\(\int \frac {e^{\text {arctanh}(a x)}}{x^2 (c-a^2 c x^2)^3} \, dx\) [953]

Optimal result

Integrand size = 23, antiderivative size = 131 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^3} \, dx=\frac {a (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {9+5 a x}{15 c^3 x \left (1-a^2 x^2\right )^{3/2}}+\frac {a (5+8 a x)}{5 c^3 \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{5 c^3 x}-\frac {a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{c^3} \] Output:

1/5*a*(a*x+1)/c^3/(-a^2*x^2+1)^(5/2)+1/15*(5*a*x+9)/c^3/x/(-a^2*x^2+1)^(3/ 
2)+1/5*a*(8*a*x+5)/c^3/(-a^2*x^2+1)^(1/2)-8/5*(-a^2*x^2+1)^(1/2)/c^3/x-a*a 
rctanh((-a^2*x^2+1)^(1/2))/c^3
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^3} \, dx=\frac {-15+38 a x+52 a^2 x^2-87 a^3 x^3-33 a^4 x^4+48 a^5 x^5-15 a x (-1+a x)^2 (1+a x) \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{15 c^3 x (-1+a x)^2 (1+a x) \sqrt {1-a^2 x^2}} \] Input:

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

Output:

(-15 + 38*a*x + 52*a^2*x^2 - 87*a^3*x^3 - 33*a^4*x^4 + 48*a^5*x^5 - 15*a*x 
*(-1 + a*x)^2*(1 + a*x)*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]])/(15* 
c^3*x*(-1 + a*x)^2*(1 + a*x)*Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.92, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {6698, 532, 25, 2336, 27, 2336, 27, 534, 243, 73, 221}

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.

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

\(\Big \downarrow \) 6698

\(\displaystyle \frac {\int \frac {a x+1}{x^2 \left (1-a^2 x^2\right )^{7/2}}dx}{c^3}\)

\(\Big \downarrow \) 532

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2336

\(\displaystyle \frac {\frac {1}{5} \left (\frac {a (9 a x+5)}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{3} \int -\frac {3 \left (6 a^2 x^2+5 a x+5\right )}{x^2 \left (1-a^2 x^2\right )^{3/2}}dx\right )+\frac {a (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{5} \left (\int \frac {6 a^2 x^2+5 a x+5}{x^2 \left (1-a^2 x^2\right )^{3/2}}dx+\frac {a (9 a x+5)}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 2336

\(\displaystyle \frac {\frac {1}{5} \left (-\int -\frac {5 (a x+1)}{x^2 \sqrt {1-a^2 x^2}}dx+\frac {a (9 a x+5)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a (11 a x+5)}{\sqrt {1-a^2 x^2}}\right )+\frac {a (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{5} \left (5 \int \frac {a x+1}{x^2 \sqrt {1-a^2 x^2}}dx+\frac {a (9 a x+5)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a (11 a x+5)}{\sqrt {1-a^2 x^2}}\right )+\frac {a (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 534

\(\displaystyle \frac {\frac {1}{5} \left (5 \left (a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{x}\right )+\frac {a (9 a x+5)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a (11 a x+5)}{\sqrt {1-a^2 x^2}}\right )+\frac {a (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\frac {1}{5} \left (5 \left (\frac {1}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2}}{x}\right )+\frac {a (9 a x+5)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a (11 a x+5)}{\sqrt {1-a^2 x^2}}\right )+\frac {a (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {1}{5} \left (5 \left (-\frac {\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {\sqrt {1-a^2 x^2}}{x}\right )+\frac {a (9 a x+5)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a (11 a x+5)}{\sqrt {1-a^2 x^2}}\right )+\frac {a (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {1}{5} \left (5 \left (-a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{x}\right )+\frac {a (9 a x+5)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a (11 a x+5)}{\sqrt {1-a^2 x^2}}\right )+\frac {a (a x+1)}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

Input:

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

Output:

((a*(1 + a*x))/(5*(1 - a^2*x^2)^(5/2)) + ((a*(5 + 9*a*x))/(3*(1 - a^2*x^2) 
^(3/2)) + (a*(5 + 11*a*x))/Sqrt[1 - a^2*x^2] + 5*(-(Sqrt[1 - a^2*x^2]/x) - 
 a*ArcTanh[Sqrt[1 - a^2*x^2]]))/5)/c^3
 

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 532

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe 
ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol 
ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) 
*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[x^m 
*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), 
x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, 
 -1] && IntegerQ[2*p]
 

rule 534

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

rule 2336

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema 
inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) 
^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a* 
b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex 
pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F 
reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(313\) vs. \(2(113)=226\).

