\(\int e^{3 \text {arctanh}(a x)} x^2 (c-a^2 c x^2) \, dx\) [1175]

Optimal result

Integrand size = 23, antiderivative size = 128 \[ \int e^{3 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right ) \, dx=-\frac {38 c \sqrt {1-a^2 x^2}}{15 a^3}-\frac {13 c x \sqrt {1-a^2 x^2}}{8 a^2}-\frac {19 c x^2 \sqrt {1-a^2 x^2}}{15 a}-\frac {3}{4} c x^3 \sqrt {1-a^2 x^2}-\frac {1}{5} a c x^4 \sqrt {1-a^2 x^2}+\frac {13 c \arcsin (a x)}{8 a^3} \] Output:

-38/15*c*(-a^2*x^2+1)^(1/2)/a^3-13/8*c*x*(-a^2*x^2+1)^(1/2)/a^2-19/15*c*x^ 
2*(-a^2*x^2+1)^(1/2)/a-3/4*c*x^3*(-a^2*x^2+1)^(1/2)-1/5*a*c*x^4*(-a^2*x^2+ 
1)^(1/2)+13/8*c*arcsin(a*x)/a^3
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.48 \[ \int e^{3 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right ) \, dx=\frac {-c \sqrt {1-a^2 x^2} \left (304+195 a x+152 a^2 x^2+90 a^3 x^3+24 a^4 x^4\right )+195 c \arcsin (a x)}{120 a^3} \] Input:

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

Output:

(-(c*Sqrt[1 - a^2*x^2]*(304 + 195*a*x + 152*a^2*x^2 + 90*a^3*x^3 + 24*a^4* 
x^4)) + 195*c*ArcSin[a*x])/(120*a^3)
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.21, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {6698, 541, 25, 2340, 25, 27, 533, 27, 533, 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.

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

\(\Big \downarrow \) 6698

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

\(\Big \downarrow \) 541

\(\displaystyle c \left (-\frac {\int -\frac {x^2 \left (15 x^2 a^4+19 x a^3+5 a^2\right )}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {1}{5} a x^4 \sqrt {1-a^2 x^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle c \left (\frac {\int \frac {x^2 \left (15 x^2 a^4+19 x a^3+5 a^2\right )}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {1}{5} a x^4 \sqrt {1-a^2 x^2}\right )\)

\(\Big \downarrow \) 2340

\(\displaystyle c \left (\frac {-\frac {\int -\frac {a^4 x^2 (76 a x+65)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {15}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{5 a^2}-\frac {1}{5} a x^4 \sqrt {1-a^2 x^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle c \left (\frac {\frac {\int \frac {a^4 x^2 (76 a x+65)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {15}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{5 a^2}-\frac {1}{5} a x^4 \sqrt {1-a^2 x^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle c \left (\frac {\frac {1}{4} a^2 \int \frac {x^2 (76 a x+65)}{\sqrt {1-a^2 x^2}}dx-\frac {15}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{5 a^2}-\frac {1}{5} a x^4 \sqrt {1-a^2 x^2}\right )\)

\(\Big \downarrow \) 533

\(\displaystyle c \left (\frac {\frac {1}{4} a^2 \left (\frac {\int \frac {a x (195 a x+152)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {76 x^2 \sqrt {1-a^2 x^2}}{3 a}\right )-\frac {15}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{5 a^2}-\frac {1}{5} a x^4 \sqrt {1-a^2 x^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle c \left (\frac {\frac {1}{4} a^2 \left (\frac {\int \frac {x (195 a x+152)}{\sqrt {1-a^2 x^2}}dx}{3 a}-\frac {76 x^2 \sqrt {1-a^2 x^2}}{3 a}\right )-\frac {15}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{5 a^2}-\frac {1}{5} a x^4 \sqrt {1-a^2 x^2}\right )\)

\(\Big \downarrow \) 533

\(\displaystyle c \left (\frac {\frac {1}{4} a^2 \left (\frac {\frac {\int \frac {a (304 a x+195)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {195 x \sqrt {1-a^2 x^2}}{2 a}}{3 a}-\frac {76 x^2 \sqrt {1-a^2 x^2}}{3 a}\right )-\frac {15}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{5 a^2}-\frac {1}{5} a x^4 \sqrt {1-a^2 x^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle c \left (\frac {\frac {1}{4} a^2 \left (\frac {\frac {\int \frac {304 a x+195}{\sqrt {1-a^2 x^2}}dx}{2 a}-\frac {195 x \sqrt {1-a^2 x^2}}{2 a}}{3 a}-\frac {76 x^2 \sqrt {1-a^2 x^2}}{3 a}\right )-\frac {15}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{5 a^2}-\frac {1}{5} a x^4 \sqrt {1-a^2 x^2}\right )\)

\(\Big \downarrow \) 455

\(\displaystyle c \left (\frac {\frac {1}{4} a^2 \left (\frac {\frac {195 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {304 \sqrt {1-a^2 x^2}}{a}}{2 a}-\frac {195 x \sqrt {1-a^2 x^2}}{2 a}}{3 a}-\frac {76 x^2 \sqrt {1-a^2 x^2}}{3 a}\right )-\frac {15}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{5 a^2}-\frac {1}{5} a x^4 \sqrt {1-a^2 x^2}\right )\)

