\(\int e^{-3 \coth ^{-1}(a x)} (c-a c x)^p \, dx\) [216]

Optimal result

Integrand size = 18, antiderivative size = 94 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {\left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{-\frac {3}{2}-p} \left (1-\frac {1}{a x}\right )^{3/2} x (c-a c x)^p \operatorname {Hypergeometric2F1}\left (-\frac {3}{2}-p,-1-p,-p,\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{(1+p) \sqrt {1+\frac {1}{a x}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 94, 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 = {6311, 6316, 134} \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {x \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{-p-\frac {3}{2}} (c-a c x)^p \operatorname {Hypergeometric2F1}\left (-p-\frac {3}{2},-p-1,-p,\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{(p+1) \sqrt {\frac {1}{a x}+1}} \]

[In]

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

[Out]

(((a - x^(-1))/(a + x^(-1)))^(-3/2 - p)*(1 - 1/(a*x))^(3/2)*x*(c - a*c*x)^p*Hypergeometric2F1[-3/2 - p, -1 - p
, -p, 2/((a + x^(-1))*x)])/((1 + p)*Sqrt[1 + 1/(a*x)])

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 6311

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

Rule 6316

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_), x_Symbol] :> Dist[(-c^p)*x^m*(1/x)^m, S
ubst[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, m,
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 (\left (1-\frac {1}{a x}\right )^{-p} x^{-p} (c-a c x)^p\right ) \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^p x^p \, dx \\ & = -\left (\left (\left (1-\frac {1}{a x}\right )^{-p} \left (\frac {1}{x}\right )^p (c-a c x)^p\right ) \text {Subst}\left (\int \frac {x^{-2-p} \left (1-\frac {x}{a}\right )^{\frac {3}{2}+p}}{\left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \frac {\left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{-\frac {3}{2}-p} \left (1-\frac {1}{a x}\right )^{3/2} x (c-a c x)^p \operatorname {Hypergeometric2F1}\left (-\frac {3}{2}-p,-1-p,-p,\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{(1+p) \sqrt {1+\frac {1}{a x}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.02 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {\sqrt {1-\frac {1}{a x}} \left (\frac {-1+a x}{1+a x}\right )^{-\frac {1}{2}-p} (1+a x) (c-a c x)^p \operatorname {Hypergeometric2F1}\left (-\frac {3}{2}-p,-1-p,-p,\frac {2}{1+a x}\right )}{a (1+p) \sqrt {1+\frac {1}{a x}}} \]

[In]

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

[Out]

(Sqrt[1 - 1/(a*x)]*((-1 + a*x)/(1 + a*x))^(-1/2 - p)*(1 + a*x)*(c - a*c*x)^p*Hypergeometric2F1[-3/2 - p, -1 -
p, -p, 2/(1 + a*x)])/(a*(1 + p)*Sqrt[1 + 1/(a*x)])

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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