\(\int e^{3 \text {arctanh}(a x)} x^2 \, dx\) [19]

Optimal result

Integrand size = 12, antiderivative size = 101 \[ \int e^{3 \text {arctanh}(a x)} x^2 \, dx=\frac {5 \sqrt {1-a^2 x^2}}{a^3}+\frac {3 x \sqrt {1-a^2 x^2}}{2 a^2}+\frac {4 \sqrt {1-a^2 x^2}}{a^3 (1-a x)}-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^3}-\frac {11 \arcsin (a x)}{2 a^3} \] Output:

5*(-a^2*x^2+1)^(1/2)/a^3+3/2*x*(-a^2*x^2+1)^(1/2)/a^2+4*(-a^2*x^2+1)^(1/2) 
/a^3/(-a*x+1)-1/3*(-a^2*x^2+1)^(3/2)/a^3-11/2*arcsin(a*x)/a^3
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.57 \[ \int e^{3 \text {arctanh}(a x)} x^2 \, dx=\frac {\frac {\sqrt {1-a^2 x^2} \left (-52+19 a x+7 a^2 x^2+2 a^3 x^3\right )}{-1+a x}-33 \arcsin (a x)}{6 a^3} \] Input:

Integrate[E^(3*ArcTanh[a*x])*x^2,x]
 

Output:

((Sqrt[1 - a^2*x^2]*(-52 + 19*a*x + 7*a^2*x^2 + 2*a^3*x^3))/(-1 + a*x) - 3 
3*ArcSin[a*x])/(6*a^3)
 

Rubi [A] (verified)

Time = 1.34 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.21, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {6674, 2164, 25, 2027, 2164, 25, 27, 563, 2346, 25, 2346, 25, 27, 455, 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.

\(\displaystyle \int x^2 e^{3 \text {arctanh}(a x)} \, dx\)

\(\Big \downarrow \) 6674

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

\(\Big \downarrow \) 2164

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2027

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

\(\Big \downarrow \) 2164

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 563

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

\(\Big \downarrow \) 2346

\(\displaystyle \frac {4 \sqrt {1-a^2 x^2}}{a^3 (1-a x)}-\frac {-\frac {\int -\frac {9 x^2 a^4+14 x a^3+12 a^2}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {1}{3} a x^2 \sqrt {1-a^2 x^2}}{a^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {4 \sqrt {1-a^2 x^2}}{a^3 (1-a x)}-\frac {\frac {\int \frac {9 x^2 a^4+14 x a^3+12 a^2}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {1}{3} a x^2 \sqrt {1-a^2 x^2}}{a^2}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {4 \sqrt {1-a^2 x^2}}{a^3 (1-a x)}-\frac {\frac {-\frac {\int -\frac {a^4 (28 a x+33)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {9}{2} a^2 x \sqrt {1-a^2 x^2}}{3 a^2}-\frac {1}{3} a x^2 \sqrt {1-a^2 x^2}}{a^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {4 \sqrt {1-a^2 x^2}}{a^3 (1-a x)}-\frac {\frac {\frac {\int \frac {a^4 (28 a x+33)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {9}{2} a^2 x \sqrt {1-a^2 x^2}}{3 a^2}-\frac {1}{3} a x^2 \sqrt {1-a^2 x^2}}{a^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 \sqrt {1-a^2 x^2}}{a^3 (1-a x)}-\frac {\frac {\frac {1}{2} a^2 \int \frac {28 a x+33}{\sqrt {1-a^2 x^2}}dx-\frac {9}{2} a^2 x \sqrt {1-a^2 x^2}}{3 a^2}-\frac {1}{3} a x^2 \sqrt {1-a^2 x^2}}{a^2}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {4 \sqrt {1-a^2 x^2}}{a^3 (1-a x)}-\frac {\frac {\frac {1}{2} a^2 \left (33 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {28 \sqrt {1-a^2 x^2}}{a}\right )-\frac {9}{2} a^2 x \sqrt {1-a^2 x^2}}{3 a^2}-\frac {1}{3} a x^2 \sqrt {1-a^2 x^2}}{a^2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {4 \sqrt {1-a^2 x^2}}{a^3 (1-a x)}-\frac {\frac {\frac {1}{2} a^2 \left (\frac {33 \arcsin (a x)}{a}-\frac {28 \sqrt {1-a^2 x^2}}{a}\right )-\frac {9}{2} a^2 x \sqrt {1-a^2 x^2}}{3 a^2}-\frac {1}{3} a x^2 \sqrt {1-a^2 x^2}}{a^2}\)

