\(\int e^{n \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx\) [782]

Optimal result

Integrand size = 24, antiderivative size = 272 \[ \int e^{n \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^{\frac {3-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)}}{(1-n) \sqrt {1-a^2 x^2}}+\frac {2 \sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-1+n),\frac {1+n}{2},\frac {1+a x}{1-a x}\right )}{(1-n) \sqrt {1-a^2 x^2}}+\frac {2^{\frac {1+n}{2}} n \sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^{\frac {3-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {3-n}{2},\frac {5-n}{2},\frac {1}{2} (1-a x)\right )}{\left (3-4 n+n^2\right ) \sqrt {1-a^2 x^2}} \] Output:

-(c-c/a^2/x^2)^(1/2)*x*(-a*x+1)^(3/2-1/2*n)*(a*x+1)^(-1/2+1/2*n)/(1-n)/(-a 
^2*x^2+1)^(1/2)+2*(c-c/a^2/x^2)^(1/2)*x*(-a*x+1)^(1/2-1/2*n)*(a*x+1)^(-1/2 
+1/2*n)*hypergeom([1, -1/2+1/2*n],[1/2+1/2*n],(a*x+1)/(-a*x+1))/(1-n)/(-a^ 
2*x^2+1)^(1/2)+2^(1/2+1/2*n)*n*(c-c/a^2/x^2)^(1/2)*x*(-a*x+1)^(3/2-1/2*n)* 
hypergeom([3/2-1/2*n, 1/2-1/2*n],[5/2-1/2*n],-1/2*a*x+1/2)/(n^2-4*n+3)/(-a 
^2*x^2+1)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.22 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.76 \[ \int e^{n \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\frac {2 \sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^{\frac {1}{2} (-1-n)} \left ((-1+n) (1+a x)^{\frac {1+n}{2}} \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{2}-\frac {n}{2},\frac {1}{2}-\frac {n}{2},\frac {1-a x}{1+a x}\right )+2^{\frac {1+n}{2}} \left (-\left ((-1+n) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-\frac {n}{2},-\frac {1}{2}-\frac {n}{2},\frac {1}{2}-\frac {n}{2},\frac {1}{2}-\frac {a x}{2}\right )\right )+(1+n) (-1+a x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-\frac {n}{2},\frac {1}{2}-\frac {n}{2},\frac {3}{2}-\frac {n}{2},\frac {1}{2}-\frac {a x}{2}\right )\right )\right )}{\left (-1+n^2\right ) \sqrt {1-a^2 x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.83, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6710, 6700, 139, 88, 79, 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.

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

\(\Big \downarrow \) 6710

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

\(\Big \downarrow \) 6700

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

\(\Big \downarrow \) 139

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

\(\Big \downarrow \) 88

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

\(\Big \downarrow \) 79

\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\int \frac {(1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-3}{2}}}{x}dx+a \left (\frac {2^{\frac {n+1}{2}} n (1-a x)^{\frac {3-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {3-n}{2},\frac {5-n}{2},\frac {1}{2} (1-a x)\right )}{a (1-n) (3-n)}-\frac {(1-a x)^{\frac {3-n}{2}} (a x+1)^{\frac {n-1}{2}}}{a (1-n)}\right )\right )}{\sqrt {1-a^2 x^2}}\)

\(\Big \downarrow \) 141

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

Input:

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

Output:

(Sqrt[c - c/(a^2*x^2)]*x*((2*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((-1 + n)/2)* 
Hypergeometric2F1[1, (-1 + n)/2, (1 + n)/2, (1 + a*x)/(1 - a*x)])/(1 - n) 
+ a*(-(((1 - a*x)^((3 - n)/2)*(1 + a*x)^((-1 + n)/2))/(a*(1 - n))) + (2^(( 
1 + n)/2)*n*(1 - a*x)^((3 - n)/2)*Hypergeometric2F1[(1 - n)/2, (3 - n)/2, 
(5 - n)/2, (1 - a*x)/2])/(a*(1 - n)*(3 - n)))))/Sqrt[1 - a^2*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 88

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], 
 x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimpl 
erQ[p, 1]
 

rule 139

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

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 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 6710

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int e^{n \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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