### 3.547 $$\int \frac{e^{n \coth ^{-1}(a x)}}{\sqrt{c-\frac{c}{a x}}} \, dx$$

Optimal. Leaf size=111 $-\frac{2^{\frac{1}{2}-\frac{n}{2}} \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} F_1\left (\frac{n+2}{2};\frac{n+1}{2},2;\frac{n+4}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right )}{a (n+2) \sqrt{c-\frac{c}{a x}}}$

[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)]))

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Rubi [A]  time = 0.149866, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {6182, 6179, 136} $-\frac{2^{\frac{1}{2}-\frac{n}{2}} \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} F_1\left (\frac{n+2}{2};\frac{n+1}{2},2;\frac{n+4}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right )}{a (n+2) \sqrt{c-\frac{c}{a x}}}$

Antiderivative was successfully veriﬁed.

[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 6182

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])

Rule 6179

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 136

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)*AppellF1[m + 1, -n, -p, m + 2, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f
))])/(b^(p + 1)*(m + 1)*(b/(b*c - a*d))^n), 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])

Rubi steps

\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{\sqrt{c-\frac{c}{a x}}} \, dx &=\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}} \operatorname{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}} F_1\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]  time = 180.006, size = 0, normalized size = 0. $\text{\Aborted}$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

\$Aborted

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Maple [F]  time = 0.172, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\rm arccoth} \left (ax\right )}}{\frac{1}{\sqrt{c-{\frac{c}{ax}}}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{\sqrt{c - \frac{c}{a x}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a x \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n} \sqrt{\frac{a c x - c}{a x}}}{a c x - c}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{n \operatorname{acoth}{\left (a x \right )}}}{\sqrt{- c \left (-1 + \frac{1}{a x}\right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{\sqrt{c - \frac{c}{a x}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)