a^{4} - 78 \, a^{3} - {\left (22 \, a^{2} + 54 \, a + 29\right )} b x - 233 \, a^{2} - 237 \, a - 80\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{8 \, {\left (b^{5} x + {\left (a + 1\right )} b^{4}\right )}} \] Input:

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

Output:

-1/8*(3*(8*a^4 + 44*a^3 + (8*a^3 + 36*a^2 + 44*a + 17)*b*x + 80*a^2 + 61*a 
 + 17)*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)) + (2*b^4*x^4 - 6*b^3*x^3 + (10*a + 11)*b^2*x^2 - 2*a^4 - 7 
8*a^3 - (22*a^2 + 54*a + 29)*b*x - 233*a^2 - 237*a - 80)*sqrt(-b^2*x^2 - 2 
*a*b*x - a^2 + 1))/(b^5*x + (a + 1)*b^4)
 

Sympy [F]

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

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

Output:

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 985, normalized size of antiderivative = 4.62 \[ \int e^{-3 \text {arctanh}(a+b x)} x^3 \, dx =\text {Too large to display} \] Input:

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

Output:

-(-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*a^3/(b^6*x^2 + 2*a*b^5*x + a^2*b^4 + 
 2*b^5*x + 2*a*b^4 + b^4) - 3*(-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*a^2/(b^ 
6*x^2 + 2*a*b^5*x + a^2*b^4 + 2*b^5*x + 2*a*b^4 + b^4) + 3/2*(-b^2*x^2 - 2 
*a*b*x - a^2 + 1)^(3/2)*a^2/(b^5*x + a*b^4 + b^4) + 6*sqrt(-b^2*x^2 - 2*a* 
b*x - a^2 + 1)*a^3/(b^5*x + a*b^4 + b^4) - 3*(-b^2*x^2 - 2*a*b*x - a^2 + 1 
)^(3/2)*a/(b^6*x^2 + 2*a*b^5*x + a^2*b^4 + 2*b^5*x + 2*a*b^4 + b^4) + 3*(- 
b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*a/(b^5*x + a*b^4 + b^4) + 18*sqrt(-b^2* 
x^2 - 2*a*b*x - a^2 + 1)*a^2/(b^5*x + a*b^4 + b^4) - (-b^2*x^2 - 2*a*b*x - 
 a^2 + 1)^(3/2)/(b^6*x^2 + 2*a*b^5*x + a^2*b^4 + 2*b^5*x + 2*a*b^4 + b^4) 
+ 3/2*(-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)/(b^5*x + a*b^4 + b^4) + 18*sqrt 
(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a/(b^5*x + a*b^4 + b^4) + 3*a^3*arcsin(b*x 
+ a)/b^4 + 6*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)/(b^5*x + a*b^4 + b^4) + 1/ 
4*(-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*x/b^3 - 3/2*sqrt(b^2*x^2 + 2*a*b*x 
+ a^2 + 4*b*x + 4*a + 3)*a*x/b^3 + 27/2*a^2*arcsin(b*x + a)/b^4 - 3/4*(-b^ 
2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*a/b^4 - 3/2*sqrt(b^2*x^2 + 2*a*b*x + a^2 
+ 4*b*x + 4*a + 3)*a^2/b^4 + 9/2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a^2/b^ 
4 - 3/2*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 4*b*x + 4*a + 3)*x/b^3 + 3/8*sqrt(- 
b^2*x^2 - 2*a*b*x - a^2 + 1)*x/b^3 + 3/2*I*a*arcsin(b*x + a + 2)/b^4 + 18* 
a*arcsin(b*x + a)/b^4 - (-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)/b^4 - 9/2*sqr 
t(b^2*x^2 + 2*a*b*x + a^2 + 4*b*x + 4*a + 3)*a/b^4 + 75/8*sqrt(-b^2*x^2...
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.99 \[ \int e^{-3 \text {arctanh}(a+b x)} x^3 \, dx=-\frac {1}{8} \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left ({\left (2 \, x {\left (\frac {x}{b} - \frac {a b^{11} + 4 \, b^{11}}{b^{13}}\right )} + \frac {2 \, a^{2} b^{10} + 20 \, a b^{10} + 19 \, b^{10}}{b^{13}}\right )} x - \frac {2 \, a^{3} b^{9} + 44 \, a^{2} b^{9} + 93 \, a b^{9} + 48 \, b^{9}}{b^{13}}\right )} - \frac {3 \, {\left (8 \, a^{3} + 36 \, a^{2} + 44 \, a + 17\right )} \arcsin \left (-b x - a\right ) \mathrm {sgn}\left (b\right )}{8 \, b^{3} {\left | b \right |}} - \frac {8 \, {\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )}}{b^{3} {\left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b}{b^{2} x + a b} + 1\right )} {\left | b \right |}} \] Input:

