Integrand size = 22, antiderivative size = 331 \[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=-\frac {4 c^2 (1-a x)^{3-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-4+n)}}{a (4-n)}-\frac {c^2 (1-a x)^{3-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-4+n)}}{3 a^4 x^3}-\frac {c^2 (10+n) (1-a x)^{3-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-4+n)}}{6 a^3 x^2}-\frac {c^2 \left (14+5 n+n^2\right ) (1-a x)^{3-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-4+n)}}{6 a^2 x}-\frac {c^2 n \left (10-n^2\right ) (1-a x)^{2-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-4+n)} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-4+n),\frac {1}{2} (-2+n),\frac {1+a x}{1-a x}\right )}{3 a (4-n)}+\frac {2^{-1+\frac {n}{2}} c^2 n (1-a x)^{3-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {4-n}{2},3-\frac {n}{2},4-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a \left (24-10 n+n^2\right )} \]
-4*c^2*(-a*x+1)^(3-1/2*n)*(a*x+1)^(-2+1/2*n)/a/(4-n)-1/3*c^2*(-a*x+1)^(3-1 /2*n)*(a*x+1)^(-2+1/2*n)/a^4/x^3-1/6*c^2*(10+n)*(-a*x+1)^(3-1/2*n)*(a*x+1) ^(-2+1/2*n)/a^3/x^2-1/6*c^2*(n^2+5*n+14)*(-a*x+1)^(3-1/2*n)*(a*x+1)^(-2+1/ 2*n)/a^2/x-1/3*c^2*n*(-n^2+10)*(-a*x+1)^(2-1/2*n)*(a*x+1)^(-2+1/2*n)*hyper geom([1, -2+1/2*n],[-1+1/2*n],(a*x+1)/(-a*x+1))/a/(4-n)+2^(-1+1/2*n)*c^2*n *(-a*x+1)^(3-1/2*n)*hypergeom([2-1/2*n, 3-1/2*n],[4-1/2*n],-1/2*a*x+1/2)/a /(n^2-10*n+24)
Time = 0.64 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.69 \[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=-\frac {c^2 e^{n \text {arctanh}(a x)} \left (4+2 n+2 a n x+a n^2 x-24 a^2 x^2-12 a^2 n x^2+2 a^2 n^2 x^2+a^2 n^3 x^2-2 a^3 n x^3-a^3 n^2 x^3+a^3 e^{2 \text {arctanh}(a x)} n \left (-10+n^2\right ) x^3 \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \text {arctanh}(a x)}\right )+a^3 \left (-20-10 n+2 n^2+n^3\right ) x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \text {arctanh}(a x)}\right )-24 a^3 e^{2 \text {arctanh}(a x)} x^3 \operatorname {Hypergeometric2F1}\left (2,1+\frac {n}{2},2+\frac {n}{2},-e^{2 \text {arctanh}(a x)}\right )\right )}{6 a^4 (2+n) x^3} \]
-1/6*(c^2*E^(n*ArcTanh[a*x])*(4 + 2*n + 2*a*n*x + a*n^2*x - 24*a^2*x^2 - 1 2*a^2*n*x^2 + 2*a^2*n^2*x^2 + a^2*n^3*x^2 - 2*a^3*n*x^3 - a^3*n^2*x^3 + a^ 3*E^(2*ArcTanh[a*x])*n*(-10 + n^2)*x^3*Hypergeometric2F1[1, 1 + n/2, 2 + n /2, E^(2*ArcTanh[a*x])] + a^3*(-20 - 10*n + 2*n^2 + n^3)*x^3*Hypergeometri c2F1[1, n/2, 1 + n/2, E^(2*ArcTanh[a*x])] - 24*a^3*E^(2*ArcTanh[a*x])*x^3* Hypergeometric2F1[2, 1 + n/2, 2 + n/2, -E^(2*ArcTanh[a*x])]))/(a^4*(2 + n) *x^3)
Time = 0.96 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {6707, 6700, 139, 88, 79, 2116, 25, 2116, 25, 27, 168, 27, 141}
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.
