\(\int e^{-2 p \coth ^{-1}(a x)} (c-\frac {c}{a x})^p \, dx\) [551]

Optimal result

Integrand size = 23, antiderivative size = 93 \[ \int e^{-2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=-\frac {4^p \left (1-\frac {1}{a x}\right )^{-p} \left (1+\frac {1}{a x}\right )^{1-p} \left (c-\frac {c}{a x}\right )^p \operatorname {AppellF1}\left (1-p,-2 p,2,2-p,\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (1-p)} \]

[Out]

-4^p*(1+1/a/x)^(1-p)*(c-c/a/x)^p*AppellF1(1-p,-2*p,2,2-p,1/2*(a+1/x)/a,1+1/a/x)/a/(1-p)/((1-1/a/x)^p)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6317, 6314, 141} \[ \int e^{-2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=-\frac {4^p \left (1-\frac {1}{a x}\right )^{-p} \left (\frac {1}{a x}+1\right )^{1-p} \operatorname {AppellF1}\left (1-p,-2 p,2,2-p,\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right ) \left (c-\frac {c}{a x}\right )^p}{a (1-p)} \]

[In]

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

[Out]

-((4^p*(1 + 1/(a*x))^(1 - p)*(c - c/(a*x))^p*AppellF1[1 - p, -2*p, 2, 2 - p, (a + x^(-1))/(2*a), 1 + 1/(a*x)])
/(a*(1 - p)*(1 - 1/(a*x))^p))

Rule 141

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(b*e - a*f
)^p*((a + b*x)^(m + 1)/(b^(p + 1)*(m + 1)*(b/(b*c - a*d))^n))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(
b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] && IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !(GtQ[d/(d*a - c*b), 0] && SimplerQ[c + d*x, a + b*x])

Rule 6314

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> Dist[-c^p, Subst[Int[(1 + d*(x/c))^p
*((1 + x/a)^(n/2)/(x^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])

Rule 6317

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Dist[(c + d/x)^p/(1 + d/(c*x))^
p, Int[u*(1 + d/(c*x))^p*E^(n*ArcCoth[a*x]), 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])

Rubi steps \begin{align*} \text {integral}& = \left (\left (1-\frac {1}{a x}\right )^{-p} \left (c-\frac {c}{a x}\right )^p\right ) \int e^{-2 p \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^p \, dx \\ & = -\left (\left (\left (1-\frac {1}{a x}\right )^{-p} \left (c-\frac {c}{a x}\right )^p\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{2 p} \left (1+\frac {x}{a}\right )^{-p}}{x^2} \, dx,x,\frac {1}{x}\right )\right ) \\ & = -\frac {4^p \left (1-\frac {1}{a x}\right )^{-p} \left (1+\frac {1}{a x}\right )^{1-p} \left (c-\frac {c}{a x}\right )^p \operatorname {AppellF1}\left (1-p,-2 p,2,2-p,\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (1-p)} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

int((c-c/a/x)^p/exp(2*p*arccoth(a*x)),x)

[Out]

int((c-c/a/x)^p/exp(2*p*arccoth(a*x)),x)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

integrate((c-c/a/x)**p/exp(2*p*acoth(a*x)),x)

[Out]

Integral((-c*(-1 + 1/(a*x)))**p*exp(-2*p*acoth(a*x)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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