### 3.133 $$\int e^{3 \coth ^{-1}(a x)} x^m \, dx$$

Optimal. Leaf size=151 $-\frac{x^m \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{m}{2},1-\frac{m}{2},\frac{1}{a^2 x^2}\right )}{a m}+\frac{4 x^m \text{Hypergeometric2F1}\left (\frac{3}{2},-\frac{m}{2},1-\frac{m}{2},\frac{1}{a^2 x^2}\right )}{a m}-\frac{3 x^{m+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} (-m-1),\frac{1-m}{2},\frac{1}{a^2 x^2}\right )}{m+1}+\frac{4 x^{m+1} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{1}{2} (-m-1),\frac{1-m}{2},\frac{1}{a^2 x^2}\right )}{m+1}$

[Out]

(-3*x^(1 + m)*Hypergeometric2F1[1/2, (-1 - m)/2, (1 - m)/2, 1/(a^2*x^2)])/(1 + m) - (x^m*Hypergeometric2F1[1/2
, -m/2, 1 - m/2, 1/(a^2*x^2)])/(a*m) + (4*x^(1 + m)*Hypergeometric2F1[3/2, (-1 - m)/2, (1 - m)/2, 1/(a^2*x^2)]
)/(1 + m) + (4*x^m*Hypergeometric2F1[3/2, -m/2, 1 - m/2, 1/(a^2*x^2)])/(a*m)

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Rubi [A]  time = 1.23086, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.417, Rules used = {6172, 6742, 364, 850, 808} $-\frac{x^m \, _2F_1\left (\frac{1}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{a^2 x^2}\right )}{a m}+\frac{4 x^m \, _2F_1\left (\frac{3}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{a^2 x^2}\right )}{a m}-\frac{3 x^{m+1} \, _2F_1\left (\frac{1}{2},\frac{1}{2} (-m-1);\frac{1-m}{2};\frac{1}{a^2 x^2}\right )}{m+1}+\frac{4 x^{m+1} \, _2F_1\left (\frac{3}{2},\frac{1}{2} (-m-1);\frac{1-m}{2};\frac{1}{a^2 x^2}\right )}{m+1}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(3*ArcCoth[a*x])*x^m,x]

[Out]

(-3*x^(1 + m)*Hypergeometric2F1[1/2, (-1 - m)/2, (1 - m)/2, 1/(a^2*x^2)])/(1 + m) - (x^m*Hypergeometric2F1[1/2
, -m/2, 1 - m/2, 1/(a^2*x^2)])/(a*m) + (4*x^(1 + m)*Hypergeometric2F1[3/2, (-1 - m)/2, (1 - m)/2, 1/(a^2*x^2)]
)/(1 + m) + (4*x^m*Hypergeometric2F1[3/2, -m/2, 1 - m/2, 1/(a^2*x^2)])/(a*m)

Rule 6172

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]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 850

Int[((x_)^(n_.)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Int[x^n*(a/d + (c*x)/e)*(a + c*x
^2)^(p - 1), x] /; FreeQ[{a, c, d, e, n, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ( !IntegerQ[n] ||
!IntegerQ[2*p] || IGtQ[n, 2] || (GtQ[p, 0] && NeQ[n, 2]))

Rule 808

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]

Rubi steps

\begin{align*} \int e^{3 \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} \left (1+\frac{x}{a}\right )^2}{\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 ) \operatorname{Subst}\left (\int \left (-\frac{3 x^{-2-m}}{\sqrt{1-\frac{x^2}{a^2}}}-\frac{x^{-1-m}}{a \sqrt{1-\frac{x^2}{a^2}}}+\frac{4 x^{-2-m}}{\left (1-\frac{x}{a}\right ) \sqrt{1-\frac{x^2}{a^2}}}\right ) \, dx,x,\frac{1}{x}\right )\right )\\ &=\left (3 \left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int \frac{x^{-2-m}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )-\left (4 \left (\frac{1}{x}\right )^m x^m\right ) \operatorname{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 )+\frac{\left (\left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int \frac{x^{-1-m}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{3 x^{1+m} \, _2F_1\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 \, _2F_1\left (\frac{1}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{a^2 x^2}\right )}{a m}-\left (4 \left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int \frac{x^{-2-m} \left (1+\frac{x}{a}\right )}{\left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{3 x^{1+m} \, _2F_1\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 \, _2F_1\left (\frac{1}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{a^2 x^2}\right )}{a m}-\left (4 \left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int \frac{x^{-2-m}}{\left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )-\frac{\left (4 \left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int \frac{x^{-1-m}}{\left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{3 x^{1+m} \, _2F_1\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 \, _2F_1\left (\frac{1}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{a^2 x^2}\right )}{a m}+\frac{4 x^{1+m} \, _2F_1\left (\frac{3}{2},\frac{1}{2} (-1-m);\frac{1-m}{2};\frac{1}{a^2 x^2}\right )}{1+m}+\frac{4 x^m \, _2F_1\left (\frac{3}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{a^2 x^2}\right )}{a m}\\ \end{align*}

Mathematica [C]  time = 0.319526, size = 228, normalized size = 1.51 $\frac{x^{m+1} \left (m \sqrt{a x-1} \sqrt{a x+1} \sqrt{x^2-\frac{1}{a^2}} \text{Hypergeometric2F1}\left (-\frac{1}{2},-\frac{m}{2}-\frac{1}{2},\frac{1}{2}-\frac{m}{2},\frac{1}{a^2 x^2}\right )+3 (m+1) \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{1-a x} \sqrt{\frac{a x+1}{a^2}} F_1\left (m;-\frac{1}{2},\frac{1}{2};m+1;-a x,a x\right )-2 (m+1) \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{1-a x} \sqrt{\frac{a x+1}{a^2}} F_1\left (m;-\frac{1}{2},\frac{3}{2};m+1;-a x,a x\right )\right )}{m (m+1) \sqrt{a x-1} \sqrt{a x+1} \sqrt{x^2-\frac{1}{a^2}}}$

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcCoth[a*x])*x^m,x]

[Out]

(x^(1 + m)*(3*(1 + m)*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[1 - a*x]*Sqrt[(1 + a*x)/a^2]*AppellF1[m, -1/2, 1/2, 1 + m, -(
a*x), a*x] - 2*(1 + m)*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[1 - a*x]*Sqrt[(1 + a*x)/a^2]*AppellF1[m, -1/2, 3/2, 1 + m, -
(a*x), a*x] + m*Sqrt[-1 + a*x]*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)]))/(m*(1 + m)*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[-a^(-2) + x^2])

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Maple [F]  time = 0.187, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+1))^(3/2)*x^m,x)

[Out]

int(1/((a*x-1)/(a*x+1))^(3/2)*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{3}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*x^m,x, algorithm="maxima")

[Out]

integrate(x^m/((a*x - 1)/(a*x + 1))^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a^{2} x^{2} + 2 \, a x + 1\right )} x^{m} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} x^{2} - 2 \, 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))^(3/2)*x^m,x, algorithm="fricas")

[Out]

integral((a^2*x^2 + 2*a*x + 1)*x^m*sqrt((a*x - 1)/(a*x + 1))/(a^2*x^2 - 2*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))**(3/2)*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{3}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*x^m,x, algorithm="giac")

[Out]

integrate(x^m/((a*x - 1)/(a*x + 1))^(3/2), x)