Integrand size = 25, antiderivative size = 376 \[ \int \frac {e^{n \text {arctanh}(a x)} x^4}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {(1-n) (3+n) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2 (2-n)}+\frac {(3+n) x (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^4 c^2}-\frac {x^3 (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^2 c^2}+\frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2 (2-n)}-\frac {(1-a x)^{-n/2} (1+a x)^{\frac {1}{2} (-2+n)}}{a^5 c^2}-\frac {(3+n) \left (2-n^2\right ) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{n/2}}{a^5 c^2 \left (4-n^2\right )}-\frac {(3+n) \left (2-n^2\right ) (1-a x)^{-n/2} (1+a x)^{n/2}}{a^5 c^2 n \left (4-n^2\right )}-\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a^5 c^2 (2-n)} \]
(1-n)*(3+n)*(-a*x+1)^(-1-1/2*n)*(a*x+1)^(-1+1/2*n)/a^5/c^2/(2-n)+(3+n)*x*( -a*x+1)^(-1-1/2*n)*(a*x+1)^(-1+1/2*n)/a^4/c^2-x^3*(-a*x+1)^(-1-1/2*n)*(a*x +1)^(-1+1/2*n)/a^2/c^2+(-a*x+1)^(1-1/2*n)*(a*x+1)^(-1+1/2*n)/a^5/c^2/(2-n) -(a*x+1)^(-1+1/2*n)/a^5/c^2/((-a*x+1)^(1/2*n))-(3+n)*(-n^2+2)*(-a*x+1)^(-1 -1/2*n)*(a*x+1)^(1/2*n)/a^5/c^2/(-n^2+4)-(3+n)*(-n^2+2)*(a*x+1)^(1/2*n)/a^ 5/c^2/n/(-n^2+4)/((-a*x+1)^(1/2*n))-2^(1/2*n)*n*(-a*x+1)^(1-1/2*n)*hyperge om([1-1/2*n, 1-1/2*n],[2-1/2*n],-1/2*a*x+1/2)/a^5/c^2/(2-n)
Time = 0.12 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.47 \[ \int \frac {e^{n \text {arctanh}(a x)} x^4}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {(1-a x)^{-1-\frac {n}{2}} \left ((1+a x)^{n/2} \left (-6+6 a^2 x^2+n^3 (-1+a x)^2 (1+a x)+n^2 \left (1-2 a^2 x^2\right )+n \left (-4+6 a x+4 a^2 x^2-4 a^3 x^3\right )\right )-2^{n/2} n^2 (2+n) (-1+a x)^2 (1+a x) \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )\right )}{a^5 c^2 (-2+n) n (2+n) (1+a x)} \]
-(((1 - a*x)^(-1 - n/2)*((1 + a*x)^(n/2)*(-6 + 6*a^2*x^2 + n^3*(-1 + a*x)^ 2*(1 + a*x) + n^2*(1 - 2*a^2*x^2) + n*(-4 + 6*a*x + 4*a^2*x^2 - 4*a^3*x^3) ) - 2^(n/2)*n^2*(2 + n)*(-1 + a*x)^2*(1 + a*x)*Hypergeometric2F1[1 - n/2, 1 - n/2, 2 - n/2, (1 - a*x)/2]))/(a^5*c^2*(-2 + n)*n*(2 + n)*(1 + a*x)))
Time = 0.59 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.98, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {6700, 111, 25, 177, 100, 25, 27, 88, 79, 101, 25, 88, 55, 48}
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.
