### 3.147 $$\int e^{\frac{1}{4} \coth ^{-1}(a x)} x^m \, dx$$

Optimal. Leaf size=41 $\frac{x^{m+1} F_1\left (-m-1;\frac{1}{8},-\frac{1}{8};-m;\frac{1}{a x},-\frac{1}{a x}\right )}{m+1}$

[Out]

(x^(1 + m)*AppellF1[-1 - m, 1/8, -1/8, -m, 1/(a*x), -(1/(a*x))])/(1 + m)

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Rubi [A]  time = 0.0352318, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.143, Rules used = {6173, 133} $\frac{x^{m+1} F_1\left (-m-1;\frac{1}{8},-\frac{1}{8};-m;\frac{1}{a x},-\frac{1}{a x}\right )}{m+1}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(ArcCoth[a*x]/4)*x^m,x]

[Out]

(x^(1 + m)*AppellF1[-1 - m, 1/8, -1/8, -m, 1/(a*x), -(1/(a*x))])/(1 + m)

Rule 6173

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]

Rule 133

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

Rubi steps

\begin{align*} \int e^{\frac{1}{4} \coth ^{-1}(a x)} x^m \, dx &=-\left (\left (\left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int \frac{x^{-2-m} \sqrt [8]{1+\frac{x}{a}}}{\sqrt [8]{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{x^{1+m} F_1\left (-1-m;\frac{1}{8},-\frac{1}{8};-m;\frac{1}{a x},-\frac{1}{a x}\right )}{1+m}\\ \end{align*}

Mathematica [F]  time = 0.275806, size = 0, normalized size = 0. $\int e^{\frac{1}{4} \coth ^{-1}(a x)} x^m \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Integrate[E^(ArcCoth[a*x]/4)*x^m,x]

[Out]

Integrate[E^(ArcCoth[a*x]/4)*x^m, x]

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Maple [F]  time = 0.131, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m}{\frac{1}{\sqrt [8]{{\frac{ax-1}{ax+1}}}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+1))^(1/8)*x^m,x)

[Out]

int(1/((a*x-1)/(a*x+1))^(1/8)*x^m,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/8)*x^m,x, algorithm="maxima")

[Out]

integrate(x^m/((a*x - 1)/(a*x + 1))^(1/8), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a x + 1\right )} x^{m} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}}}{a x - 1}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/8)*x^m,x, algorithm="fricas")

[Out]

integral((a*x + 1)*x^m*((a*x - 1)/(a*x + 1))^(7/8)/(a*x - 1), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))**(1/8)*x**m,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/8)*x^m,x, algorithm="giac")

[Out]

integrate(x^m/((a*x - 1)/(a*x + 1))^(1/8), x)