\(\int e^{n \coth ^{-1}(a x)} (c-a c x)^{-1+\frac {n}{2}} \, dx\) [301]

Optimal result

Integrand size = 24, antiderivative size = 80 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{-1+\frac {n}{2}} \, dx=\frac {2 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{n/2} x (c-a c x)^{\frac {1}{2} (-2+n)} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{n} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{-1+\frac {n}{2}} \, dx=-\frac {2 \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} (c-a c x)^{n/2} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {2}{1+a x}\right )}{a c n} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.20, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6727, 27, 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.

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

\(\Big \downarrow \) 6727

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 141

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 6727

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

Maple [F]

\[\int {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (-a c x +c \right )^{-1+\frac {n}{2}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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