Time = 0.27 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.40

method	result	size
default	\(-\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{x}+a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{4 a}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{4 a \left (x -\frac {1}{a}\right )^{2}}+\frac {27 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{16 \left (x -\frac {1}{a}\right )}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{24 a \left (x +\frac {1}{a}\right )^{2}}+\frac {23 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{48 \left (x +\frac {1}{a}\right )}}{c^{3}}\)	\(314\)
risch	\(\frac {a^{2} x^{2}-1}{x \sqrt {-a^{2} x^{2}+1}\, c^{3}}-\frac {a \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{4 a^{2}}-\frac {3 \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{4 a}+\frac {23 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{16 a \left (x -\frac {1}{a}\right )}-\frac {-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 a \left (x +\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 \left (x +\frac {1}{a}\right )}}{8 a}+\frac {7 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{16 a \left (x +\frac {1}{a}\right )}\right )}{c^{3}}\)	\(414\)

Input:

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

Output:

-1/c^3*((-a^2*x^2+1)^(1/2)/x+a*arctanh(1/(-a^2*x^2+1)^(1/2))+1/4/a*(1/5/a/ 
(x-1/a)^3*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)-2/5*a*(1/3/a/(x-1/a)^2*(-(x-1 
/a)^2*a^2-2*a*(x-1/a))^(1/2)-1/3/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2 
)))-1/4/a/(x-1/a)^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)+27/16/(x-1/a)*(-(x- 
1/a)^2*a^2-2*a*(x-1/a))^(1/2)+1/24/a/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a) 
)^(1/2)+23/48/(x+1/a)*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.70 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^3} \, dx=\frac {23 \, a^{6} x^{6} - 23 \, a^{5} x^{5} - 46 \, a^{4} x^{4} + 46 \, a^{3} x^{3} + 23 \, a^{2} x^{2} - 23 \, a x + 15 \, {\left (a^{6} x^{6} - a^{5} x^{5} - 2 \, a^{4} x^{4} + 2 \, a^{3} x^{3} + a^{2} x^{2} - a x\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (48 \, a^{5} x^{5} - 33 \, a^{4} x^{4} - 87 \, a^{3} x^{3} + 52 \, a^{2} x^{2} + 38 \, a x - 15\right )} \sqrt {-a^{2} x^{2} + 1}}{15 \, {\left (a^{5} c^{3} x^{6} - a^{4} c^{3} x^{5} - 2 \, a^{3} c^{3} x^{4} + 2 \, a^{2} c^{3} x^{3} + a c^{3} x^{2} - c^{3} x\right )}} \] Input:

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

Output:

1/15*(23*a^6*x^6 - 23*a^5*x^5 - 46*a^4*x^4 + 46*a^3*x^3 + 23*a^2*x^2 - 23* 
a*x + 15*(a^6*x^6 - a^5*x^5 - 2*a^4*x^4 + 2*a^3*x^3 + a^2*x^2 - a*x)*log(( 
sqrt(-a^2*x^2 + 1) - 1)/x) - (48*a^5*x^5 - 33*a^4*x^4 - 87*a^3*x^3 + 52*a^ 
2*x^2 + 38*a*x - 15)*sqrt(-a^2*x^2 + 1))/(a^5*c^3*x^6 - a^4*c^3*x^5 - 2*a^ 
3*c^3*x^4 + 2*a^2*c^3*x^3 + a*c^3*x^2 - c^3*x)
 

Sympy [F]

\[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^3} \, dx=\frac {\int \frac {a}{- a^{6} x^{7} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{5} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{3} \sqrt {- a^{2} x^{2} + 1} + x \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{6} x^{8} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{6} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} + x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \] Input:

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

Output:

(Integral(a/(-a**6*x**7*sqrt(-a**2*x**2 + 1) + 3*a**4*x**5*sqrt(-a**2*x**2 
 + 1) - 3*a**2*x**3*sqrt(-a**2*x**2 + 1) + x*sqrt(-a**2*x**2 + 1)), x) + I 
ntegral(1/(-a**6*x**8*sqrt(-a**2*x**2 + 1) + 3*a**4*x**6*sqrt(-a**2*x**2 + 
 1) - 3*a**2*x**4*sqrt(-a**2*x**2 + 1) + x**2*sqrt(-a**2*x**2 + 1)), x))/c 
**3
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.36 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^3} \, dx=-\frac {\frac {15 \, a^{2} \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right )}{c^{3}} - \frac {15 \, a^{2} \log \left (\sqrt {-a^{2} x^{2} + 1} - 1\right )}{c^{3}} - \frac {2 \, {\left (15 \, {\left (a^{2} x^{2} - 1\right )}^{2} a^{2} - 5 \, {\left (a^{2} x^{2} - 1\right )} a^{2} + 3 \, a^{2}\right )}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} c^{3}}}{30 \, a} + \frac {16 \, a^{6} x^{6} - 40 \, a^{4} x^{4} + 30 \, a^{2} x^{2} - 5}{5 \, {\left (a^{4} c^{3} x^{5} - 2 \, a^{2} c^{3} x^{3} + c^{3} x\right )} \sqrt {a x + 1} \sqrt {-a x + 1}} \] Input:

