\(\int \frac {e^{n \text {arctanh}(a x)}}{(c-\frac {c}{a^2 x^2})^2} \, dx\) [765]

Optimal result

Integrand size = 22, antiderivative size = 373 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {(1-n) (3+n) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a c^2 (2-n)}+\frac {(3+n) x (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2}-\frac {a^2 x^3 (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2}+\frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a c^2 (2-n)}-\frac {(1-a x)^{-n/2} (1+a x)^{\frac {1}{2} (-2+n)}}{a c^2}-\frac {(3+n) \left (2-n^2\right ) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{n/2}}{a c^2 \left (4-n^2\right )}-\frac {(3+n) \left (2-n^2\right ) (1-a x)^{-n/2} (1+a x)^{n/2}}{a c^2 n \left (4-n^2\right )}-\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a c^2 (2-n)} \] Output:

(1-n)*(3+n)*(-a*x+1)^(-1-1/2*n)*(a*x+1)^(-1+1/2*n)/a/c^2/(2-n)+(3+n)*x*(-a 
*x+1)^(-1-1/2*n)*(a*x+1)^(-1+1/2*n)/c^2-a^2*x^3*(-a*x+1)^(-1-1/2*n)*(a*x+1 
)^(-1+1/2*n)/c^2+(-a*x+1)^(1-1/2*n)*(a*x+1)^(-1+1/2*n)/a/c^2/(2-n)-(a*x+1) 
^(-1+1/2*n)/a/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/c^2/(-n^2+4)-(3+n)*(-n^2+2)*(a*x+1)^(1/2*n)/a/c^2/n/(-n^2+ 
4)/((-a*x+1)^(1/2*n))-2^(1/2*n)*n*(-a*x+1)^(1-1/2*n)*hypergeom([1-1/2*n, 1 
-1/2*n],[2-1/2*n],-1/2*a*x+1/2)/a/c^2/(2-n)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.48 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=-\frac {(1-a x)^{-1-\frac {n}{2}} \left ((1+a x)^{n/2} \left (-6+6 a^2 x^2+n^3 (-1+a x)^2 (1+a x)+n^2 \left (1-2 a^2 x^2\right )+n \left (-4+6 a x+4 a^2 x^2-4 a^3 x^3\right )\right )-2^{n/2} n^2 (2+n) (-1+a x)^2 (1+a x) \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )\right )}{a c^2 (-2+n) n (2+n) (1+a x)} \] Input:

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

Output:

-(((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*c^2*(-2 + n)*n*(2 + n)*(1 + a*x)))
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 370, normalized size of antiderivative = 0.99, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {6707, 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.

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

\(\Big \downarrow \) 6707

\(\displaystyle \frac {a^4 \int \frac {e^{n \text {arctanh}(a x)} x^4}{\left (1-a^2 x^2\right )^2}dx}{c^2}\)

\(\Big \downarrow \) 6700

\(\displaystyle \frac {a^4 \int x^4 (1-a x)^{-\frac {n}{2}-2} (a x+1)^{\frac {n-4}{2}}dx}{c^2}\)

\(\Big \downarrow \) 111

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 177

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

\(\Big \downarrow \) 100

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 88

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

\(\Big \downarrow \) 79

\(\displaystyle \frac {a^4 \left (\frac {(n+3) \int x^2 (1-a x)^{-\frac {n}{2}-2} (a x+1)^{\frac {n-4}{2}}dx-n \left (\frac {(1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{a^3 n}+\frac {\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (2-n)}-\frac {(1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n-2}{2}}}{a (2-n)}}{a^2 n}\right )}{a^2}-\frac {x^3 (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}\right )}{c^2}\)

\(\Big \downarrow \) 101

\(\displaystyle \frac {a^4 \left (\frac {(n+3) \left (\frac {\int -(1-a x)^{-\frac {n}{2}-2} (a x+1)^{\frac {n-4}{2}} (a n x+1)dx}{a^2}+\frac {x (a x+1)^{\frac {n-2}{2}} (1-a x)^{-\frac {n}{2}-1}}{a^2}\right )-n \left (\frac {(1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{a^3 n}+\frac {\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (2-n)}-\frac {(1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n-2}{2}}}{a (2-n)}}{a^2 n}\right )}{a^2}-\frac {x^3 (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}\right )}{c^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {a^4 \left (\frac {(n+3) \left (\frac {x (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}-\frac {\int (1-a x)^{-\frac {n}{2}-2} (a x+1)^{\frac {n-4}{2}} (a n x+1)dx}{a^2}\right )-n \left (\frac {(1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{a^3 n}+\frac {\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (2-n)}-\frac {(1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n-2}{2}}}{a (2-n)}}{a^2 n}\right )}{a^2}-\frac {x^3 (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}\right )}{c^2}\)

