\(\int \frac {e^{n \text {arctanh}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx\) [1360]

Optimal result

Integrand size = 24, antiderivative size = 104 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=-\frac {2^{\frac {1+n}{2}} (1-a x)^{\frac {1-n}{2}} \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {1-n}{2},\frac {3-n}{2},\frac {1}{2} (1-a x)\right )}{a (1-n) \sqrt {c-a^2 c x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.97 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\frac {2^{\frac {1+n}{2}} (1-a x)^{\frac {1}{2}-\frac {n}{2}} \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {n}{2},\frac {1}{2}-\frac {n}{2},\frac {3}{2}-\frac {n}{2},\frac {1}{2}-\frac {a x}{2}\right )}{a (-1+n) \sqrt {c-a^2 c x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6693, 6690, 79}

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.

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

\(\Big \downarrow \) 6693

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

\(\Big \downarrow \) 6690

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

\(\Big \downarrow \) 79

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 79

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

rule 6690

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

rule 6693

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(exp(n*arctanh(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]

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

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

Output:

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

Maxima [F]

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

integrate(exp(n*arctanh(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]

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

integrate(exp(n*arctanh(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. \[ \int \frac {e^{n \text {arctanh}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{\sqrt {c-a^2\,c\,x^2}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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