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

Output:

-1/30*(15*a^2*log(sqrt(-a^2*x^2 + 1) + 1)/c^3 - 15*a^2*log(sqrt(-a^2*x^2 + 
 1) - 1)/c^3 - 2*(15*(a^2*x^2 - 1)^2*a^2 - 5*(a^2*x^2 - 1)*a^2 + 3*a^2)/(( 
-a^2*x^2 + 1)^(5/2)*c^3))/a + 1/5*(16*a^6*x^6 - 40*a^4*x^4 + 30*a^2*x^2 - 
5)/((a^4*c^3*x^5 - 2*a^2*c^3*x^3 + c^3*x)*sqrt(a*x + 1)*sqrt(-a*x + 1))
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 23.56 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.56 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^3} \, dx=\frac {17\,a^3\,\sqrt {1-a^2\,x^2}}{60\,\left (a^4\,c^3\,x^2-2\,a^3\,c^3\,x+a^2\,c^3\right )}-\frac {a^3\,\sqrt {1-a^2\,x^2}}{24\,\left (a^4\,c^3\,x^2+2\,a^3\,c^3\,x+a^2\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{c^3\,x}+\frac {23\,a^2\,\sqrt {1-a^2\,x^2}}{48\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}+\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {413\,a^2\,\sqrt {1-a^2\,x^2}}{240\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {a^2\,\sqrt {1-a^2\,x^2}}{20\,\sqrt {-a^2}\,\left (3\,c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}+a^2\,c^3\,x^3\,\sqrt {-a^2}-3\,a\,c^3\,x^2\,\sqrt {-a^2}\right )}+\frac {a\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{c^3} \] Input:

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

Output:

(17*a^3*(1 - a^2*x^2)^(1/2))/(60*(a^2*c^3 - 2*a^3*c^3*x + a^4*c^3*x^2)) - 
(a^3*(1 - a^2*x^2)^(1/2))/(24*(a^2*c^3 + 2*a^3*c^3*x + a^4*c^3*x^2)) - (1 
- a^2*x^2)^(1/2)/(c^3*x) + (a*atan((1 - a^2*x^2)^(1/2)*1i)*1i)/c^3 + (23*a 
^2*(1 - a^2*x^2)^(1/2))/(48*(-a^2)^(1/2)*(c^3*x*(-a^2)^(1/2) + (c^3*(-a^2) 
^(1/2))/a)) + (413*a^2*(1 - a^2*x^2)^(1/2))/(240*(-a^2)^(1/2)*(c^3*x*(-a^2 
)^(1/2) - (c^3*(-a^2)^(1/2))/a)) + (a^2*(1 - a^2*x^2)^(1/2))/(20*(-a^2)^(1 
/2)*(3*c^3*x*(-a^2)^(1/2) - (c^3*(-a^2)^(1/2))/a + a^2*c^3*x^3*(-a^2)^(1/2 
) - 3*a*c^3*x^2*(-a^2)^(1/2)))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.97 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^3} \, dx=\frac {60 \sqrt {-a^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{4} x^{4}-60 \sqrt {-a^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{3} x^{3}-60 \sqrt {-a^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{2} x^{2}+60 \sqrt {-a^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a x -133 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+133 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+133 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-133 \sqrt {-a^{2} x^{2}+1}\, a x +192 a^{5} x^{5}-132 a^{4} x^{4}-348 a^{3} x^{3}+208 a^{2} x^{2}+152 a x -60}{60 \sqrt {-a^{2} x^{2}+1}\, c^{3} x \left (a^{3} x^{3}-a^{2} x^{2}-a x +1\right )} \] Input:

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

Output:

(60*sqrt( - a**2*x**2 + 1)*log(tan(asin(a*x)/2))*a**4*x**4 - 60*sqrt( - a* 
*2*x**2 + 1)*log(tan(asin(a*x)/2))*a**3*x**3 - 60*sqrt( - a**2*x**2 + 1)*l 
og(tan(asin(a*x)/2))*a**2*x**2 + 60*sqrt( - a**2*x**2 + 1)*log(tan(asin(a* 
x)/2))*a*x - 133*sqrt( - a**2*x**2 + 1)*a**4*x**4 + 133*sqrt( - a**2*x**2 
+ 1)*a**3*x**3 + 133*sqrt( - a**2*x**2 + 1)*a**2*x**2 - 133*sqrt( - a**2*x 
**2 + 1)*a*x + 192*a**5*x**5 - 132*a**4*x**4 - 348*a**3*x**3 + 208*a**2*x* 
*2 + 152*a*x - 60)/(60*sqrt( - a**2*x**2 + 1)*c**3*x*(a**3*x**3 - a**2*x** 
2 - a*x + 1))