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

Output:

-1/8*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*((2*x*(x/b - (a*b^11 + 4*b^11)/b^1 
3) + (2*a^2*b^10 + 20*a*b^10 + 19*b^10)/b^13)*x - (2*a^3*b^9 + 44*a^2*b^9 
+ 93*a*b^9 + 48*b^9)/b^13) - 3/8*(8*a^3 + 36*a^2 + 44*a + 17)*arcsin(-b*x 
- a)*sgn(b)/(b^3*abs(b)) - 8*(a^3 + 3*a^2 + 3*a + 1)/(b^3*((sqrt(-b^2*x^2 
- 2*a*b*x - a^2 + 1)*abs(b) + b)/(b^2*x + a*b) + 1)*abs(b))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 703, normalized size of antiderivative = 3.30 \[ \int e^{-3 \text {arctanh}(a+b x)} x^3 \, dx=\frac {102+235 a +24 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \mathit {asin} \left (b x +a \right ) a^{3}+108 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \mathit {asin} \left (b x +a \right ) a^{2}+132 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \mathit {asin} \left (b x +a \right ) a -51 \mathit {asin} \left (b x +a \right ) b x +2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b^{4} x^{4}-53 a \,b^{2} x^{2}+51 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \mathit {asin} \left (b x +a \right )-24 \mathit {asin} \left (b x +a \right ) a^{4}-217 a^{2} b x +4 a \,b^{3} x^{3}-22 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2} b x +10 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a \,b^{2} x^{2}-54 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a b x -12 a^{2} b^{2} x^{2}-40 b^{2} x^{2}-132 \mathit {asin} \left (b x +a \right ) a^{3}-240 \mathit {asin} \left (b x +a \right ) a^{2}-183 \mathit {asin} \left (b x +a \right ) a -102 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{3}-281 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2}-293 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a -2 a^{5}-2 a^{4} b x +2 a \,b^{4} x^{4}-2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{4}+2 b^{5} x^{5}-8 b^{4} x^{4}+17 b^{3} x^{3}-76 a^{3} b x +100 a^{2}-83 a^{3}-6 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b^{3} x^{3}+11 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b^{2} x^{2}-29 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x -24 \mathit {asin} \left (b x +a \right ) a^{3} b x -108 \mathit {asin} \left (b x +a \right ) a^{2} b x -132 \mathit {asin} \left (b x +a \right ) a b x -52 a^{4}-51 \mathit {asin} \left (b x +a \right )-102 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}-156 a b x -29 b x}{8 b^{4} \left (\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}-a -b x -1\right )} \] Input:

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

Output:

(24*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)*a**3 + 108*sqrt( 
 - a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)*a**2 + 132*sqrt( - a**2 - 
 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)*a + 51*sqrt( - a**2 - 2*a*b*x - b* 
*2*x**2 + 1)*asin(a + b*x) - 24*asin(a + b*x)*a**4 - 24*asin(a + b*x)*a**3 
*b*x - 132*asin(a + b*x)*a**3 - 108*asin(a + b*x)*a**2*b*x - 240*asin(a + 
b*x)*a**2 - 132*asin(a + b*x)*a*b*x - 183*asin(a + b*x)*a - 51*asin(a + b* 
x)*b*x - 51*asin(a + b*x) - 2*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a**4 
 - 102*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a**3 - 22*sqrt( - a**2 - 2* 
a*b*x - b**2*x**2 + 1)*a**2*b*x - 281*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 
 1)*a**2 + 10*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a*b**2*x**2 - 54*sqr 
t( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a*b*x - 293*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)*b**4*x**4 - 6 
*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*b**3*x**3 + 11*sqrt( - a**2 - 2*a 
*b*x - b**2*x**2 + 1)*b**2*x**2 - 29*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 
1)*b*x - 102*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1) - 2*a**5 - 2*a**4*b*x 
 - 52*a**4 - 76*a**3*b*x - 83*a**3 - 12*a**2*b**2*x**2 - 217*a**2*b*x + 10 
0*a**2 + 2*a*b**4*x**4 + 4*a*b**3*x**3 - 53*a*b**2*x**2 - 156*a*b*x + 235* 
a + 2*b**5*x**5 - 8*b**4*x**4 + 17*b**3*x**3 - 40*b**2*x**2 - 29*b*x + 102 
)/(8*b**4*(sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1) - a - b*x - 1))