\(\int \frac {e^{\text {arctanh}(a+b x)} x^2}{1-a^2-2 a b x-b^2 x^2} \, dx\) [915]

Optimal result

Integrand size = 34, antiderivative size = 78 \[ \int \frac {e^{\text {arctanh}(a+b x)} x^2}{1-a^2-2 a b x-b^2 x^2} \, dx=\frac {(1-a)^2 \sqrt {1+a+b x}}{b^3 \sqrt {1-a-b x}}+\frac {\sqrt {1-a-b x} \sqrt {1+a+b x}}{b^3}-\frac {(1-2 a) \arcsin (a+b x)}{b^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.82 \[ \int \frac {e^{\text {arctanh}(a+b x)} x^2}{1-a^2-2 a b x-b^2 x^2} \, dx=-\frac {\frac {\left (2-3 a+a^2-b x\right ) \sqrt {1-a^2-2 a b x-b^2 x^2}}{-1+a+b x}-(-1+2 a) \arcsin (a+b x)}{b^3} \] Input:

Integrate[(E^ArcTanh[a + b*x]*x^2)/(1 - a^2 - 2*a*b*x - b^2*x^2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.26, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {6714, 100, 27, 90, 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.

\(\displaystyle \int \frac {x^2 e^{\text {arctanh}(a+b x)}}{-a^2-2 a b x-b^2 x^2+1} \, dx\)

\(\Big \downarrow \) 6714

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

\(\Big \downarrow \) 100

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 62

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

\(\Big \downarrow \) 1090

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

\(\Big \downarrow \) 223

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

Input:

Int[(E^ArcTanh[a + b*x]*x^2)/(1 - a^2 - 2*a*b*x - b^2*x^2),x]
 

Output:

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

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 90

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

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

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 6714

Int[E^(ArcTanh[(a_) + (b_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_) + (e_.)*( 
x_)^2)^(p_.), x_Symbol] :> Simp[(c/(1 - a^2))^p   Int[u*(1 - a - b*x)^(p - 
n/2)*(1 + a + b*x)^(p + n/2), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && 
EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c + e*(1 - a^2), 0] && (IntegerQ[p] || GtQ[c 
/(1 - a^2), 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(193\) vs. \(2(70)=140\).

Time = 0.43 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.49

method	result	size
risch	\(-\frac {b^{2} x^{2}+2 a b x +a^{2}-1}{b^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {\frac {\left (-a^{2}+2 a -1\right ) \sqrt {-\left (x +\frac {-1+a}{b}\right )^{2} b^{2}-2 \left (x +\frac {-1+a}{b}\right ) b}}{b^{2} \left (x +\frac {-1+a}{b}\right )}-\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\sqrt {b^{2}}}+\frac {2 a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\sqrt {b^{2}}}}{b^{2}}\)	\(194\)
default	\(b \left (-\frac {x^{2}}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {3 a \left (\frac {x}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {a \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 a b \right )}{b \left (-4 b^{2} \left (-a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b}-\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b^{2} \sqrt {b^{2}}}\right )}{b}+\frac {2 \left (-a^{2}+1\right ) \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 a b \right )}{b \left (-4 b^{2} \left (-a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b^{2}}\right )+\left (a +1\right ) \left (\frac {x}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {a \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 a b \right )}{b \left (-4 b^{2} \left (-a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b}-\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b^{2} \sqrt {b^{2}}}\right )\)	\(484\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.54 \[ \int \frac {e^{\text {arctanh}(a+b x)} x^2}{1-a^2-2 a b x-b^2 x^2} \, dx=-\frac {{\left ({\left (2 \, a - 1\right )} b x + 2 \, a^{2} - 3 \, a + 1\right )} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - b x - 3 \, a + 2\right )}}{b^{4} x + {\left (a - 1\right )} b^{3}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (67) = 134\).

Time = 0.18 (sec) , antiderivative size = 329, normalized size of antiderivative = 4.22 \[ \int \frac {e^{\text {arctanh}(a+b x)} x^2}{1-a^2-2 a b x-b^2 x^2} \, dx=-\frac {{\left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2}}{b^{5} x + a b^{4} - b^{4}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a b}{b^{6} x + a b^{5} + b^{5}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a b}{b^{6} x + a b^{5} - b^{5}} + \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b^{5} x + a b^{4} + b^{4}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b^{5} x + a b^{4} - b^{4}} + \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{5} x + a b^{4} - b^{4}} - \frac {2 \, a \arcsin \left (b x + a\right )}{b^{4}} + \frac {\arcsin \left (b x + a\right )}{b^{4}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{4}}\right )} b^{2}}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}} \] Input:

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

Output:

-(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a^2/(b^5*x + a*b^4 - b^4) - sqrt(-b^2 
*x^2 - 2*a*b*x - a^2 + 1)*a*b/(b^6*x + a*b^5 + b^5) - sqrt(-b^2*x^2 - 2*a* 
b*x - a^2 + 1)*a*b/(b^6*x + a*b^5 - b^5) + sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 
 1)*a/(b^5*x + a*b^4 + b^4) - sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a/(b^5*x 
+ a*b^4 - b^4) + sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)/(b^5*x + a*b^4 - b^4) 
- 2*a*arcsin(b*x + a)/b^4 + arcsin(b*x + a)/b^4 - sqrt(-b^2*x^2 - 2*a*b*x 
- a^2 + 1)/b^4)*b^2/sqrt(a^2*b^2 - (a^2 - 1)*b^2)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.44 \[ \int \frac {e^{\text {arctanh}(a+b x)} x^2}{1-a^2-2 a b x-b^2 x^2} \, dx=-\frac {{\left (2 \, a - 1\right )} \arcsin \left (-b x - a\right ) \mathrm {sgn}\left (b\right )}{b^{2} {\left | b \right |}} + \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{3}} + \frac {2 \, {\left (a^{2} - 2 \, a + 1\right )}}{b^{2} {\left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b}{b^{2} x + a b} - 1\right )} {\left | b \right |}} \] Input:

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

Output:

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 260, normalized size of antiderivative = 3.33 \[ \int \frac {e^{\text {arctanh}(a+b x)} x^2}{1-a^2-2 a b x-b^2 x^2} \, dx=\frac {2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \mathit {asin} \left (b x +a \right ) a -\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \mathit {asin} \left (b x +a \right )+2 \mathit {asin} \left (b x +a \right ) a^{2}+2 \mathit {asin} \left (b x +a \right ) a b x -3 \mathit {asin} \left (b x +a \right ) a -\mathit {asin} \left (b x +a \right ) b x +\mathit {asin} \left (b x +a \right )-2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2}+5 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a +\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x -2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}+a^{2}-2 a b x -3 a -b^{2} x^{2}+b x +2}{b^{3} \left (\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}+a +b x -1\right )} \] Input:

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

Output:

(2*sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)*a - sqrt( - a**2 
- 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x) + 2*asin(a + b*x)*a**2 + 2*asin(a 
 + b*x)*a*b*x - 3*asin(a + b*x)*a - asin(a + b*x)*b*x + asin(a + b*x) - 2* 
sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*a**2 + 5*sqrt( - a**2 - 2*a*b*x - 
b**2*x**2 + 1)*a + sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*b*x - 2*sqrt( - 
 a**2 - 2*a*b*x - b**2*x**2 + 1) + a**2 - 2*a*b*x - 3*a - b**2*x**2 + b*x 
+ 2)/(b**3*(sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1) + a + b*x - 1))