\(\int \frac {e^{\coth ^{-1}(a x)} (e x)^m}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx\) [906]

Optimal result

Integrand size = 27, antiderivative size = 58 \[ \int \frac {e^{\coth ^{-1}(a x)} (e x)^m}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=-\frac {a \sqrt {1-\frac {1}{a^2 x^2}} (e x)^{2+m} \operatorname {Hypergeometric2F1}(1,2+m,3+m,a x)}{e^2 (2+m) \sqrt {c-\frac {c}{a^2 x^2}}} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\coth ^{-1}(a x)} (e x)^m}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=-\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x^2 (e x)^m \operatorname {Hypergeometric2F1}(1,2+m,3+m,a x)}{(2+m) \sqrt {c-\frac {c}{a^2 x^2}}} \] Input:

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

Output:

-((a*Sqrt[1 - 1/(a^2*x^2)]*x^2*(e*x)^m*Hypergeometric2F1[1, 2 + m, 3 + m, 
a*x])/((2 + m)*Sqrt[c - c/(a^2*x^2)]))
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6751, 6747, 8, 25, 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.

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

\(\Big \downarrow \) 6751

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

\(\Big \downarrow \) 6747

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

\(\Big \downarrow \) 8

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 74

\(\displaystyle -\frac {a \sqrt {1-\frac {1}{a^2 x^2}} (e x)^{m+2} \operatorname {Hypergeometric2F1}(1,m+2,m+3,a x)}{e^2 (m+2) \sqrt {c-\frac {c}{a^2 x^2}}}\)

Input:

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

Output:

-((a*Sqrt[1 - 1/(a^2*x^2)]*(e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m 
, a*x])/(e^2*(2 + m)*Sqrt[c - c/(a^2*x^2)]))
 

Defintions of rubi rules used

rule 8

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_))^(p_), x_Symbol] :> Simp[1/a^m   Int[u*(a* 
x)^(m + p), x], x] /; FreeQ[{a, m, p}, x] && IntegerQ[m]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 6747

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symb 
ol] :> Simp[c^p/a^(2*p)   Int[(u/x^(2*p))*(-1 + a*x)^(p - n/2)*(1 + a*x)^(p 
 + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] &&  !Inte 
gerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + n/2]
 

rule 6751

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo 
l] :> Simp[c^IntPart[p]*((c + d/x^2)^FracPart[p]/(1 - 1/(a^2*x^2))^FracPart 
[p])   Int[u*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, 
 d, n, p}, x] && EqQ[c + a^2*d, 0] &&  !IntegerQ[n/2] &&  !(IntegerQ[p] || 
GtQ[c, 0])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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