\(\int \frac {e^{\text {arctanh}(a x)} x^m}{(c-a^2 c x^2)^2} \, dx\) [1027]

Optimal result

Integrand size = 23, antiderivative size = 80 \[ \int \frac {e^{\text {arctanh}(a x)} x^m}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{c^2 (1+m)}+\frac {a x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {2+m}{2},\frac {4+m}{2},a^2 x^2\right )}{c^2 (2+m)} \] Output:

x^(1+m)*hypergeom([5/2, 1/2+1/2*m],[3/2+1/2*m],a^2*x^2)/c^2/(1+m)+a*x^(2+m 
)*hypergeom([5/2, 1+1/2*m],[2+1/2*m],a^2*x^2)/c^2/(2+m)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.02 \[ \int \frac {e^{\text {arctanh}(a x)} x^m}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1+m}{2},1+\frac {1+m}{2},a^2 x^2\right )}{1+m}+\frac {a x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {2+m}{2},1+\frac {2+m}{2},a^2 x^2\right )}{2+m}}{c^2} \] Input:

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

Output:

((x^(1 + m)*Hypergeometric2F1[5/2, (1 + m)/2, 1 + (1 + m)/2, a^2*x^2])/(1 
+ m) + (a*x^(2 + m)*Hypergeometric2F1[5/2, (2 + m)/2, 1 + (2 + m)/2, a^2*x 
^2])/(2 + m))/c^2
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6698, 557, 278}

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.

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

\(\Big \downarrow \) 6698

\(\displaystyle \frac {\int \frac {x^m (a x+1)}{\left (1-a^2 x^2\right )^{5/2}}dx}{c^2}\)

\(\Big \downarrow \) 557

\(\displaystyle \frac {a \int \frac {x^{m+1}}{\left (1-a^2 x^2\right )^{5/2}}dx+\int \frac {x^m}{\left (1-a^2 x^2\right )^{5/2}}dx}{c^2}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {\frac {x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {m+1}{2},\frac {m+3}{2},a^2 x^2\right )}{m+1}+\frac {a x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {m+2}{2},\frac {m+4}{2},a^2 x^2\right )}{m+2}}{c^2}\)

Input:

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

Output:

((x^(1 + m)*Hypergeometric2F1[5/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/(1 + m) 
 + (a*x^(2 + m)*Hypergeometric2F1[5/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(2 
+ m))/c^2
 

Defintions of rubi rules used

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 557

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[c   Int[(e*x)^m*(a + b*x^2)^p, x], x] + Simp[d/e   Int[(e*x)^( 
m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x]
 

rule 6698

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(-sqrt(-a^2*x^2 + 1)*x^m/(a^5*c^2*x^5 - a^4*c^2*x^4 - 2*a^3*c^2*x^ 
3 + 2*a^2*c^2*x^2 + a*c^2*x - c^2), x)
 

Sympy [F]

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

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

Output:

(Integral(x**m/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x* 
*2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x*x**m/(a**4*x**4*sqrt(-a 
**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), 
x))/c**2
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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