\(\Big \downarrow \) 223

\(\displaystyle c \left (\frac {\frac {1}{4} a^2 \left (\frac {\frac {\frac {195 \arcsin (a x)}{a}-\frac {304 \sqrt {1-a^2 x^2}}{a}}{2 a}-\frac {195 x \sqrt {1-a^2 x^2}}{2 a}}{3 a}-\frac {76 x^2 \sqrt {1-a^2 x^2}}{3 a}\right )-\frac {15}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{5 a^2}-\frac {1}{5} a x^4 \sqrt {1-a^2 x^2}\right )\)

Input:

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

Output:

c*(-1/5*(a*x^4*Sqrt[1 - a^2*x^2]) + ((-15*a^2*x^3*Sqrt[1 - a^2*x^2])/4 + ( 
a^2*((-76*x^2*Sqrt[1 - a^2*x^2])/(3*a) + ((-195*x*Sqrt[1 - a^2*x^2])/(2*a) 
 + ((-304*Sqrt[1 - a^2*x^2])/a + (195*ArcSin[a*x])/a)/(2*a))/(3*a)))/4)/(5 
*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 533

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* 
p + 2))   Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], 
 x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer 
Q[2*p]
 

rule 541

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

rule 2340

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 
)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m 
+ q + 2*p + 1))   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) 
*Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; 
GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ 
[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
 

rule 6698

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[c^p   Int[x^m*(1 - a^2*x^2)^(p - n/2)*(1 + a*x)^n, x], x] 
/; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 
 0]) && IGtQ[(n + 1)/2, 0] &&  !IntegerQ[p - n/2]
 

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.70

method	result	size
risch	\(\frac {\left (24 a^{4} x^{4}+90 a^{3} x^{3}+152 a^{2} x^{2}+195 a x +304\right ) \left (a^{2} x^{2}-1\right ) c}{120 a^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {13 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c}{8 a^{2} \sqrt {a^{2}}}\)	\(90\)
meijerg	\(\frac {2 c \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (-5 a^{2} x^{2}+15\right )}{10 a^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{2 a^{5}}\right )}{a^{2} \sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {c \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{a^{2} \sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {c \left (-\frac {16 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (-8 x^{6} a^{6}-16 a^{4} x^{4}-64 a^{2} x^{2}+128\right )}{40 \sqrt {-a^{2} x^{2}+1}}\right )}{a^{3} \sqrt {\pi }}+\frac {2 c \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-2 a^{4} x^{4}-8 a^{2} x^{2}+16\right )}{6 \sqrt {-a^{2} x^{2}+1}}\right )}{a^{3} \sqrt {\pi }}+\frac {3 c \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {7}{2}} \left (-14 a^{4} x^{4}-35 a^{2} x^{2}+105\right )}{56 a^{6} \sqrt {-a^{2} x^{2}+1}}-\frac {15 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {7}{2}} \arcsin \left (a x \right )}{8 a^{7}}\right )}{a^{2} \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {3 c \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}\right )}{a^{3} \sqrt {\pi }}\)	\(371\)
default	\(-c \left (a^{5} \left (-\frac {x^{6}}{5 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {2 x^{4}}{5 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {6 \left (-\frac {4 x^{2}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {8}{3 a^{4} \sqrt {-a^{2} x^{2}+1}}\right )}{5 a^{2}}}{a^{2}}\right )-\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}-3 a \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )-2 a^{2} \left (-\frac {x^{3}}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )+2 a^{3} \left (-\frac {x^{4}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {4 x^{2}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {8}{3 a^{4} \sqrt {-a^{2} x^{2}+1}}}{a^{2}}\right )+3 a^{4} \left (-\frac {x^{5}}{4 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {5 x^{3}}{8 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {5 \left (\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )}{4 a^{2}}}{a^{2}}\right )\right )\)	\(440\)

Input:

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

Output:

1/120*(24*a^4*x^4+90*a^3*x^3+152*a^2*x^2+195*a*x+304)*(a^2*x^2-1)/a^3/(-a^ 
2*x^2+1)^(1/2)*c+13/8/a^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1 
/2))*c
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.62 \[ \int e^{3 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right ) \, dx=-\frac {390 \, c \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (24 \, a^{4} c x^{4} + 90 \, a^{3} c x^{3} + 152 \, a^{2} c x^{2} + 195 \, a c x + 304 \, c\right )} \sqrt {-a^{2} x^{2} + 1}}{120 \, a^{3}} \] Input:

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

Output:

-1/120*(390*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (24*a^4*c*x^4 + 90* 
a^3*c*x^3 + 152*a^2*c*x^2 + 195*a*c*x + 304*c)*sqrt(-a^2*x^2 + 1))/a^3
 

Sympy [A] (verification not implemented)

