\(\int e^{n \coth ^{-1}(a x)} (c-\frac {c}{a x})^p \, dx\) [549]

Optimal result

Integrand size = 22, antiderivative size = 110 \[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=-\frac {2^{1-\frac {n}{2}+p} \left (1-\frac {1}{a x}\right )^{-p} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} \left (c-\frac {c}{a x}\right )^p \operatorname {AppellF1}\left (\frac {2+n}{2},\frac {1}{2} (n-2 p),2,\frac {4+n}{2},\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (2+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 110, 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 = {6317, 6314, 141} \[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=-\frac {2^{-\frac {n}{2}+p+1} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-p} \left (c-\frac {c}{a x}\right )^p \operatorname {AppellF1}\left (\frac {n+2}{2},\frac {1}{2} (n-2 p),2,\frac {n+4}{2},\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (n+2)} \]

[In]

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

[Out]

-((2^(1 - n/2 + p)*(1 + 1/(a*x))^((2 + n)/2)*(c - c/(a*x))^p*AppellF1[(2 + n)/2, (n - 2*p)/2, 2, (4 + n)/2, (a
+ x^(-1))/(2*a), 1 + 1/(a*x)])/(a*(2 + n)*(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^{n \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 )^{-\frac {n}{2}+p} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx,x,\frac {1}{x}\right )\right ) \\ & = -\frac {2^{1-\frac {n}{2}+p} \left (1-\frac {1}{a x}\right )^{-p} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} \left (c-\frac {c}{a x}\right )^p \operatorname {AppellF1}\left (\frac {2+n}{2},\frac {1}{2} (n-2 p),2,\frac {4+n}{2},\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (2+n)} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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