\(\int e^{-\coth ^{-1}(a x)} x^m \, dx\) [136]

Optimal result

Integrand size = 12, antiderivative size = 75 \[ \int e^{-\coth ^{-1}(a x)} x^m \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-1-m),\frac {1-m}{2},\frac {1}{a^2 x^2}\right )}{1+m}-\frac {x^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {m}{2},1-\frac {m}{2},\frac {1}{a^2 x^2}\right )}{a m} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6307, 822, 371} \[ \int e^{-\coth ^{-1}(a x)} x^m \, dx=\frac {x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-m-1),\frac {1-m}{2},\frac {1}{a^2 x^2}\right )}{m+1}-\frac {x^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {m}{2},1-\frac {m}{2},\frac {1}{a^2 x^2}\right )}{a m} \]

[In]

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

[Out]

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

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 822

Int[((e_.)*(x_))^(m_)*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[f, Int[(e*x)^m*(a + c*
x^2)^p, x], x] + Dist[g/e, Int[(e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, p}, x] &&  !Ration
alQ[m] &&  !IGtQ[p, 0]

Rule 6307

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(x_)^(m_), x_Symbol] :> Dist[(-x^m)*(1/x)^m, Subst[Int[(1 + x/a)^((n + 1)/2)
/(x^(m + 2)*(1 - x/a)^((n - 1)/2)*Sqrt[1 - x^2/a^2]), x], x, 1/x], x] /; FreeQ[{a, m}, x] && IntegerQ[(n - 1)/
2] &&  !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} \left (1-\frac {x}{a}\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\right ) \\ & = -\left (\left (\left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int \frac {x^{-2-m}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\right )+\frac {\left (\left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int \frac {x^{-1-m}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a} \\ & = \frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-1-m),\frac {1-m}{2},\frac {1}{a^2 x^2}\right )}{1+m}-\frac {x^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {m}{2},1-\frac {m}{2},\frac {1}{a^2 x^2}\right )}{a m} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.30 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.53 \[ \int e^{-\coth ^{-1}(a x)} x^m \, dx=x^{1+m} \left (-\frac {\sqrt {1-\frac {1}{a^2 x^2}} \sqrt {\frac {-1+a x}{a^2}} \operatorname {AppellF1}\left (m,-\frac {1}{2},\frac {1}{2},1+m,a x,-a x\right )}{m \sqrt {1-a x} \sqrt {-\frac {1}{a^2}+x^2}}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2},\frac {1}{a^2 x^2}\right )}{1+m}\right ) \]

[In]

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

[Out]

x^(1 + m)*(-((Sqrt[1 - 1/(a^2*x^2)]*Sqrt[(-1 + a*x)/a^2]*AppellF1[m, -1/2, 1/2, 1 + m, a*x, -(a*x)])/(m*Sqrt[1
- a*x]*Sqrt[-a^(-2) + x^2])) + Hypergeometric2F1[-1/2, -1/2 - m/2, 1/2 - m/2, 1/(a^2*x^2)]/(1 + m))

Maple [F]

\[\int x^{m} \sqrt {\frac {a x -1}{a x +1}}d x\]

[In]

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

[Out]

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

Fricas [F]

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

[In]

integrate(x^m*((a*x-1)/(a*x+1))^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate(x^m*((a*x-1)/(a*x+1))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate(x^m*((a*x-1)/(a*x+1))^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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