| \(\displaystyle \int \frac {x^4 e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6700 |
| \(\displaystyle \frac {\int x^4 (1-a x)^{-\frac {n}{2}-2} (a x+1)^{\frac {n-4}{2}}dx}{c^2}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {-\frac {\int -x^2 (1-a x)^{-\frac {n}{2}-2} (a x+1)^{\frac {n-4}{2}} (a n x+3)dx}{a^2}-\frac {x^3 (a x+1)^{\frac {n-2}{2}} (1-a x)^{-\frac {n}{2}-1}}{a^2}}{c^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int x^2 (1-a x)^{-\frac {n}{2}-2} (a x+1)^{\frac {n-4}{2}} (a n x+3)dx}{a^2}-\frac {x^3 (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}}{c^2}\) |
| \(\Big \downarrow \) 177 |
| \(\displaystyle \frac {\frac {(n+3) \int x^2 (1-a x)^{-\frac {n}{2}-2} (a x+1)^{\frac {n-4}{2}}dx-n \int x^2 (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-4}{2}}dx}{a^2}-\frac {x^3 (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}}{c^2}\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {\frac {(n+3) \int x^2 (1-a x)^{-\frac {n}{2}-2} (a x+1)^{\frac {n-4}{2}}dx-n \left (\frac {(1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{a^3 n}-\frac {\int -a (1-a x)^{-n/2} (a x+1)^{\frac {n-4}{2}} (-a x n-n+1)dx}{a^3 n}\right )}{a^2}-\frac {x^3 (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}}{c^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {(n+3) \int x^2 (1-a x)^{-\frac {n}{2}-2} (a x+1)^{\frac {n-4}{2}}dx-n \left (\frac {\int a (1-a x)^{-n/2} (a x+1)^{\frac {n-4}{2}} (-a x n-n+1)dx}{a^3 n}+\frac {(1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{a^3 n}\right )}{a^2}-\frac {x^3 (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}}{c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {(n+3) \int x^2 (1-a x)^{-\frac {n}{2}-2} (a x+1)^{\frac {n-4}{2}}dx-n \left (\frac {\int (1-a x)^{-n/2} (a x+1)^{\frac {n-4}{2}} (-a x n-n+1)dx}{a^2 n}+\frac {(1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{a^3 n}\right )}{a^2}-\frac {x^3 (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}}{c^2}\) |
| \(\Big \downarrow \) 88 |
| \(\displaystyle \frac {\frac {(n+3) \int x^2 (1-a x)^{-\frac {n}{2}-2} (a x+1)^{\frac {n-4}{2}}dx-n \left (\frac {-n \int (1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}dx-\frac {(a x+1)^{\frac {n-2}{2}} (1-a x)^{1-\frac {n}{2}}}{a (2-n)}}{a^2 n}+\frac {(1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{a^3 n}\right )}{a^2}-\frac {x^3 (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}}{c^2}\) |
| \(\Big \downarrow \) 79 |
| \(\displaystyle \frac {\frac {(n+3) \int x^2 (1-a x)^{-\frac {n}{2}-2} (a x+1)^{\frac {n-4}{2}}dx-n \left (\frac {(1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{a^3 n}+\frac {\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (2-n)}-\frac {(1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n-2}{2}}}{a (2-n)}}{a^2 n}\right )}{a^2}-\frac {x^3 (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}}{c^2}\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {\frac {(n+3) \left (\frac {\int -(1-a x)^{-\frac {n}{2}-2} (a x+1)^{\frac {n-4}{2}} (a n x+1)dx}{a^2}+\frac {x (a x+1)^{\frac {n-2}{2}} (1-a x)^{-\frac {n}{2}-1}}{a^2}\right )-n \left (\frac {(1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{a^3 n}+\frac {\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (2-n)}-\frac {(1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n-2}{2}}}{a (2-n)}}{a^2 n}\right )}{a^2}-\frac {x^3 (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}}{c^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {(n+3) \left (\frac {x (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}-\frac {\int (1-a x)^{-\frac {n}{2}-2} (a x+1)^{\frac {n-4}{2}} (a n x+1)dx}{a^2}\right )-n \left (\frac {(1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{a^3 n}+\frac {\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (2-n)}-\frac {(1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n-2}{2}}}{a (2-n)}}{a^2 n}\right )}{a^2}-\frac {x^3 (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}}{c^2}\) |
| \(\Big \downarrow \) 88 |
