\(\int \frac {e^{\text {arctanh}(a x)} x^m}{\sqrt {c-a^2 c x^2}} \, dx\) [1038]

Optimal result

Integrand size = 25, antiderivative size = 51 \[ \int \frac {e^{\text {arctanh}(a x)} x^m}{\sqrt {c-a^2 c x^2}} \, dx=\frac {x^{1+m} \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}(1,1+m,2+m,a x)}{(1+m) \sqrt {c-a^2 c x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\text {arctanh}(a x)} x^m}{\sqrt {c-a^2 c x^2}} \, dx=\frac {x^{1+m} \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}(1,1+m,2+m,a x)}{(1+m) \sqrt {c-a^2 c x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 51, 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, 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.

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

\(\Big \downarrow \) 6703

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

\(\Big \downarrow \) 6700

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

\(\Big \downarrow \) 74

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

Input:

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

Output:

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

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])
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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