\(\int \frac {e^{n \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx\) [720]

Optimal result

Integrand size = 24, antiderivative size = 111 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x \operatorname {Hypergeometric2F1}\left (1,\frac {1-n}{2},\frac {3-n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{(1-n) \sqrt {c-a^2 c x^2}} \] Output:

2*(1-1/a^2/x^2)^(1/2)*(1-1/a/x)^(1/2-1/2*n)*(1+1/a/x)^(-1/2+1/2*n)*x*hyper 
geom([1, 1/2-1/2*n],[3/2-1/2*n],(a-1/x)/(a+1/x))/(1-n)/(-a^2*c*x^2+c)^(1/2 
)
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.73 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=-\frac {2 e^{(1+n) \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},e^{2 \coth ^{-1}(a x)}\right )}{\sqrt {1-\frac {1}{a^2 x^2}} \left (a^2 c x+a^2 c n x\right )} \] Input:

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

Output:

(-2*E^((1 + n)*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[1, (1 + 
 n)/2, (3 + n)/2, E^(2*ArcCoth[a*x])])/(Sqrt[1 - 1/(a^2*x^2)]*(a^2*c*x + a 
^2*c*n*x))
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6746, 6749, 141}

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.

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

\(\Big \downarrow \) 6746

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

\(\Big \downarrow \) 6749

\(\displaystyle -\frac {x \sqrt {1-\frac {1}{a^2 x^2}} \int \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)} \left (1+\frac {1}{a x}\right )^{\frac {n-1}{2}} xd\frac {1}{x}}{\sqrt {c-a^2 c x^2}}\)

\(\Big \downarrow \) 141

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 141

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^( 
n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f 
))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, 
p}, x] && EqQ[m + n + p + 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !Su 
mSimplerQ[p, 1]) &&  !ILtQ[m, 0]
 

rule 6746

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

rule 6749

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

Maple [F]

\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{\sqrt {-a^{2} c \,x^{2}+c}}d x\]

Input:

int(exp(n*arccoth(a*x))/(-a^2*c*x^2+c)^(1/2),x)
 

Output:

int(exp(n*arccoth(a*x))/(-a^2*c*x^2+c)^(1/2),x)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

integrate(exp(n*acoth(a*x))/(-a**2*c*x**2+c)**(1/2),x)
 

Output:

Integral(exp(n*acoth(a*x))/sqrt(-c*(a*x - 1)*(a*x + 1)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

int(exp(n*acoth(a*x))/(c - a^2*c*x^2)^(1/2),x)
 

Output:

int(exp(n*acoth(a*x))/(c - a^2*c*x^2)^(1/2), x)
 

Reduce [F]

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

int(exp(n*acoth(a*x))/(-a^2*c*x^2+c)^(1/2),x)
 

Output:

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