\(\int \frac {e^{n \text {arctanh}(a x)} x^m}{(c-a^2 c x^2)^2} \, dx\) [1397]

Optimal result

Integrand size = 25, antiderivative size = 42 \[ \int \frac {e^{n \text {arctanh}(a x)} x^m}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {x^{1+m} \operatorname {AppellF1}\left (1+m,\frac {4+n}{2},2-\frac {n}{2},2+m,a x,-a x\right )}{c^2 (1+m)} \] Output:

x^(1+m)*AppellF1(1+m,2+1/2*n,2-1/2*n,2+m,a*x,-a*x)/c^2/(1+m)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.05, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6700, 150}

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.

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

\(\Big \downarrow \) 6700

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

\(\Big \downarrow \) 150

\(\displaystyle \frac {x^{m+1} \operatorname {AppellF1}\left (m+1,\frac {n+4}{2},\frac {4-n}{2},m+2,a x,-a x\right )}{c^2 (m+1)}\)

Input:

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

Output:

(x^(1 + m)*AppellF1[1 + m, (4 + n)/2, (4 - n)/2, 2 + m, a*x, -(a*x)])/(c^2 
*(1 + m))
 

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 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])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(x^m*(-(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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {e^{n \text {arctanh}(a x)} x^m}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {-x^{m} e^{\mathit {atanh} \left (a x \right ) n}+\left (\int \frac {x^{m} e^{\mathit {atanh} \left (a x \right ) n} x}{a^{4} x^{4}-2 a^{2} x^{2}+1}d x \right ) a^{4} m \,x^{2}-2 \left (\int \frac {x^{m} e^{\mathit {atanh} \left (a x \right ) n} x}{a^{4} x^{4}-2 a^{2} x^{2}+1}d x \right ) a^{4} x^{2}-\left (\int \frac {x^{m} e^{\mathit {atanh} \left (a x \right ) n} x}{a^{4} x^{4}-2 a^{2} x^{2}+1}d x \right ) a^{2} m +2 \left (\int \frac {x^{m} e^{\mathit {atanh} \left (a x \right ) n} x}{a^{4} x^{4}-2 a^{2} x^{2}+1}d x \right ) a^{2}-\left (\int \frac {x^{m} e^{\mathit {atanh} \left (a x \right ) n}}{a^{4} x^{5}-2 a^{2} x^{3}+x}d x \right ) a^{2} m \,x^{2}+\left (\int \frac {x^{m} e^{\mathit {atanh} \left (a x \right ) n}}{a^{4} x^{5}-2 a^{2} x^{3}+x}d x \right ) m}{a \,c^{2} n \left (a^{2} x^{2}-1\right )} \] Input:

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

Output:

( - x**m*e**(atanh(a*x)*n) + int((x**m*e**(atanh(a*x)*n)*x)/(a**4*x**4 - 2 
*a**2*x**2 + 1),x)*a**4*m*x**2 - 2*int((x**m*e**(atanh(a*x)*n)*x)/(a**4*x* 
*4 - 2*a**2*x**2 + 1),x)*a**4*x**2 - int((x**m*e**(atanh(a*x)*n)*x)/(a**4* 
x**4 - 2*a**2*x**2 + 1),x)*a**2*m + 2*int((x**m*e**(atanh(a*x)*n)*x)/(a**4 
*x**4 - 2*a**2*x**2 + 1),x)*a**2 - int((x**m*e**(atanh(a*x)*n))/(a**4*x**5 
 - 2*a**2*x**3 + x),x)*a**2*m*x**2 + int((x**m*e**(atanh(a*x)*n))/(a**4*x* 
*5 - 2*a**2*x**3 + x),x)*m)/(a*c**2*n*(a**2*x**2 - 1))