| \(\displaystyle \frac {\frac {(n+3) \left (\frac {x (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}-\frac {\frac {\left (2-n^2\right ) \int (1-a x)^{-\frac {n}{2}-2} (a x+1)^{\frac {n-2}{2}}dx}{2-n}-\frac {(1-n) (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a (2-n)}}{a^2}\right )-n \left (\frac {(1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{a^3 n}+\frac {\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (2-n)}-\frac {(1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n-2}{2}}}{a (2-n)}}{a^2 n}\right )}{a^2}-\frac {x^3 (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}}{c^2}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {\frac {(n+3) \left (\frac {x (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}-\frac {\frac {\left (2-n^2\right ) \left (\frac {\int (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}dx}{n+2}+\frac {(a x+1)^{n/2} (1-a x)^{-\frac {n}{2}-1}}{a (n+2)}\right )}{2-n}-\frac {(1-n) (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a (2-n)}}{a^2}\right )-n \left (\frac {(1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{a^3 n}+\frac {\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (2-n)}-\frac {(1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n-2}{2}}}{a (2-n)}}{a^2 n}\right )}{a^2}-\frac {x^3 (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}}{c^2}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\frac {(n+3) \left (\frac {x (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}-\frac {\frac {\left (2-n^2\right ) \left (\frac {(a x+1)^{n/2} (1-a x)^{-\frac {n}{2}-1}}{a (n+2)}+\frac {(a x+1)^{n/2} (1-a x)^{-n/2}}{a n (n+2)}\right )}{2-n}-\frac {(1-n) (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a (2-n)}}{a^2}\right )-n \left (\frac {(1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{a^3 n}+\frac {\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (2-n)}-\frac {(1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n-2}{2}}}{a (2-n)}}{a^2 n}\right )}{a^2}-\frac {x^3 (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}}{c^2}\) |
(-((x^3*(1 - a*x)^(-1 - n/2)*(1 + a*x)^((-2 + n)/2))/a^2) + ((3 + n)*((x*( 1 - a*x)^(-1 - n/2)*(1 + a*x)^((-2 + n)/2))/a^2 - (-(((1 - n)*(1 - a*x)^(- 1 - n/2)*(1 + a*x)^((-2 + n)/2))/(a*(2 - n))) + ((2 - n^2)*(((1 - a*x)^(-1 - n/2)*(1 + a*x)^(n/2))/(a*(2 + n)) + (1 + a*x)^(n/2)/(a*n*(2 + n)*(1 - a *x)^(n/2))))/(2 - n))/a^2) - n*((1 + a*x)^((-2 + n)/2)/(a^3*n*(1 - a*x)^(n /2)) + (-(((1 - a*x)^(1 - n/2)*(1 + a*x)^((-2 + n)/2))/(a*(2 - n))) + (2^( n/2)*n*(1 - a*x)^(1 - n/2)*Hypergeometric2F1[(2 - n)/2, 1 - n/2, 2 - n/2, (1 - a*x)/2])/(a*(2 - n)))/(a^2*n)))/a^2)/c^2
3.14.18.3.1 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)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
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_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h/b Int[(a + b*x)^(m + 1)*(c + d *x)^n*(e + f*x)^p, x], x] + Simp[(b*g - a*h)/b Int[(a + b*x)^m*(c + d*x)^ n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && (Su mSimplerQ[m, 1] || ( !SumSimplerQ[n, 1] && !SumSimplerQ[p, 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])
\[\int \frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} x^{4}}{\left (-a^{2} c \,x^{2}+c \right )^{2}}d x\]
\[ \int \frac {e^{n \text {arctanh}(a x)} x^4}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {x^{4} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{2}} \,d x } \]
\[ \int \frac {e^{n \text {arctanh}(a x)} x^4}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {\int \frac {x^{4} e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
\[ \int \frac {e^{n \text {arctanh}(a x)} x^4}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {x^{4} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{2}} \,d x } \]
\[ \int \frac {e^{n \text {arctanh}(a x)} x^4}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {x^{4} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {e^{n \text {arctanh}(a x)} x^4}{\left (c-a^2 c x^2\right )^2} \, dx=\int \frac {x^4\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{{\left (c-a^2\,c\,x^2\right )}^2} \,d x \]