Time = 5.86 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.14 \[ \int e^{3 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right ) \, dx=\begin {cases} \sqrt {- a^{2} x^{2} + 1} \left (- \frac {a c x^{4}}{5} - \frac {3 c x^{3}}{4} - \frac {19 c x^{2}}{15 a} - \frac {13 c x}{8 a^{2}} - \frac {38 c}{15 a^{3}}\right ) + \frac {13 c \log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{8 a^{2} \sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {a^{3} c x^{6}}{6} + \frac {3 a^{2} c x^{5}}{5} + \frac {3 a c x^{4}}{4} + \frac {c x^{3}}{3} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((sqrt(-a**2*x**2 + 1)*(-a*c*x**4/5 - 3*c*x**3/4 - 19*c*x**2/(15* 
a) - 13*c*x/(8*a**2) - 38*c/(15*a**3)) + 13*c*log(-2*a**2*x + 2*sqrt(-a**2 
)*sqrt(-a**2*x**2 + 1))/(8*a**2*sqrt(-a**2)), Ne(a**2, 0)), (a**3*c*x**6/6 
 + 3*a**2*c*x**5/5 + 3*a*c*x**4/4 + c*x**3/3, True))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.16 \[ \int e^{3 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right ) \, dx=\frac {a^{3} c x^{6}}{5 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {3 \, a^{2} c x^{5}}{4 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {16 \, a c x^{4}}{15 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {7 \, c x^{3}}{8 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {19 \, c x^{2}}{15 \, \sqrt {-a^{2} x^{2} + 1} a} - \frac {13 \, c x}{8 \, \sqrt {-a^{2} x^{2} + 1} a^{2}} + \frac {13 \, c \arcsin \left (a x\right )}{8 \, a^{3}} - \frac {38 \, c}{15 \, \sqrt {-a^{2} x^{2} + 1} a^{3}} \] Input:

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

Output:

1/5*a^3*c*x^6/sqrt(-a^2*x^2 + 1) + 3/4*a^2*c*x^5/sqrt(-a^2*x^2 + 1) + 16/1 
5*a*c*x^4/sqrt(-a^2*x^2 + 1) + 7/8*c*x^3/sqrt(-a^2*x^2 + 1) + 19/15*c*x^2/ 
(sqrt(-a^2*x^2 + 1)*a) - 13/8*c*x/(sqrt(-a^2*x^2 + 1)*a^2) + 13/8*c*arcsin 
(a*x)/a^3 - 38/15*c/(sqrt(-a^2*x^2 + 1)*a^3)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.54 \[ \int e^{3 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right ) \, dx=-\frac {1}{120} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, {\left (3 \, {\left (4 \, a c x + 15 \, c\right )} x + \frac {76 \, c}{a}\right )} x + \frac {195 \, c}{a^{2}}\right )} x + \frac {304 \, c}{a^{3}}\right )} + \frac {13 \, c \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{8 \, a^{2} {\left | a \right |}} \] Input:

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

Output:

-1/120*sqrt(-a^2*x^2 + 1)*((2*(3*(4*a*c*x + 15*c)*x + 76*c/a)*x + 195*c/a^ 
2)*x + 304*c/a^3) + 13/8*c*arcsin(a*x)*sgn(a)/(a^2*abs(a))
 

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.93 \[ \int e^{3 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right ) \, dx=\frac {13\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,a^2\,\sqrt {-a^2}}-\frac {3\,c\,x^3\,\sqrt {1-a^2\,x^2}}{4}-\frac {19\,c\,x^2\,\sqrt {1-a^2\,x^2}}{15\,a}-\frac {13\,c\,x\,\sqrt {1-a^2\,x^2}}{8\,a^2}-\frac {a\,c\,x^4\,\sqrt {1-a^2\,x^2}}{5}-\frac {38\,c\,\sqrt {1-a^2\,x^2}}{15\,a^3} \] Input:

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

Output:

(13*c*asinh(x*(-a^2)^(1/2)))/(8*a^2*(-a^2)^(1/2)) - (3*c*x^3*(1 - a^2*x^2) 
^(1/2))/4 - (19*c*x^2*(1 - a^2*x^2)^(1/2))/(15*a) - (13*c*x*(1 - a^2*x^2)^ 
(1/2))/(8*a^2) - (a*c*x^4*(1 - a^2*x^2)^(1/2))/5 - (38*c*(1 - a^2*x^2)^(1/ 
2))/(15*a^3)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.77 \[ \int e^{3 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right ) \, dx=\frac {c \left (195 \mathit {asin} \left (a x \right )-24 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-90 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-152 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-195 \sqrt {-a^{2} x^{2}+1}\, a x -304 \sqrt {-a^{2} x^{2}+1}+304\right )}{120 a^{3}} \] Input:

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

Output:

(c*(195*asin(a*x) - 24*sqrt( - a**2*x**2 + 1)*a**4*x**4 - 90*sqrt( - a**2* 
x**2 + 1)*a**3*x**3 - 152*sqrt( - a**2*x**2 + 1)*a**2*x**2 - 195*sqrt( - a 
**2*x**2 + 1)*a*x - 304*sqrt( - a**2*x**2 + 1) + 304))/(120*a**3)