\(\int \frac {e^{2 \text {arctanh}(a x)} x^m}{c-a^2 c x^2} \, dx\) [1150]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {e^{2 \text {arctanh}(a x)} x^m}{c-a^2 c x^2} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}(2,1+m,2+m,a x)}{c (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^{2 \text {arctanh}(a x)} x^m}{c-a^2 c x^2} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}(2,1+m,2+m,a x)}{c (1+m)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6700, 74}

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.

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

\(\Big \downarrow \) 6700

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

\(\Big \downarrow \) 74

\(\displaystyle \frac {x^{m+1} \operatorname {Hypergeometric2F1}(2,m+1,m+2,a x)}{c (m+1)}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 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]

\[\int \frac {\left (a x +1\right )^{2} x^{m}}{\left (-a^{2} x^{2}+1\right ) \left (-a^{2} c \,x^{2}+c \right )}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate((a*x+1)^2/(-a^2*x^2+1)*x^m/(-a^2*c*x^2+c),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

Integral(x**m/(a**2*x**2 - 2*a*x + 1), x)/c
 

Maxima [F]

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

integrate((a*x+1)^2/(-a^2*x^2+1)*x^m/(-a^2*c*x^2+c),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((a*x+1)^2/(-a^2*x^2+1)*x^m/(-a^2*c*x^2+c),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(x**m + int(x**m/(a**2*m*x**3 - a**2*x**3 - 2*a*m*x**2 + 2*a*x**2 + m*x - 
x),x)*a*m**2*x - int(x**m/(a**2*m*x**3 - a**2*x**3 - 2*a*m*x**2 + 2*a*x**2 
 + m*x - x),x)*a*m*x - int(x**m/(a**2*m*x**3 - a**2*x**3 - 2*a*m*x**2 + 2* 
a*x**2 + m*x - x),x)*m**2 + int(x**m/(a**2*m*x**3 - a**2*x**3 - 2*a*m*x**2 
 + 2*a*x**2 + m*x - x),x)*m)/(a*c*(a*m*x - a*x - m + 1))