\(\int e^{-\coth ^{-1}(a x)} (c-a c x)^p \, dx\) [197]

Optimal result

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

[Out]

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

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

[In]

Int[(c - a*c*x)^p/E^ArcCoth[a*x],x]

[Out]

(((a - x^(-1))/(a + x^(-1)))^(-1/2 - p)*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x*(c - a*c*x)^p*Hypergeometric2F1[
-1 - p, -1/2 - p, -p, 2/((a + x^(-1))*x)])/(1 + 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 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^{-\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 {1}{2}+p}}{\sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \frac {\left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{-\frac {1}{2}-p} \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x (c-a c x)^p \operatorname {Hypergeometric2F1}\left (-1-p,-\frac {1}{2}-p,-p,\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{1+p} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.81 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (\frac {-1+a x}{1+a x}\right )^{-\frac {1}{2}-p} (c-a c x)^p \operatorname {Hypergeometric2F1}\left (-1-p,-\frac {1}{2}-p,-p,\frac {2}{1+a x}\right )}{1+p} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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