\(\Big \downarrow \) 88

\(\displaystyle \frac {a^4 \left (\frac {(n+3) \left (\frac {x (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}-\frac {\frac {\left (2-n^2\right ) \int (1-a x)^{-\frac {n}{2}-2} (a x+1)^{\frac {n-2}{2}}dx}{2-n}-\frac {(1-n) (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a (2-n)}}{a^2}\right )-n \left (\frac {(1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{a^3 n}+\frac {\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (2-n)}-\frac {(1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n-2}{2}}}{a (2-n)}}{a^2 n}\right )}{a^2}-\frac {x^3 (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}\right )}{c^2}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {a^4 \left (\frac {(n+3) \left (\frac {x (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}-\frac {\frac {\left (2-n^2\right ) \left (\frac {\int (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}dx}{n+2}+\frac {(a x+1)^{n/2} (1-a x)^{-\frac {n}{2}-1}}{a (n+2)}\right )}{2-n}-\frac {(1-n) (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a (2-n)}}{a^2}\right )-n \left (\frac {(1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{a^3 n}+\frac {\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (2-n)}-\frac {(1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n-2}{2}}}{a (2-n)}}{a^2 n}\right )}{a^2}-\frac {x^3 (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}\right )}{c^2}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {a^4 \left (\frac {(n+3) \left (\frac {x (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}-\frac {\frac {\left (2-n^2\right ) \left (\frac {(a x+1)^{n/2} (1-a x)^{-\frac {n}{2}-1}}{a (n+2)}+\frac {(a x+1)^{n/2} (1-a x)^{-n/2}}{a n (n+2)}\right )}{2-n}-\frac {(1-n) (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a (2-n)}}{a^2}\right )-n \left (\frac {(1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{a^3 n}+\frac {\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (2-n)}-\frac {(1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n-2}{2}}}{a (2-n)}}{a^2 n}\right )}{a^2}-\frac {x^3 (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{a^2}\right )}{c^2}\)

Input:

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

Output:

(a^4*(-((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
 

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 48

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]
 

rule 55

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])
 

rule 79

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]))
 

rule 88

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]
 

rule 100

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])))
 

rule 101

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]
 

rule 111

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]
 

rule 177

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]))
 

rule 6700

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])
 

rule 6707

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]
 

Maple [F]

\[\int \frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{\left (c -\frac {c}{a^{2} x^{2}}\right )^{2}}d x\]

Input:

int(exp(n*arctanh(a*x))/(c-c/a^2/x^2)^2,x)
 

Output:

int(exp(n*arctanh(a*x))/(c-c/a^2/x^2)^2,x)
 

Fricas [F]

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

integrate(exp(n*arctanh(a*x))/(c-c/a^2/x^2)^2,x, algorithm="fricas")
 

Output:

integral(a^4*x^4*(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^2*x^4 - 2*a^2*c^2*x 
^2 + c^2), x)
 

Sympy [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {a^{4} \int \frac {x^{4} e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \] Input:

integrate(exp(n*atanh(a*x))/(c-c/a**2/x**2)**2,x)
 

Output:

a**4*Integral(x**4*exp(n*atanh(a*x))/(a**4*x**4 - 2*a**2*x**2 + 1), x)/c** 
2
                                                                                    
                                                                                    
 

Maxima [F]

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

integrate(exp(n*arctanh(a*x))/(c-c/a^2/x^2)^2,x, algorithm="maxima")
 

Output:

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

Giac [F(-2)]

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

integrate(exp(n*arctanh(a*x))/(c-c/a^2/x^2)^2,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{1,[0,1,0]%%%} / %%%{1,[0,0,2]%%%} Error: Bad Argument Valu 
e
 

Mupad [F(-1)]

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

int(exp(n*atanh(a*x))/(c - c/(a^2*x^2))^2,x)
 

Output:

int(exp(n*atanh(a*x))/(c - c/(a^2*x^2))^2, x)
 

Reduce [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {\left (\int \frac {e^{\mathit {atanh} \left (a x \right ) n} x^{4}}{a^{4} x^{4}-2 a^{2} x^{2}+1}d x \right ) a^{4}}{c^{2}} \] Input:

int(exp(n*atanh(a*x))/(c-c/a^2/x^2)^2,x)
 

Output:

(int((e**(atanh(a*x)*n)*x**4)/(a**4*x**4 - 2*a**2*x**2 + 1),x)*a**4)/c**2