\(\int e^{n \text {arctanh}(a x)} x^m (c-a^2 c x^2)^2 \, dx\) [1394]

Optimal result

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

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

Mathematica [F]

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

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

\(\Big \downarrow \) 6700

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

\(\Big \downarrow \) 150

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

Input:

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

Output:

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

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

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]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

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