\(\int e^{\text {arctanh}(a x)} x^2 (c-a c x)^2 \, dx\) [317]

Optimal result

Integrand size = 19, antiderivative size = 112 \[ \int e^{\text {arctanh}(a x)} x^2 (c-a c x)^2 \, dx=-\frac {c^2 x \sqrt {1-a^2 x^2}}{8 a^2}+\frac {1}{4} c^2 x^3 \sqrt {1-a^2 x^2}+\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a^3}-\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{5 a^3}+\frac {c^2 \arcsin (a x)}{8 a^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.67 \[ \int e^{\text {arctanh}(a x)} x^2 (c-a c x)^2 \, dx=-\frac {c^2 \left (\sqrt {1-a^2 x^2} \left (-16+15 a x-8 a^2 x^2-30 a^3 x^3+24 a^4 x^4\right )+30 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{120 a^3} \] Input:

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

Output:

-1/120*(c^2*(Sqrt[1 - a^2*x^2]*(-16 + 15*a*x - 8*a^2*x^2 - 30*a^3*x^3 + 24 
*a^4*x^4) + 30*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/a^3
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {6678, 27, 533, 25, 27, 533, 25, 27, 455, 211, 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.

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

\(\Big \downarrow \) 6678

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 533

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 533

\(\displaystyle c^2 \left (\frac {x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac {\frac {\int -a (5-8 a x) \sqrt {1-a^2 x^2}dx}{4 a^2}+\frac {5 x \left (1-a^2 x^2\right )^{3/2}}{4 a}}{5 a}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle c^2 \left (\frac {x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac {\frac {5 x \left (1-a^2 x^2\right )^{3/2}}{4 a}-\frac {\int a (5-8 a x) \sqrt {1-a^2 x^2}dx}{4 a^2}}{5 a}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle c^2 \left (\frac {x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac {\frac {5 x \left (1-a^2 x^2\right )^{3/2}}{4 a}-\frac {\int (5-8 a x) \sqrt {1-a^2 x^2}dx}{4 a}}{5 a}\right )\)

\(\Big \downarrow \) 455

\(\displaystyle c^2 \left (\frac {x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac {\frac {5 x \left (1-a^2 x^2\right )^{3/2}}{4 a}-\frac {5 \int \sqrt {1-a^2 x^2}dx+\frac {8 \left (1-a^2 x^2\right )^{3/2}}{3 a}}{4 a}}{5 a}\right )\)

\(\Big \downarrow \) 211

\(\displaystyle c^2 \left (\frac {x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac {\frac {5 x \left (1-a^2 x^2\right )^{3/2}}{4 a}-\frac {5 \left (\frac {1}{2} \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {1}{2} x \sqrt {1-a^2 x^2}\right )+\frac {8 \left (1-a^2 x^2\right )^{3/2}}{3 a}}{4 a}}{5 a}\right )\)

\(\Big \downarrow \) 223

\(\displaystyle c^2 \left (\frac {x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}-\frac {\frac {5 x \left (1-a^2 x^2\right )^{3/2}}{4 a}-\frac {5 \left (\frac {1}{2} x \sqrt {1-a^2 x^2}+\frac {\arcsin (a x)}{2 a}\right )+\frac {8 \left (1-a^2 x^2\right )^{3/2}}{3 a}}{4 a}}{5 a}\right )\)

Input:

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

Output:

c^2*((x^2*(1 - a^2*x^2)^(3/2))/(5*a) - ((5*x*(1 - a^2*x^2)^(3/2))/(4*a) - 
((8*(1 - a^2*x^2)^(3/2))/(3*a) + 5*((x*Sqrt[1 - a^2*x^2])/2 + ArcSin[a*x]/ 
(2*a)))/(4*a))/(5*a))
 

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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 6678

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

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.84

method	result	size
risch	\(\frac {\left (24 a^{4} x^{4}-30 a^{3} x^{3}-8 a^{2} x^{2}+15 a x -16\right ) \left (a^{2} x^{2}-1\right ) c^{2}}{120 a^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{2}}{8 a^{2} \sqrt {a^{2}}}\)	\(94\)
meijerg	\(-\frac {c^{2} \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (6 a^{4} x^{4}+8 a^{2} x^{2}+16\right ) \sqrt {-a^{2} x^{2}+1}}{15}\right )}{2 a^{3} \sqrt {\pi }}-\frac {c^{2} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (10 a^{2} x^{2}+15\right ) \sqrt {-a^{2} x^{2}+1}}{20 a^{4}}+\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{4 a^{5}}\right )}{2 a^{2} \sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {c^{2} \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{6}\right )}{2 a^{3} \sqrt {\pi }}-\frac {c^{2} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{2 a^{2} \sqrt {\pi }\, \sqrt {-a^{2}}}\)	\(239\)
default	\(c^{2} \left (-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}+a^{3} \left (-\frac {x^{4} \sqrt {-a^{2} x^{2}+1}}{5 a^{2}}+\frac {-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}}{a^{2}}\right )-a \left (-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}\right )-a^{2} \left (-\frac {x^{3} \sqrt {-a^{2} x^{2}+1}}{4 a^{2}}+\frac {-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )\right )\)	\(243\)

