\(\int e^{n \coth ^{-1}(a x)} x^m \, dx\) [148]

Optimal result

Integrand size = 12, antiderivative size = 45 \[ \int e^{n \coth ^{-1}(a x)} x^m \, dx=\frac {x^{1+m} \operatorname {AppellF1}\left (-1-m,\frac {n}{2},-\frac {n}{2},-m,\frac {1}{a x},-\frac {1}{a x}\right )}{1+m} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6308, 138} \[ \int e^{n \coth ^{-1}(a x)} x^m \, dx=\frac {x^{m+1} \operatorname {AppellF1}\left (-m-1,\frac {n}{2},-\frac {n}{2},-m,\frac {1}{a x},-\frac {1}{a x}\right )}{m+1} \]

[In]

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

[Out]

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

Rule 138

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[c^n*e^p*((b*x)^(m +
1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rule 6308

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

Rubi steps \begin{align*} \text {integral}& = -\left (\left (\left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int x^{-2-m} \left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \frac {x^{1+m} \operatorname {AppellF1}\left (-1-m,\frac {n}{2},-\frac {n}{2},-m,\frac {1}{a x},-\frac {1}{a x}\right )}{1+m} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

Integrate[E^(n*ArcCoth[a*x])*x^m, x]

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(x**m*exp(n*acoth(a*x)), x)

Maxima [F]

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

[In]

integrate(exp(n*arccoth(a*x))*x^m,x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate(exp(n*arccoth(a*x))*x^m,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^m*exp(n*acoth(a*x)), x)