### 3.931 $$\int e^{n \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} \, dx$$

Optimal. Leaf size=295 $\frac{2 n \sqrt{c-\frac{c}{a^2 x^2}} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \text{Hypergeometric2F1}\left (1,\frac{1-n}{2},\frac{3-n}{2},\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a (1-n) \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{2^{\frac{n+1}{2}} \sqrt{c-\frac{c}{a^2 x^2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \text{Hypergeometric2F1}\left (\frac{1-n}{2},\frac{1-n}{2},\frac{3-n}{2},\frac{a-\frac{1}{x}}{2 a}\right )}{a (1-n) \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left (\frac{1}{a x}+1\right )^{\frac{n+1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}$

[Out]

(Sqrt[c - c/(a^2*x^2)]*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1 + n)/2)*x)/Sqrt[1 - 1/(a^2*x^2)] + (2*n*Sqr
t[c - c/(a^2*x^2)]*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((-1 + n)/2)*Hypergeometric2F1[1, (1 - n)/2, (3 - n
)/2, (a - x^(-1))/(a + x^(-1))])/(a*(1 - n)*Sqrt[1 - 1/(a^2*x^2)]) - (2^((1 + n)/2)*Sqrt[c - c/(a^2*x^2)]*(1 -
1/(a*x))^((1 - n)/2)*Hypergeometric2F1[(1 - n)/2, (1 - n)/2, (3 - n)/2, (a - x^(-1))/(2*a)])/(a*(1 - n)*Sqrt[
1 - 1/(a^2*x^2)])

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Rubi [C]  time = 0.15462, antiderivative size = 111, normalized size of antiderivative = 0.38, 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 = {6197, 6194, 136} $-\frac{2^{\frac{3}{2}-\frac{n}{2}} \sqrt{c-\frac{c}{a^2 x^2}} \left (\frac{1}{a x}+1\right )^{\frac{n+3}{2}} F_1\left (\frac{n+3}{2};\frac{n-1}{2},2;\frac{n+5}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right )}{a (n+3) \sqrt{1-\frac{1}{a^2 x^2}}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((2^(3/2 - n/2)*Sqrt[c - c/(a^2*x^2)]*(1 + 1/(a*x))^((3 + n)/2)*AppellF1[(3 + n)/2, (-1 + n)/2, 2, (5 + n)/2,
(a + x^(-1))/(2*a), 1 + 1/(a*x)])/(a*(3 + n)*Sqrt[1 - 1/(a^2*x^2)]))

Rule 6197

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbol] :> Dist[(c^IntPart[p]*(c + d/x^2
)^FracPart[p])/(1 - 1/(a^2*x^2))^FracPart[p], Int[u*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a
, c, d, n, p}, x] && EqQ[c + a^2*d, 0] &&  !IntegerQ[n/2] &&  !(IntegerQ[p] || GtQ[c, 0])

Rule 6194

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> -Dist[c^p, Subst[Int[((1 - x/a)^(p
- n/2)*(1 + x/a)^(p + n/2))/x^2, x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] &&  !Integ
erQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) &&  !IntegersQ[2*p, p + n/2]

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 e^{n \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} \, dx &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} \int e^{n \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a^2 x^2}} \, dx}{\sqrt{1-\frac{1}{a^2 x^2}}}\\ &=-\frac{\sqrt{c-\frac{c}{a^2 x^2}} \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{\frac{1}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{\frac{1}{2}+\frac{n}{2}}}{x^2} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a^2 x^2}}}\\ &=-\frac{2^{\frac{3}{2}-\frac{n}{2}} \sqrt{c-\frac{c}{a^2 x^2}} \left (1+\frac{1}{a x}\right )^{\frac{3+n}{2}} F_1\left (\frac{3+n}{2};\frac{1}{2} (-1+n),2;\frac{5+n}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right )}{a (3+n) \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}

Mathematica [A]  time = 0.463257, size = 146, normalized size = 0.49 $\frac{a x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a^2 x^2}} e^{n \coth ^{-1}(a x)} \left (2 e^{\coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+3}{2},-e^{2 \coth ^{-1}(a x)}\right )+2 n e^{\coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+3}{2},e^{2 \coth ^{-1}(a x)}\right )+a (n+1) x \sqrt{1-\frac{1}{a^2 x^2}}\right )}{(n+1) \left (a^2 x^2-1\right )}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*E^(n*ArcCoth[a*x])*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a^2*x^2)]*x^2*(a*(1 + n)*Sqrt[1 - 1/(a^2*x^2)]*x + 2*E
^ArcCoth[a*x]*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -E^(2*ArcCoth[a*x])] + 2*E^ArcCoth[a*x]*n*Hypergeomet
ric2F1[1, (1 + n)/2, (3 + n)/2, E^(2*ArcCoth[a*x])]))/((1 + n)*(-1 + a^2*x^2))

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Maple [F]  time = 0.181, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\rm arccoth} \left (ax\right )}}\sqrt{c-{\frac{c}{{a}^{2}{x}^{2}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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