\(\int e^{n \coth ^{-1}(a x)} (c-a c x)^2 \, dx\) [285]

Optimal result

Integrand size = 18, antiderivative size = 81 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {16 c^2 \left (1-\frac {1}{a x}\right )^{3-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-6+n)} \operatorname {Hypergeometric2F1}\left (4,3-\frac {n}{2},4-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (6-n)} \] Output:

16*c^2*(1-1/a/x)^(3-1/2*n)*(1+1/a/x)^(-3+1/2*n)*hypergeom([4, 3-1/2*n],[4- 
1/2*n],(a-1/x)/(a+1/x))/a/(6-n)
 

Mathematica [A] (verified)

Time = 1.23 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.78 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {c^2 e^{n \coth ^{-1}(a x)} \left (e^{2 \coth ^{-1}(a x)} n \left (8-6 n+n^2\right ) \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )+(2+n) \left (6+6 a x+a n^2 x-6 a^2 x^2+2 a^3 x^3+n \left (-1-6 a x+a^2 x^2\right )+\left (8-6 n+n^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )\right )}{6 a (2+n)} \] Input:

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

Output:

(c^2*E^(n*ArcCoth[a*x])*(E^(2*ArcCoth[a*x])*n*(8 - 6*n + n^2)*Hypergeometr 
ic2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (2 + n)*(6 + 6*a*x + a*n^ 
2*x - 6*a^2*x^2 + 2*a^3*x^3 + n*(-1 - 6*a*x + a^2*x^2) + (8 - 6*n + n^2)*H 
ypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])])))/(6*a*(2 + n))
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.

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

\(\Big \downarrow \) 6725

\(\displaystyle -a^2 c^2 \int \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{n/2} x^4d\frac {1}{x}\)

\(\Big \downarrow \) 141

\(\displaystyle \frac {16 c^2 \left (1-\frac {1}{a x}\right )^{3-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-6}{2}} \operatorname {Hypergeometric2F1}\left (4,3-\frac {n}{2},4-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (6-n)}\)

Input:

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

Output:

(16*c^2*(1 - 1/(a*x))^(3 - n/2)*(1 + 1/(a*x))^((-6 + n)/2)*Hypergeometric2 
F1[4, 3 - n/2, 4 - n/2, (a - x^(-1))/(a + x^(-1))])/(a*(6 - 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

c**2*(Integral(-2*a*x*exp(n*acoth(a*x)), x) + Integral(a**2*x**2*exp(n*aco 
th(a*x)), x) + Integral(exp(n*acoth(a*x)), x))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

c**2*(int(e**(acoth(a*x)*n),x) + int(e**(acoth(a*x)*n)*x**2,x)*a**2 - 2*in 
t(e**(acoth(a*x)*n)*x,x)*a)