| \(\displaystyle \int \left (c-\frac {c}{a^2 x^2}\right )^2 e^{n \text {arctanh}(a x)} \, dx\) |
| \(\Big \downarrow \) 6707 |
| \(\displaystyle \frac {c^2 \int \frac {e^{n \text {arctanh}(a x)} \left (1-a^2 x^2\right )^2}{x^4}dx}{a^4}\) |
| \(\Big \downarrow \) 6700 |
| \(\displaystyle \frac {c^2 \int \frac {(1-a x)^{2-\frac {n}{2}} (a x+1)^{\frac {n+4}{2}}}{x^4}dx}{a^4}\) |
| \(\Big \downarrow \) 139 |
| \(\displaystyle \frac {c^2 \left (a^4 \int (1-a x)^{2-\frac {n}{2}} (a x+1)^{\frac {n-6}{2}} (a x+5)dx+\int \frac {(1-a x)^{2-\frac {n}{2}} (a x+1)^{\frac {n-6}{2}} \left (10 a^3 x^3+10 a^2 x^2+5 a x+1\right )}{x^4}dx\right )}{a^4}\) |
| \(\Big \downarrow \) 88 |
| \(\displaystyle \frac {c^2 \left (a^4 \left (-\frac {n \int (1-a x)^{2-\frac {n}{2}} (a x+1)^{\frac {n-4}{2}}dx}{4-n}-\frac {4 (a x+1)^{\frac {n-4}{2}} (1-a x)^{3-\frac {n}{2}}}{a (4-n)}\right )+\int \frac {(1-a x)^{2-\frac {n}{2}} (a x+1)^{\frac {n-6}{2}} \left (10 a^3 x^3+10 a^2 x^2+5 a x+1\right )}{x^4}dx\right )}{a^4}\) |
| \(\Big \downarrow \) 79 |
| \(\displaystyle \frac {c^2 \left (\int \frac {(1-a x)^{2-\frac {n}{2}} (a x+1)^{\frac {n-6}{2}} \left (10 a^3 x^3+10 a^2 x^2+5 a x+1\right )}{x^4}dx+a^4 \left (\frac {2^{\frac {n}{2}-1} n (1-a x)^{3-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {4-n}{2},3-\frac {n}{2},4-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (4-n) (6-n)}-\frac {4 (1-a x)^{3-\frac {n}{2}} (a x+1)^{\frac {n-4}{2}}}{a (4-n)}\right )\right )}{a^4}\) |
| \(\Big \downarrow \) 2116 |
| \(\displaystyle \frac {c^2 \left (-\frac {1}{3} \int -\frac {(1-a x)^{2-\frac {n}{2}} (a x+1)^{\frac {n-6}{2}} \left (30 x^2 a^3+32 x a^2+(n+10) a\right )}{x^3}dx+a^4 \left (\frac {2^{\frac {n}{2}-1} n (1-a x)^{3-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {4-n}{2},3-\frac {n}{2},4-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (4-n) (6-n)}-\frac {4 (1-a x)^{3-\frac {n}{2}} (a x+1)^{\frac {n-4}{2}}}{a (4-n)}\right )-\frac {(a x+1)^{\frac {n-4}{2}} (1-a x)^{3-\frac {n}{2}}}{3 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {c^2 \left (\frac {1}{3} \int \frac {(1-a x)^{2-\frac {n}{2}} (a x+1)^{\frac {n-6}{2}} \left (30 x^2 a^3+32 x a^2+(n+10) a\right )}{x^3}dx+a^4 \left (\frac {2^{\frac {n}{2}-1} n (1-a x)^{3-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {4-n}{2},3-\frac {n}{2},4-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (4-n) (6-n)}-\frac {4 (1-a x)^{3-\frac {n}{2}} (a x+1)^{\frac {n-4}{2}}}{a (4-n)}\right )-\frac {(a x+1)^{\frac {n-4}{2}} (1-a x)^{3-\frac {n}{2}}}{3 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 2116 |
| \(\displaystyle \frac {c^2 \left (\frac {1}{3} \left (-\frac {1}{2} \int -\frac {a^2 (1-a x)^{2-\frac {n}{2}} (a x+1)^{\frac {n-6}{2}} \left (n^2+5 n+a (n+70) x+14\right )}{x^2}dx-\frac {a (n+10) (a x+1)^{\frac {n-4}{2}} (1-a x)^{3-\frac {n}{2}}}{2 x^2}\right )+a^4 \left (\frac {2^{\frac {n}{2}-1} n (1-a x)^{3-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {4-n}{2},3-\frac {n}{2},4-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (4-n) (6-n)}-\frac {4 (1-a x)^{3-\frac {n}{2}} (a x+1)^{\frac {n-4}{2}}}{a (4-n)}\right )-\frac {(a x+1)^{\frac {n-4}{2}} (1-a x)^{3-\frac {n}{2}}}{3 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {c^2 \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {a^2 (1-a x)^{2-\frac {n}{2}} (a x+1)^{\frac {n-6}{2}} \left (n^2+5 n+a (n+70) x+14\right )}{x^2}dx-\frac {a (n+10) (1-a x)^{3-\frac {n}{2}} (a x+1)^{\frac {n-4}{2}}}{2 x^2}\right )+a^4 \left (\frac {2^{\frac {n}{2}-1} n (1-a x)^{3-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {4-n}{2},3-\frac {n}{2},4-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (4-n) (6-n)}-\frac {4 (1-a x)^{3-\frac {n}{2}} (a x+1)^{\frac {n-4}{2}}}{a (4-n)}\right )-\frac {(a x+1)^{\frac {n-4}{2}} (1-a x)^{3-\frac {n}{2}}}{3 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c^2 \left (\frac {1}{3} \left (\frac {1}{2} a^2 \int \frac {(1-a x)^{2-\frac {n}{2}} (a x+1)^{\frac {n-6}{2}} \left (n^2+5 n+a (n+70) x+14\right )}{x^2}dx-\frac {a (n+10) (1-a x)^{3-\frac {n}{2}} (a x+1)^{\frac {n-4}{2}}}{2 x^2}\right )+a^4 \left (\frac {2^{\frac {n}{2}-1} n (1-a x)^{3-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {4-n}{2},3-\frac {n}{2},4-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (4-n) (6-n)}-\frac {4 (1-a x)^{3-\frac {n}{2}} (a x+1)^{\frac {n-4}{2}}}{a (4-n)}\right )-\frac {(a x+1)^{\frac {n-4}{2}} (1-a x)^{3-\frac {n}{2}}}{3 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {c^2 \left (\frac {1}{3} \left (\frac {1}{2} a^2 \left (-\int \frac {a n \left (10-n^2\right ) (1-a x)^{2-\frac {n}{2}} (a x+1)^{\frac {n-6}{2}}}{x}dx-\frac {\left (n^2+5 n+14\right ) (a x+1)^{\frac {n-4}{2}} (1-a x)^{3-\frac {n}{2}}}{x}\right )-\frac {a (n+10) (1-a x)^{3-\frac {n}{2}} (a x+1)^{\frac {n-4}{2}}}{2 x^2}\right )+a^4 \left (\frac {2^{\frac {n}{2}-1} n (1-a x)^{3-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {4-n}{2},3-\frac {n}{2},4-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (4-n) (6-n)}-\frac {4 (1-a x)^{3-\frac {n}{2}} (a x+1)^{\frac {n-4}{2}}}{a (4-n)}\right )-\frac {(a x+1)^{\frac {n-4}{2}} (1-a x)^{3-\frac {n}{2}}}{3 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c^2 \left (\frac {1}{3} \left (\frac {1}{2} a^2 \left (-a n \left (10-n^2\right ) \int \frac {(1-a x)^{2-\frac {n}{2}} (a x+1)^{\frac {n-6}{2}}}{x}dx-\frac {\left (n^2+5 n+14\right ) (a x+1)^{\frac {n-4}{2}} (1-a x)^{3-\frac {n}{2}}}{x}\right )-\frac {a (n+10) (1-a x)^{3-\frac {n}{2}} (a x+1)^{\frac {n-4}{2}}}{2 x^2}\right )+a^4 \left (\frac {2^{\frac {n}{2}-1} n (1-a x)^{3-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {4-n}{2},3-\frac {n}{2},4-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (4-n) (6-n)}-\frac {4 (1-a x)^{3-\frac {n}{2}} (a x+1)^{\frac {n-4}{2}}}{a (4-n)}\right )-\frac {(a x+1)^{\frac {n-4}{2}} (1-a x)^{3-\frac {n}{2}}}{3 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 141 |
| \(\displaystyle \frac {c^2 \left (a^4 \left (\frac {2^{\frac {n}{2}-1} n (1-a x)^{3-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {4-n}{2},3-\frac {n}{2},4-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (4-n) (6-n)}-\frac {4 (1-a x)^{3-\frac {n}{2}} (a x+1)^{\frac {n-4}{2}}}{a (4-n)}\right )+\frac {1}{3} \left (\frac {1}{2} a^2 \left (-\frac {2 a n \left (10-n^2\right ) (a x+1)^{\frac {n-4}{2}} (1-a x)^{\frac {4-n}{2}} \operatorname {Hypergeometric2F1}\left (1,\frac {n-4}{2},\frac {n-2}{2},\frac {a x+1}{1-a x}\right )}{4-n}-\frac {\left (n^2+5 n+14\right ) (a x+1)^{\frac {n-4}{2}} (1-a x)^{3-\frac {n}{2}}}{x}\right )-\frac {a (n+10) (1-a x)^{3-\frac {n}{2}} (a x+1)^{\frac {n-4}{2}}}{2 x^2}\right )-\frac {(a x+1)^{\frac {n-4}{2}} (1-a x)^{3-\frac {n}{2}}}{3 x^3}\right )}{a^4}\) |
