\(\int \frac {e^{n \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx\) [547]

Optimal result

Integrand size = 24, antiderivative size = 111 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=-\frac {2^{\frac {1}{2}-\frac {n}{2}} \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} \operatorname {AppellF1}\left (\frac {2+n}{2},\frac {1+n}{2},2,\frac {4+n}{2},\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (2+n) \sqrt {c-\frac {c}{a x}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6317, 6314, 141} \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=-\frac {2^{\frac {1}{2}-\frac {n}{2}} \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \operatorname {AppellF1}\left (\frac {n+2}{2},\frac {n+1}{2},2,\frac {n+4}{2},\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (n+2) \sqrt {c-\frac {c}{a x}}} \]

[In]

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

[Out]

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

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}& = \frac {\sqrt {1-\frac {1}{a x}} \int \frac {e^{n \coth ^{-1}(a x)}}{\sqrt {1-\frac {1}{a x}}} \, dx}{\sqrt {c-\frac {c}{a x}}} \\ & = -\frac {\sqrt {1-\frac {1}{a x}} \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-\frac {1}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx,x,\frac {1}{x}\right )}{\sqrt {c-\frac {c}{a x}}} \\ & = -\frac {2^{\frac {1}{2}-\frac {n}{2}} \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} \operatorname {AppellF1}\left (\frac {2+n}{2},\frac {1+n}{2},2,\frac {4+n}{2},\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (2+n) \sqrt {c-\frac {c}{a x}}} \\ \end{align*}

Mathematica [F(-1)]

Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\text {\$Aborted} \]

[In]

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

[Out]

$Aborted

Maple [F]

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

[In]

int(exp(n*arccoth(a*x))/(c-c/a/x)^(1/2),x)

[Out]

int(exp(n*arccoth(a*x))/(c-c/a/x)^(1/2),x)

Fricas [F]

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

[In]

integrate(exp(n*arccoth(a*x))/(c-c/a/x)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate(exp(n*acoth(a*x))/(c-c/a/x)**(1/2),x)

[Out]

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

Maxima [F]

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

[In]

integrate(exp(n*arccoth(a*x))/(c-c/a/x)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate(exp(n*arccoth(a*x))/(c-c/a/x)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int(exp(n*acoth(a*x))/(c - c/(a*x))^(1/2),x)

[Out]

int(exp(n*acoth(a*x))/(c - c/(a*x))^(1/2), x)