\(\int e^{n \coth ^{-1}(a x)} (c-a c x) \, dx\) [286]

Optimal result

Integrand size = 16, antiderivative size = 79 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x) \, dx=-\frac {8 c \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-4+n)} \operatorname {Hypergeometric2F1}\left (3,2-\frac {n}{2},3-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (4-n)} \] Output:

-8*c*(1-1/a/x)^(2-1/2*n)*(1+1/a/x)^(-2+1/2*n)*hypergeom([3, 2-1/2*n],[3-1/ 
2*n],(a-1/x)/(a+1/x))/a/(4-n)
 

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.32 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x) \, dx=-\frac {c e^{n \coth ^{-1}(a x)} \left (e^{2 \coth ^{-1}(a x)} (-2+n) n \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )+(2+n) \left (-1+a (-2+n) x+a^2 x^2+(-2+n) \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )\right )}{2 a (2+n)} \] Input:

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

Output:

-1/2*(c*E^(n*ArcCoth[a*x])*(E^(2*ArcCoth[a*x])*(-2 + n)*n*Hypergeometric2F 
1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (2 + n)*(-1 + a*(-2 + n)*x + 
a^2*x^2 + (-2 + n)*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])]) 
))/(a*(2 + n))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6725, 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.

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

\(\Big \downarrow \) 6725

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

\(\Big \downarrow \) 141

\(\displaystyle -\frac {8 c \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-4}{2}} \operatorname {Hypergeometric2F1}\left (3,2-\frac {n}{2},3-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (4-n)}\)

Input:

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

Output:

(-8*c*(1 - 1/(a*x))^(2 - n/2)*(1 + 1/(a*x))^((-4 + n)/2)*Hypergeometric2F1 
[3, 2 - n/2, 3 - n/2, (a - x^(-1))/(a + x^(-1))])/(a*(4 - n))
 

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 6725

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> S 
imp[-d^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]

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

Input:

int(exp(n*arccoth(a*x))*(-a*c*x+c),x)
 

Output:

int(exp(n*arccoth(a*x))*(-a*c*x+c),x)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

int(exp(n*acoth(a*x))*(c - a*c*x),x)
 

Output:

int(exp(n*acoth(a*x))*(c - a*c*x), x)
 

Reduce [F]

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

int(exp(n*acoth(a*x))*(-a*c*x+c),x)
 

Output:

c*(int(e**(acoth(a*x)*n),x) - int(e**(acoth(a*x)*n)*x,x)*a)