(c^2*(-1/3*((1 - a*x)^(3 - n/2)*(1 + a*x)^((-4 + n)/2))/x^3 + (-1/2*(a*(10 + n)*(1 - a*x)^(3 - n/2)*(1 + a*x)^((-4 + n)/2))/x^2 + (a^2*(-(((14 + 5*n + n^2)*(1 - a*x)^(3 - n/2)*(1 + a*x)^((-4 + n)/2))/x) - (2*a*n*(10 - n^2) *(1 - a*x)^((4 - n)/2)*(1 + a*x)^((-4 + n)/2)*Hypergeometric2F1[1, (-4 + n )/2, (-2 + n)/2, (1 + a*x)/(1 - a*x)])/(4 - n)))/2)/3 + a^4*((-4*(1 - a*x) ^(3 - n/2)*(1 + a*x)^((-4 + n)/2))/(a*(4 - n)) + (2^(-1 + n/2)*n*(1 - a*x) ^(3 - n/2)*Hypergeometric2F1[(4 - n)/2, 3 - n/2, 4 - n/2, (1 - a*x)/2])/(a *(4 - n)*(6 - n)))))/a^4
3.8.91.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && !RationalQ[p] && SumSimpl erQ[p, 1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[f^(p - 1)/d^p Int[(a + b*x)^m*((d*e*p - c*f*(p - 1) + d*f*x)/(c + d*x)^(m + 1)), x], x] + Simp[f^(p - 1) Int[(a + b*x)^m*((e + f*x)^p/(c + d*x)^(m + 1))*ExpandToSum[f^(-p + 1)*(c + d*x)^(-p + 1) - (d*e *p - c*f*(p - 1) + d*f*x)/(d^p*(e + f*x)^p), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m + n + p, 0] && ILtQ[p, 0] && (LtQ[m, 0] || SumS implerQ[m, 1] || !(LtQ[n, 0] || SumSimplerQ[n, 1]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^( n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f ))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] || !Su mSimplerQ[p, 1]) && !ILtQ[m, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_ .)*(x_))^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[b*R*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Si mp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n* (e + f*x)^p*ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f*R*(m + 1 ) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x] , x], x]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && PolyQ[Px, x] && ILtQ[m, -1]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symb ol] :> Simp[d^p Int[(u/x^(2*p))*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x ] /; FreeQ[{a, c, d, n}, x] && EqQ[c + a^2*d, 0] && IntegerQ[p]
\[\int {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} \left (c -\frac {c}{a^{2} x^{2}}\right )^{2}d x\]
\[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\int { {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{2} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
\[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\frac {c^{2} \left (\int a^{4} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx + \int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{x^{4}}\, dx + \int \left (- \frac {2 a^{2} e^{n \operatorname {atanh}{\left (a x \right )}}}{x^{2}}\right )\, dx\right )}{a^{4}} \]
c**2*(Integral(a**4*exp(n*atanh(a*x)), x) + Integral(exp(n*atanh(a*x))/x** 4, x) + Integral(-2*a**2*exp(n*atanh(a*x))/x**2, x))/a**4
\[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\int { {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{2} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
\[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\int { {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{2} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
Timed out. \[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-\frac {c}{a^2\,x^2}\right )}^2 \,d x \]