\(\int e^{-3 \coth ^{-1}(a x)} (c-a^2 c x^2)^p \, dx\) [771]

Optimal result

Integrand size = 22, antiderivative size = 118 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{-\frac {3}{2}-p} \left (1-\frac {1}{a x}\right )^{\frac {3}{2}+p} \left (1+\frac {1}{a x}\right )^{-\frac {1}{2}+p} x \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-\frac {3}{2}-p,-2 p,\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{1+2 p} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6327, 6331, 134} \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {x \left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{-p-\frac {3}{2}} \left (1-\frac {1}{a x}\right )^{p+\frac {3}{2}} \left (\frac {1}{a x}+1\right )^{p-\frac {1}{2}} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-2 p-1,-p-\frac {3}{2},-2 p,\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{2 p+1} \]

[In]

Int[(c - a^2*c*x^2)^p/E^(3*ArcCoth[a*x]),x]

[Out]

(((a - x^(-1))/(a + x^(-1)))^(-3/2 - p)*(1 - 1/(a*x))^(3/2 + p)*(1 + 1/(a*x))^(-1/2 + p)*x*(c - a^2*c*x^2)^p*H
ypergeometric2F1[-1 - 2*p, -3/2 - p, -2*p, 2/((a + x^(-1))*x)])/((1 + 2*p)*(1 - 1/(a^2*x^2))^p)

Rule 134

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x
)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((b*e - a*f)*(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c
*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))])/((b*e - a*f)*((c + d*x)/((b*c - a*d)*(e + f*x))))^n, x] /; FreeQ[{a
, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 0] &&  !IntegerQ[n]

Rule 6327

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

Rule 6331

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

Rubi steps \begin{align*} \text {integral}& = \left (\left (1-\frac {1}{a^2 x^2}\right )^{-p} x^{-2 p} \left (c-a^2 c x^2\right )^p\right ) \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^p x^{2 p} \, dx \\ & = -\left (\left (\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (\frac {1}{x}\right )^{2 p} \left (c-a^2 c x^2\right )^p\right ) \text {Subst}\left (\int x^{-2-2 p} \left (1-\frac {x}{a}\right )^{\frac {3}{2}+p} \left (1+\frac {x}{a}\right )^{-\frac {3}{2}+p} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \frac {\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{-\frac {3}{2}-p} \left (1-\frac {1}{a x}\right )^{\frac {3}{2}+p} \left (1+\frac {1}{a x}\right )^{-\frac {1}{2}+p} x \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-\frac {3}{2}-p,-2 p,\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{1+2 p} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.01 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {4^{1+p} e^{-\coth ^{-1}(a x)} \left (1-e^{2 \coth ^{-1}(a x)}\right )^{2 p} \left (\frac {e^{\coth ^{-1}(a x)}}{-1+e^{2 \coth ^{-1}(a x)}}\right )^{2 p} \left (a \sqrt {1-\frac {1}{a^2 x^2}} x\right )^{-2 p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}+p,2+2 p,\frac {1}{2}+p,e^{2 \coth ^{-1}(a x)}\right )}{a-2 a p} \]

[In]

Integrate[(c - a^2*c*x^2)^p/E^(3*ArcCoth[a*x]),x]

[Out]

(4^(1 + p)*(1 - E^(2*ArcCoth[a*x]))^(2*p)*(E^ArcCoth[a*x]/(-1 + E^(2*ArcCoth[a*x])))^(2*p)*(c - a^2*c*x^2)^p*H
ypergeometric2F1[-1/2 + p, 2 + 2*p, 1/2 + p, E^(2*ArcCoth[a*x])])/(E^ArcCoth[a*x]*(a - 2*a*p)*(a*Sqrt[1 - 1/(a
^2*x^2)]*x)^(2*p))

Maple [F]

\[\int \left (-a^{2} c \,x^{2}+c \right )^{p} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}d x\]

[In]

int((-a^2*c*x^2+c)^p*((a*x-1)/(a*x+1))^(3/2),x)

[Out]

int((-a^2*c*x^2+c)^p*((a*x-1)/(a*x+1))^(3/2),x)

Fricas [F]

\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{p} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]

[In]

integrate((-a^2*c*x^2+c)^p*((a*x-1)/(a*x+1))^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\text {Timed out} \]

[In]

integrate((-a**2*c*x**2+c)**p*((a*x-1)/(a*x+1))**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{p} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]

[In]

integrate((-a^2*c*x^2+c)^p*((a*x-1)/(a*x+1))^(3/2),x, algorithm="maxima")

[Out]

integrate((-a^2*c*x^2 + c)^p*((a*x - 1)/(a*x + 1))^(3/2), x)

Giac [F]

\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{p} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]

[In]

integrate((-a^2*c*x^2+c)^p*((a*x-1)/(a*x+1))^(3/2),x, algorithm="giac")

[Out]

integrate((-a^2*c*x^2 + c)^p*((a*x - 1)/(a*x + 1))^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int {\left (c-a^2\,c\,x^2\right )}^p\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \]

[In]

int((c - a^2*c*x^2)^p*((a*x - 1)/(a*x + 1))^(3/2),x)

[Out]

int((c - a^2*c*x^2)^p*((a*x - 1)/(a*x + 1))^(3/2), x)