Input:

Int[E^(3*ArcTanh[a*x])*x^2,x]
 

Output:

(4*Sqrt[1 - a^2*x^2])/(a^3*(1 - a*x)) - (-1/3*(a*x^2*Sqrt[1 - a^2*x^2]) + 
((-9*a^2*x*Sqrt[1 - a^2*x^2])/2 + (a^2*((-28*Sqrt[1 - a^2*x^2])/a + (33*Ar 
cSin[a*x])/a))/2)/(3*a^2))/a^2
 

Defintions of rubi rules used

rule 25

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

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 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 455

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

rule 563

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Simp[(-(-c)^(m - n - 2))*d^(2*n - m + 3)*(Sqrt[a + b*x^2]/(2^(n + 1)* 
b^(n + 2)*(c + d*x))), x] - Simp[d^(2*n - m + 2)/b^(n + 1)   Int[(1/Sqrt[a 
+ b*x^2])*ExpandToSum[(2^(-n - 1)*(-c)^(m - n - 1) - d^m*x^m*(-c + d*x)^(-n 
 - 1))/(c + d*x), x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2 
, 0] && IGtQ[m, 0] && ILtQ[n, 0] && EqQ[n + p, -3/2]
 

rule 2027

Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ 
(p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & 
& PosQ[s - r] &&  !(EqQ[p, 1] && EqQ[u, 1])
 

rule 2164

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] 
:> Simp[d*e   Int[(d + e*x)^(m - 1)*PolynomialQuotient[Pq, a*e + b*d*x, x]* 
(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, d, e, m, p}, x] && PolyQ[Pq, x] 
 && EqQ[b*d^2 + a*e^2, 0] && EqQ[PolynomialRemainder[Pq, a*e + b*d*x, x], 0 
]
 

rule 2346

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], 
e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( 
q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1))   Int[(a + b*x^2)^p*ExpandToS 
um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], 
x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] &&  !LeQ[p, -1]
 

rule 6674

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x 
)^m*((1 + a*x)^((n + 1)/2)/((1 - a*x)^((n - 1)/2)*Sqrt[1 - a^2*x^2])), x] / 
; FreeQ[{a, c, m}, x] && IntegerQ[(n - 1)/2]
 

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.12

method	result	size
risch	\(-\frac {\left (2 a^{2} x^{2}+9 a x +28\right ) \left (a^{2} x^{2}-1\right )}{6 a^{3} \sqrt {-a^{2} x^{2}+1}}-\frac {11 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}-\frac {4 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{4} \left (x -\frac {1}{a}\right )}\)	\(113\)
meijerg	\(-\frac {\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}}{a^{2} \sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-2 a^{4} x^{4}-8 a^{2} x^{2}+16\right )}{6 \sqrt {-a^{2} x^{2}+1}}}{a^{3} \sqrt {\pi }}+\frac {\frac {3 \sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (-5 a^{2} x^{2}+15\right )}{10 a^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {9 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{2 a^{5}}}{a^{2} \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {-6 \sqrt {\pi }+\frac {3 \sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}}{a^{3} \sqrt {\pi }}\)	\(227\)
default	\(\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}+a^{3} \left (-\frac {x^{4}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {4 x^{2}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {8}{3 a^{4} \sqrt {-a^{2} x^{2}+1}}}{a^{2}}\right )+3 a \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )+3 a^{2} \left (-\frac {x^{3}}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )\)	\(237\)

Input:

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

Output:

-1/6*(2*a^2*x^2+9*a*x+28)*(a^2*x^2-1)/a^3/(-a^2*x^2+1)^(1/2)-11/2/a^2/(a^2 
)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-4/a^4/(x-1/a)*(-(x-1/a)^2 
*a^2-2*a*(x-1/a))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.84 \[ \int e^{3 \text {arctanh}(a x)} x^2 \, dx=\frac {52 \, a x + 66 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (2 \, a^{3} x^{3} + 7 \, a^{2} x^{2} + 19 \, a x - 52\right )} \sqrt {-a^{2} x^{2} + 1} - 52}{6 \, {\left (a^{4} x - a^{3}\right )}} \] Input:

