3.8.39 \(\int e^{n \coth ^{-1}(a x)} \, dx\) [739]

Optimal. Leaf size=78 \[ \frac {4 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (2-n)} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6305, 133} \begin {gather*} \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)} \end {gather*}

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 133

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)/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2,
(-(d*e - c*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) &&  !ILtQ[m, 0]

Rule 6305

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]

Rubi steps

\begin {align*} \int e^{n \coth ^{-1}(a x)} \, dx &=-\text {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*}

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Mathematica [A]
time = 0.03, size = 82, normalized size = 1.05 \begin {gather*} \frac {e^{n \coth ^{-1}(a x)} \left (e^{2 \coth ^{-1}(a x)} n \, _2F_1\left (1,1+\frac {n}{2};2+\frac {n}{2};e^{2 \coth ^{-1}(a x)}\right )+(2+n) \left (a x+\, _2F_1\left (1,\frac {n}{2};1+\frac {n}{2};e^{2 \coth ^{-1}(a x)}\right )\right )\right )}{a (2+n)} \end {gather*}

Antiderivative was successfully verified.

[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 = 0.01, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int e^{n \operatorname {acoth}{\left (a x \right )}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(exp(n*acoth(a*x)), x)

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