\(\int e^{n \text {arctanh}(a x)} x^m (c-a^2 c x^2)^p \, dx\) [1398]

Optimal result

Integrand size = 25, antiderivative size = 70 \[ \int e^{n \text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^p \, dx=\frac {x^{1+m} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {AppellF1}\left (1+m,\frac {1}{2} (n-2 p),-\frac {n}{2}-p,2+m,a x,-a x\right )}{1+m} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6703, 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.

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

\(\Big \downarrow \) 6703

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int e^{n \text {arctanh}(a x)} x^m \left (1-a^2 x^2\right )^pdx\)

\(\Big \downarrow \) 6700

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

\(\Big \downarrow \) 150

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

Input:

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

Output:

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

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])
 

rule 6703

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_ 
Symbol] :> Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPar 
t[p])   Int[x^m*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, 
d, m, n, p}, x] && EqQ[a^2*c + d, 0] &&  !(IntegerQ[p] || GtQ[c, 0]) &&  !I 
ntegerQ[n/2]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral(x**m*(-c*(a*x - 1)*(a*x + 1))**p*exp(n*atanh(a*x)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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