\(\int \frac {e^{2 \text {arctanh}(a x)} x^3}{(c-a^2 c x^2)^{3/2}} \, dx\) [1153]

Optimal result

Integrand size = 27, antiderivative size = 117 \[ \int \frac {e^{2 \text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {(1+a x)^2}{3 a^4 \left (c-a^2 c x^2\right )^{3/2}}-\frac {8 (1+a x)}{3 a^4 c \sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{a^4 c^2}+\frac {2 \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a^4 c^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.77 \[ \int \frac {e^{2 \text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {\frac {\left (-10+14 a x-3 a^2 x^2\right ) \sqrt {c-a^2 c x^2}}{(-1+a x)^2}-6 \sqrt {c} \arctan \left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )}{3 a^4 c^2} \] Input:

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

Output:

(((-10 + 14*a*x - 3*a^2*x^2)*Sqrt[c - a^2*c*x^2])/(-1 + a*x)^2 - 6*Sqrt[c] 
*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))])/(3*a^4*c^2)
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.17, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6701, 529, 2166, 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.

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

\(\Big \downarrow \) 6701

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

\(\Big \downarrow \) 529

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

\(\Big \downarrow \) 2166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 455

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 216

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

Input:

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

Output:

c*((1 + a*x)^2/(3*a^4*c*(c - a^2*c*x^2)^(3/2)) - ((8*(1 + a*x))/(a^4*c*Sqr 
t[c - a^2*c*x^2]) - (3*(-(Sqrt[c - a^2*c*x^2]/(a*c)) + (2*ArcTan[(a*Sqrt[c 
]*x)/Sqrt[c - a^2*c*x^2]])/(a*Sqrt[c])))/(a^3*c))/(3*c))
 

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 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 529

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

rule 2166

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

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 = 0.19 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.40

method	result	size
risch	\(\frac {a^{2} x^{2}-1}{a^{4} \sqrt {-c \left (a^{2} x^{2}-1\right )}\, c}+\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}{3 a^{6} c \left (x -\frac {1}{a}\right )^{2}}+\frac {8 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}{3 a^{5} c \left (x -\frac {1}{a}\right )}+\frac {2 \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{a^{3} \sqrt {a^{2} c}}}{c}\)	\(164\)
default	\(\frac {x^{2}}{a^{2} c \sqrt {-a^{2} c \,x^{2}+c}}-\frac {4}{c \,a^{4} \sqrt {-a^{2} c \,x^{2}+c}}-\frac {2 x}{a^{3} c \sqrt {-a^{2} c \,x^{2}+c}}-\frac {2 \left (\frac {x}{a^{2} c \sqrt {-a^{2} c \,x^{2}+c}}-\frac {\arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{a^{2} c \sqrt {a^{2} c}}\right )}{a}-\frac {2 \left (\frac {1}{3 a c \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}+\frac {-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c}{3 a \,c^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}\right )}{a^{4}}\)	\(240\)

Input:

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

Output:

1/a^4*(a^2*x^2-1)/(-c*(a^2*x^2-1))^(1/2)/c+(1/3/a^6/c/(x-1/a)^2*(-(x-1/a)^ 
2*a^2*c-2*(x-1/a)*a*c)^(1/2)+8/3/a^5/c/(x-1/a)*(-(x-1/a)^2*a^2*c-2*(x-1/a) 
*a*c)^(1/2)+2/a^3/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2 
)))/c
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.96 \[ \int \frac {e^{2 \text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + \sqrt {-a^{2} c x^{2} + c} {\left (3 \, a^{2} x^{2} - 14 \, a x + 10\right )}}{3 \, {\left (a^{6} c^{2} x^{2} - 2 \, a^{5} c^{2} x + a^{4} c^{2}\right )}}, -\frac {6 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} c x^{2} + c} {\left (3 \, a^{2} x^{2} - 14 \, a x + 10\right )}}{3 \, {\left (a^{6} c^{2} x^{2} - 2 \, a^{5} c^{2} x + a^{4} c^{2}\right )}}\right ] \] Input:

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

Output:

[-1/3*(3*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^ 
2 + c)*a*sqrt(-c)*x - c) + sqrt(-a^2*c*x^2 + c)*(3*a^2*x^2 - 14*a*x + 10)) 
/(a^6*c^2*x^2 - 2*a^5*c^2*x + a^4*c^2), -1/3*(6*(a^2*x^2 - 2*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)*(3*a^2*x^2 - 14*a*x + 10))/(a^6*c^2*x^2 - 2*a^5*c^2*x + a^4*c^2) 
]
 