Input:

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

Output:

1/120*(24*a^4*x^4-30*a^3*x^3-8*a^2*x^2+15*a*x-16)*(a^2*x^2-1)/a^3/(-a^2*x^ 
2+1)^(1/2)*c^2+1/8/a^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2) 
)*c^2
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.82 \[ \int e^{\text {arctanh}(a x)} x^2 (c-a c x)^2 \, dx=-\frac {30 \, c^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (24 \, a^{4} c^{2} x^{4} - 30 \, a^{3} c^{2} x^{3} - 8 \, a^{2} c^{2} x^{2} + 15 \, a c^{2} x - 16 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{120 \, a^{3}} \] Input:

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

Output:

-1/120*(30*c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (24*a^4*c^2*x^4 - 
30*a^3*c^2*x^3 - 8*a^2*c^2*x^2 + 15*a*c^2*x - 16*c^2)*sqrt(-a^2*x^2 + 1))/ 
a^3
 

Sympy [A] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.35 \[ \int e^{\text {arctanh}(a x)} x^2 (c-a c x)^2 \, dx=\begin {cases} \sqrt {- a^{2} x^{2} + 1} \left (- \frac {a c^{2} x^{4}}{5} + \frac {c^{2} x^{3}}{4} + \frac {c^{2} x^{2}}{15 a} - \frac {c^{2} x}{8 a^{2}} + \frac {2 c^{2}}{15 a^{3}}\right ) + \frac {c^{2} \log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{8 a^{2} \sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {a^{3} c^{2} x^{6}}{6} - \frac {a^{2} c^{2} x^{5}}{5} - \frac {a c^{2} x^{4}}{4} + \frac {c^{2} x^{3}}{3} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((sqrt(-a**2*x**2 + 1)*(-a*c**2*x**4/5 + c**2*x**3/4 + c**2*x**2/ 
(15*a) - c**2*x/(8*a**2) + 2*c**2/(15*a**3)) + c**2*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* 
*2*x**6/6 - a**2*c**2*x**5/5 - a*c**2*x**4/4 + c**2*x**3/3, True))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.05 \[ \int e^{\text {arctanh}(a x)} x^2 (c-a c x)^2 \, dx=-\frac {1}{5} \, \sqrt {-a^{2} x^{2} + 1} a c^{2} x^{4} + \frac {1}{4} \, \sqrt {-a^{2} x^{2} + 1} c^{2} x^{3} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{2} x^{2}}{15 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2} x}{8 \, a^{2}} + \frac {c^{2} \arcsin \left (a x\right )}{8 \, a^{3}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} c^{2}}{15 \, a^{3}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.72 \[ \int e^{\text {arctanh}(a x)} x^2 (c-a c x)^2 \, dx=-\frac {1}{120} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, {\left (3 \, {\left (4 \, a c^{2} x - 5 \, c^{2}\right )} x - \frac {4 \, c^{2}}{a}\right )} x + \frac {15 \, c^{2}}{a^{2}}\right )} x - \frac {16 \, c^{2}}{a^{3}}\right )} + \frac {c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{8 \, a^{2} {\left | a \right |}} \] Input:

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

Output:

-1/120*sqrt(-a^2*x^2 + 1)*((2*(3*(4*a*c^2*x - 5*c^2)*x - 4*c^2/a)*x + 15*c 
^2/a^2)*x - 16*c^2/a^3) + 1/8*c^2*arcsin(a*x)*sgn(a)/(a^2*abs(a))
 

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.17 \[ \int e^{\text {arctanh}(a x)} x^2 (c-a c x)^2 \, dx=\frac {2\,c^2\,\sqrt {1-a^2\,x^2}}{15\,a^3}+\frac {c^2\,x^3\,\sqrt {1-a^2\,x^2}}{4}-\frac {c^2\,x\,\sqrt {1-a^2\,x^2}}{8\,a^2}-\frac {a\,c^2\,x^4\,\sqrt {1-a^2\,x^2}}{5}+\frac {c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,a^2\,\sqrt {-a^2}}+\frac {c^2\,x^2\,\sqrt {1-a^2\,x^2}}{15\,a} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.90 \[ \int e^{\text {arctanh}(a x)} x^2 (c-a c x)^2 \, dx=\frac {c^{2} \left (15 \mathit {asin} \left (a x \right )-24 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+30 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+8 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-15 \sqrt {-a^{2} x^{2}+1}\, a x +16 \sqrt {-a^{2} x^{2}+1}-16\right )}{120 a^{3}} \] Input:

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

Output:

(c**2*(15*asin(a*x) - 24*sqrt( - a**2*x**2 + 1)*a**4*x**4 + 30*sqrt( - a** 
2*x**2 + 1)*a**3*x**3 + 8*sqrt( - a**2*x**2 + 1)*a**2*x**2 - 15*sqrt( - a* 
*2*x**2 + 1)*a*x + 16*sqrt( - a**2*x**2 + 1) - 16))/(120*a**3)