\(\int e^{-\frac {1}{2} \coth ^{-1}(a x)} x^m \, dx\) [142]

Optimal result

Integrand size = 14, antiderivative size = 41 \[ \int e^{-\frac {1}{2} \coth ^{-1}(a x)} x^m \, dx=\frac {x^{1+m} \operatorname {AppellF1}\left (-1-m,-\frac {1}{4},\frac {1}{4},-m,\frac {1}{a x},-\frac {1}{a x}\right )}{1+m} \]

[Out]

x^(1+m)*AppellF1(-1-m,-1/4,1/4,-m,1/a/x,-1/a/x)/(1+m)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6308, 138} \[ \int e^{-\frac {1}{2} \coth ^{-1}(a x)} x^m \, dx=\frac {x^{m+1} \operatorname {AppellF1}\left (-m-1,-\frac {1}{4},\frac {1}{4},-m,\frac {1}{a x},-\frac {1}{a x}\right )}{m+1} \]

[In]

Int[x^m/E^(ArcCoth[a*x]/2),x]

[Out]

(x^(1 + m)*AppellF1[-1 - m, -1/4, 1/4, -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 \frac {x^{-2-m} \sqrt [4]{1-\frac {x}{a}}}{\sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \frac {x^{1+m} \operatorname {AppellF1}\left (-1-m,-\frac {1}{4},\frac {1}{4},-m,\frac {1}{a x},-\frac {1}{a x}\right )}{1+m} \\ \end{align*}

Mathematica [F]

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

[In]

Integrate[x^m/E^(ArcCoth[a*x]/2),x]

[Out]

Integrate[x^m/E^(ArcCoth[a*x]/2), x]

Maple [F]

\[\int x^{m} \left (\frac {a x -1}{a x +1}\right )^{\frac {1}{4}}d x\]

[In]

int(x^m*((a*x-1)/(a*x+1))^(1/4),x)

[Out]

int(x^m*((a*x-1)/(a*x+1))^(1/4),x)

Fricas [F]

\[ \int e^{-\frac {1}{2} \coth ^{-1}(a x)} x^m \, dx=\int { x^{m} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} \,d x } \]

[In]

integrate(x^m*((a*x-1)/(a*x+1))^(1/4),x, algorithm="fricas")

[Out]

integral(x^m*((a*x - 1)/(a*x + 1))^(1/4), x)

Sympy [F]

\[ \int e^{-\frac {1}{2} \coth ^{-1}(a x)} x^m \, dx=\int x^{m} \sqrt [4]{\frac {a x - 1}{a x + 1}}\, dx \]

[In]

integrate(x**m*((a*x-1)/(a*x+1))**(1/4),x)

[Out]

Integral(x**m*((a*x - 1)/(a*x + 1))**(1/4), x)

Maxima [F]

\[ \int e^{-\frac {1}{2} \coth ^{-1}(a x)} x^m \, dx=\int { x^{m} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} \,d x } \]

[In]

integrate(x^m*((a*x-1)/(a*x+1))^(1/4),x, algorithm="maxima")

[Out]

integrate(x^m*((a*x - 1)/(a*x + 1))^(1/4), x)

Giac [F]

\[ \int e^{-\frac {1}{2} \coth ^{-1}(a x)} x^m \, dx=\int { x^{m} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} \,d x } \]

[In]

integrate(x^m*((a*x-1)/(a*x+1))^(1/4),x, algorithm="giac")

[Out]

integrate(x^m*((a*x - 1)/(a*x + 1))^(1/4), x)

Mupad [F(-1)]

Timed out. \[ \int e^{-\frac {1}{2} \coth ^{-1}(a x)} x^m \, dx=\int x^m\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4} \,d x \]

[In]

int(x^m*((a*x - 1)/(a*x + 1))^(1/4),x)

[Out]

int(x^m*((a*x - 1)/(a*x + 1))^(1/4), x)