Sympy [F]

\[ \int \frac {e^{2 \text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=- \int \frac {x^{3}}{- a^{3} c x^{3} \sqrt {- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt {- a^{2} c x^{2} + c} + a c x \sqrt {- a^{2} c x^{2} + c} - c \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \frac {a x^{4}}{- a^{3} c x^{3} \sqrt {- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt {- a^{2} c x^{2} + c} + a c x \sqrt {- a^{2} c x^{2} + c} - c \sqrt {- a^{2} c x^{2} + c}}\, dx \] Input:

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

Output:

-Integral(x**3/(-a**3*c*x**3*sqrt(-a**2*c*x**2 + c) + a**2*c*x**2*sqrt(-a* 
*2*c*x**2 + c) + a*c*x*sqrt(-a**2*c*x**2 + c) - c*sqrt(-a**2*c*x**2 + c)), 
 x) - Integral(a*x**4/(-a**3*c*x**3*sqrt(-a**2*c*x**2 + c) + a**2*c*x**2*s 
qrt(-a**2*c*x**2 + c) + a*c*x*sqrt(-a**2*c*x**2 + c) - c*sqrt(-a**2*c*x**2 
 + c)), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 260 vs. \(2 (101) = 202\).

Time = 0.21 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.22 \[ \int \frac {e^{2 \text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {1}{3} \, {\left (\frac {a^{3}}{\sqrt {-a^{2} c x^{2} + c} a^{9} c x + \sqrt {-a^{2} c x^{2} + c} a^{8} c} - \frac {a^{3}}{\sqrt {-a^{2} c x^{2} + c} a^{9} c x - \sqrt {-a^{2} c x^{2} + c} a^{8} c} - \frac {a}{\sqrt {-a^{2} c x^{2} + c} a^{7} c x + \sqrt {-a^{2} c x^{2} + c} a^{6} c} - \frac {a}{\sqrt {-a^{2} c x^{2} + c} a^{7} c x - \sqrt {-a^{2} c x^{2} + c} a^{6} c} + \frac {3 \, x^{2}}{\sqrt {-a^{2} c x^{2} + c} a^{3} c} - \frac {8 \, x}{\sqrt {-a^{2} c x^{2} + c} a^{4} c} + \frac {6 \, \arcsin \left (a x\right )}{a^{5} c^{\frac {3}{2}}} - \frac {12}{\sqrt {-a^{2} c x^{2} + c} a^{5} c}\right )} a \] Input:

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

Output:

1/3*(a^3/(sqrt(-a^2*c*x^2 + c)*a^9*c*x + sqrt(-a^2*c*x^2 + c)*a^8*c) - a^3 
/(sqrt(-a^2*c*x^2 + c)*a^9*c*x - sqrt(-a^2*c*x^2 + c)*a^8*c) - a/(sqrt(-a^ 
2*c*x^2 + c)*a^7*c*x + sqrt(-a^2*c*x^2 + c)*a^6*c) - a/(sqrt(-a^2*c*x^2 + 
c)*a^7*c*x - sqrt(-a^2*c*x^2 + c)*a^6*c) + 3*x^2/(sqrt(-a^2*c*x^2 + c)*a^3 
*c) - 8*x/(sqrt(-a^2*c*x^2 + c)*a^4*c) + 6*arcsin(a*x)/(a^5*c^(3/2)) - 12/ 
(sqrt(-a^2*c*x^2 + c)*a^5*c))*a
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{2 \text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.56 \[ \int \frac {e^{2 \text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \left (6 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a x -6 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )+6 \mathit {asin} \left (a x \right ) a^{2} x^{2}-12 \mathit {asin} \left (a x \right ) a x +6 \mathit {asin} \left (a x \right )-3 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+8 \sqrt {-a^{2} x^{2}+1}\, a x -4 \sqrt {-a^{2} x^{2}+1}+3 a^{3} x^{3}-17 a^{2} x^{2}+8 a x +4\right )}{3 a^{4} c^{2} \left (\sqrt {-a^{2} x^{2}+1}\, a x -\sqrt {-a^{2} x^{2}+1}+a^{2} x^{2}-2 a x +1\right )} \] Input:

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

Output:

(sqrt(c)*(6*sqrt( - a**2*x**2 + 1)*asin(a*x)*a*x - 6*sqrt( - a**2*x**2 + 1 
)*asin(a*x) + 6*asin(a*x)*a**2*x**2 - 12*asin(a*x)*a*x + 6*asin(a*x) - 3*s 
qrt( - a**2*x**2 + 1)*a**2*x**2 + 8*sqrt( - a**2*x**2 + 1)*a*x - 4*sqrt( - 
 a**2*x**2 + 1) + 3*a**3*x**3 - 17*a**2*x**2 + 8*a*x + 4))/(3*a**4*c**2*(s 
qrt( - a**2*x**2 + 1)*a*x - sqrt( - a**2*x**2 + 1) + a**2*x**2 - 2*a*x + 1 
))