\(\int \frac {e^{n \text {arctanh}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx\) [779]

Optimal result

Integrand size = 22, antiderivative size = 125 \[ \int \frac {e^{n \text {arctanh}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=-\frac {(1-n) (1-a x)^{-n/2} (1+a x)^{n/2}}{a c n}+\frac {x (1-a x)^{-n/2} (1+a x)^{n/2}}{c}-\frac {2^{1+\frac {n}{2}} (1-a x)^{-n/2} \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},-\frac {n}{2},1-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a c} \] Output:

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.66 \[ \int \frac {e^{n \text {arctanh}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {(1-a x)^{-n/2} \left ((1+a x)^{n/2} (-1+n+a n x)-2^{1+\frac {n}{2}} n \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},-\frac {n}{2},1-\frac {n}{2},\frac {1}{2} (1-a x)\right )\right )}{a c n} \] Input:

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

Output:

((1 + a*x)^(n/2)*(-1 + n + a*n*x) - 2^(1 + n/2)*n*Hypergeometric2F1[-1/2*n 
, -1/2*n, 1 - n/2, (1 - a*x)/2])/(a*c*n*(1 - a*x)^(n/2))
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.01, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6707, 6700, 101, 25, 88}

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.

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

\(\Big \downarrow \) 6707

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

\(\Big \downarrow \) 6700

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

\(\Big \downarrow \) 101

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 88

\(\displaystyle \text {Indeterminate}\)

Input:

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

Output:

Indeterminate
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

integrate(exp(n*arctanh(a*x))/(c-c/a^2/x^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,[0,0,1]%%%} Error: Bad Argument Valu 
e
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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