\(\int \frac {e^{2 \text {arctanh}(a x)} x^2}{\sqrt {c-a^2 c x^2}} \, dx\) [1146]

Optimal result

Integrand size = 27, antiderivative size = 111 \[ \int \frac {e^{2 \text {arctanh}(a x)} x^2}{\sqrt {c-a^2 c x^2}} \, dx=\frac {2 (1+a x)}{a^3 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {c-a^2 c x^2}}{a^3 c}+\frac {x \sqrt {c-a^2 c x^2}}{2 a^2 c}-\frac {5 \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{2 a^3 \sqrt {c}} \] Output:

2*(a*x+1)/a^3/(-a^2*c*x^2+c)^(1/2)+2*(-a^2*c*x^2+c)^(1/2)/a^3/c+1/2*x*(-a^ 
2*c*x^2+c)^(1/2)/a^2/c-5/2*arctan(a*c^(1/2)*x/(-a^2*c*x^2+c)^(1/2))/a^3/c^ 
(1/2)
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.85 \[ \int \frac {e^{2 \text {arctanh}(a x)} x^2}{\sqrt {c-a^2 c x^2}} \, dx=\frac {\left (-8+3 a x+a^2 x^2\right ) \sqrt {c-a^2 c x^2}+5 \sqrt {c} (-1+a x) \arctan \left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )}{2 a^3 c (-1+a x)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6701, 527, 2346, 25, 27, 455, 224, 216}

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.

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

\(\Big \downarrow \) 6701

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

\(\Big \downarrow \) 527

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

\(\Big \downarrow \) 2346

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 455

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 216

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

Input:

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

Output:

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

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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

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 527

Int[((x_)^(m_.)*((c_) + (d_.)*(x_))^(n_.))/((a_) + (b_.)*(x_)^2)^(3/2), x_S 
ymbol] :> Simp[(-2^(n - 1))*c^(m + n - 2)*((c + d*x)/(b*d^(m - 1)*Sqrt[a + 
b*x^2])), x] + Simp[1/(b*d^(m - 2))   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] && IGtQ[n, 0] && IGtQ[m, 0] && EqQ[b*c^2 + a*d^2, 
 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 6701

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Simp[c^(n/2)   Int[x^m*(c + d*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/2, 0]
 

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.05

method	result	size
risch	\(-\frac {\left (a x +4\right ) \left (a^{2} x^{2}-1\right )}{2 a^{3} \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\frac {5 \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 a^{2} \sqrt {a^{2} c}}-\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}{a^{4} c \left (x -\frac {1}{a}\right )}\)	\(116\)
default	\(\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2 a^{2} c}-\frac {5 \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 a^{2} \sqrt {a^{2} c}}+\frac {2 \sqrt {-a^{2} c \,x^{2}+c}}{a^{3} c}-\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}{a^{4} c \left (x -\frac {1}{a}\right )}\)	\(126\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.66 \[ \int \frac {e^{2 \text {arctanh}(a x)} x^2}{\sqrt {c-a^2 c x^2}} \, dx=\left [-\frac {5 \, {\left (a x - 1\right )} \sqrt {-c} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - 2 \, \sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} + 3 \, a x - 8\right )}}{4 \, {\left (a^{4} c x - a^{3} c\right )}}, \frac {5 \, {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} + 3 \, a x - 8\right )}}{2 \, {\left (a^{4} c x - a^{3} c\right )}}\right ] \] Input:

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

Output:

[-1/4*(5*(a*x - 1)*sqrt(-c)*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqr 
t(-c)*x - c) - 2*sqrt(-a^2*c*x^2 + c)*(a^2*x^2 + 3*a*x - 8))/(a^4*c*x - a^ 
3*c), 1/2*(5*(a*x - 1)*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^ 
2*c*x^2 - c)) + sqrt(-a^2*c*x^2 + c)*(a^2*x^2 + 3*a*x - 8))/(a^4*c*x - a^3 
*c)]
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.80 \[ \int \frac {e^{2 \text {arctanh}(a x)} x^2}{\sqrt {c-a^2 c x^2}} \, dx=-\frac {1}{2} \, a {\left (\frac {4 \, \sqrt {-a^{2} c x^{2} + c}}{a^{5} c x - a^{4} c} - \frac {\sqrt {-a^{2} c x^{2} + c} x}{a^{3} c} + \frac {5 \, \arcsin \left (a x\right )}{a^{4} \sqrt {c}} - \frac {4 \, \sqrt {-a^{2} c x^{2} + c}}{a^{4} c}\right )} \] Input:

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{2 \text {arctanh}(a x)} x^2}{\sqrt {c-a^2 c x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.14 \[ \int \frac {e^{2 \text {arctanh}(a x)} x^2}{\sqrt {c-a^2 c x^2}} \, dx=\frac {\sqrt {c}\, \left (-5 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )-5 \mathit {asin} \left (a x \right ) a x +5 \mathit {asin} \left (a x \right )+\sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+3 \sqrt {-a^{2} x^{2}+1}\, a x -10 \sqrt {-a^{2} x^{2}+1}-a^{3} x^{3}-4 a^{2} x^{2}+3 a x +10\right )}{2 a^{3} c \left (\sqrt {-a^{2} x^{2}+1}+a x -1\right )} \] Input:

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

Output:

(sqrt(c)*( - 5*sqrt( - a**2*x**2 + 1)*asin(a*x) - 5*asin(a*x)*a*x + 5*asin 
(a*x) + sqrt( - a**2*x**2 + 1)*a**2*x**2 + 3*sqrt( - a**2*x**2 + 1)*a*x - 
10*sqrt( - a**2*x**2 + 1) - a**3*x**3 - 4*a**2*x**2 + 3*a*x + 10))/(2*a**3 
*c*(sqrt( - a**2*x**2 + 1) + a*x - 1))