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

Output:

1/6*(52*a*x + 66*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (2*a^3 
*x^3 + 7*a^2*x^2 + 19*a*x - 52)*sqrt(-a^2*x^2 + 1) - 52)/(a^4*x - a^3)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.99 \[ \int e^{3 \text {arctanh}(a x)} x^2 \, dx=-\frac {a x^{4}}{3 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {3 \, x^{3}}{2 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {13 \, x^{2}}{3 \, \sqrt {-a^{2} x^{2} + 1} a} + \frac {11 \, x}{2 \, \sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {11 \, \arcsin \left (a x\right )}{2 \, a^{3}} + \frac {26}{3 \, \sqrt {-a^{2} x^{2} + 1} a^{3}} \] Input:

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

Output:

-1/3*a*x^4/sqrt(-a^2*x^2 + 1) - 3/2*x^3/sqrt(-a^2*x^2 + 1) - 13/3*x^2/(sqr 
t(-a^2*x^2 + 1)*a) + 11/2*x/(sqrt(-a^2*x^2 + 1)*a^2) - 11/2*arcsin(a*x)/a^ 
3 + 26/3/(sqrt(-a^2*x^2 + 1)*a^3)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.86 \[ \int e^{3 \text {arctanh}(a x)} x^2 \, dx=\frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} {\left (x {\left (\frac {2 \, x}{a} + \frac {9}{a^{2}}\right )} + \frac {28}{a^{3}}\right )} - \frac {11 \, \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{2 \, a^{2} {\left | a \right |}} + \frac {8}{a^{2} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \] Input:

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

Output:

1/6*sqrt(-a^2*x^2 + 1)*(x*(2*x/a + 9/a^2) + 28/a^3) - 11/2*arcsin(a*x)*sgn 
(a)/(a^2*abs(a)) + 8/(a^2*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)*ab 
s(a))
 

Mupad [B] (verification not implemented)

Time = 13.89 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.42 \[ \int e^{3 \text {arctanh}(a x)} x^2 \, dx=\frac {4\,\sqrt {1-a^2\,x^2}}{a^2\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {2}{3\,a\,\sqrt {-a^2}}-\frac {4\,\sqrt {-a^2}}{a^3}+\frac {a\,x^2}{3\,\sqrt {-a^2}}-\frac {3\,x\,\sqrt {-a^2}}{2\,a^2}\right )}{\sqrt {-a^2}}-\frac {11\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^2\,\sqrt {-a^2}} \] Input:

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

Output:

(4*(1 - a^2*x^2)^(1/2))/(a^2*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)*(-a^2)^(1/2 
)) - ((1 - a^2*x^2)^(1/2)*(2/(3*a*(-a^2)^(1/2)) - (4*(-a^2)^(1/2))/a^3 + ( 
a*x^2)/(3*(-a^2)^(1/2)) - (3*x*(-a^2)^(1/2))/(2*a^2)))/(-a^2)^(1/2) - (11* 
asinh(x*(-a^2)^(1/2)))/(2*a^2*(-a^2)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.49 \[ \int e^{3 \text {arctanh}(a x)} x^2 \, dx=\frac {-33 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )-33 \mathit {asin} \left (a x \right ) a x +33 \mathit {asin} \left (a x \right )+2 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+7 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+19 \sqrt {-a^{2} x^{2}+1}\, a x -66 \sqrt {-a^{2} x^{2}+1}-2 a^{4} x^{4}-9 a^{3} x^{3}-26 a^{2} x^{2}+19 a x +66}{6 a^{3} \left (\sqrt {-a^{2} x^{2}+1}+a x -1\right )} \] Input:

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

Output:

( - 33*sqrt( - a**2*x**2 + 1)*asin(a*x) - 33*asin(a*x)*a*x + 33*asin(a*x) 
+ 2*sqrt( - a**2*x**2 + 1)*a**3*x**3 + 7*sqrt( - a**2*x**2 + 1)*a**2*x**2 
+ 19*sqrt( - a**2*x**2 + 1)*a*x - 66*sqrt( - a**2*x**2 + 1) - 2*a**4*x**4 
- 9*a**3*x**3 - 26*a**2*x**2 + 19*a*x + 66)/(6*a**3*(sqrt( - a**2*x**2 + 1 
) + a*x - 1))