### 3.739 $$\int e^{n \coth ^{-1}(a x)} \, dx$$

Optimal. Leaf size=78 $\frac{4 \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \left (\frac{1}{a x}+1\right )^{\frac{n-2}{2}} \text{Hypergeometric2F1}\left (2,1-\frac{n}{2},2-\frac{n}{2},\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a (2-n)}$

[Out]

(4*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((-2 + n)/2)*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (a - x^(-1))/(a +
x^(-1))])/(a*(2 - n))

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Rubi [A]  time = 0.0229457, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {6170, 131} $\frac{4 \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \left (\frac{1}{a x}+1\right )^{\frac{n-2}{2}} \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a (2-n)}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(n*ArcCoth[a*x]),x]

[Out]

(4*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((-2 + n)/2)*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (a - x^(-1))/(a +
x^(-1))])/(a*(2 - n))

Rule 6170

Int[E^(ArcCoth[(a_.)*(x_)]*(n_)), x_Symbol] :> -Subst[Int[(1 + x/a)^(n/2)/(x^2*(1 - x/a)^(n/2)), x], x, 1/x] /
; FreeQ[{a, n}, x] &&  !IntegerQ[n]

Rule 131

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*c -
a*d)^n*(a + b*x)^(m + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))
)])/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0]

Rubi steps

\begin{align*} \int e^{n \coth ^{-1}(a x)} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{-n/2} \left (1+\frac{x}{a}\right )^{n/2}}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=\frac{4 \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-2+n)} \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a (2-n)}\\ \end{align*}

Mathematica [A]  time = 0.175268, size = 82, normalized size = 1.05 $\frac{e^{n \coth ^{-1}(a x)} \left (n e^{2 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n}{2}+1,\frac{n}{2}+2,e^{2 \coth ^{-1}(a x)}\right )+(n+2) \left (\text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n}{2}+1,e^{2 \coth ^{-1}(a x)}\right )+a x\right )\right )}{a (n+2)}$

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(n*ArcCoth[a*x]),x]

[Out]

(E^(n*ArcCoth[a*x])*(E^(2*ArcCoth[a*x])*n*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (2 + n)
*(a*x + Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])])))/(a*(2 + n))

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\rm arccoth} \left (ax\right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arccoth(a*x)),x)

[Out]

int(exp(n*arccoth(a*x)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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)),x, algorithm="maxima")

[Out]

integrate(((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}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arccoth(a*x)),x, algorithm="fricas")

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*acoth(a*x)),x)

[Out]

Integral(exp(n*acoth(a*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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)),x, algorithm="giac")

[Out]

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