\(\int e^{n \coth ^{-1}(a x)} (c-a c x)^2 \, dx\) [366]

Optimal result

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

[Out]

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)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6310, 6315, 133} \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {16 c^2 \left (1-\frac {1}{a x}\right )^{3-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-6}{2}} \operatorname {Hypergeometric2F1}\left (4,3-\frac {n}{2},4-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (6-n)} \]

[In]

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

[Out]

(16*c^2*(1 - 1/(a*x))^(3 - n/2)*(1 + 1/(a*x))^((-6 + n)/2)*Hypergeometric2F1[4, 3 - n/2, 4 - n/2, (a - x^(-1))
/(a + x^(-1))])/(a*(6 - n))

Rule 133

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> 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] ||  !SumSimplerQ[p, 1]) &&  !ILtQ[m, 0]

Rule 6310

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

Rule 6315

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

Rubi steps \begin{align*} \text {integral}& = \left (a^2 c^2\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^2 x^2 \, dx \\ & = -\left (\left (a^2 c^2\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{2-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{x^4} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \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)} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.18 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.78 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {c^2 e^{n \coth ^{-1}(a x)} \left (e^{2 \coth ^{-1}(a x)} n \left (8-6 n+n^2\right ) \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )+(2+n) \left (6+6 a x+a n^2 x-6 a^2 x^2+2 a^3 x^3+n \left (-1-6 a x+a^2 x^2\right )+\left (8-6 n+n^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )\right )}{6 a (2+n)} \]

[In]

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

[Out]

(c^2*E^(n*ArcCoth[a*x])*(E^(2*ArcCoth[a*x])*n*(8 - 6*n + n^2)*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcC
oth[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)
*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])])))/(6*a*(2 + n))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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