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

Optimal. Leaf size=150 \[ -\frac {3 x^{m+1} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-m-1);\frac {1-m}{2};\frac {1}{a^2 x^2}\right )}{m+1}+\frac {4 x^{m+1} \, _2F_1\left (\frac {3}{2},\frac {1}{2} (-m-1);\frac {1-m}{2};\frac {1}{a^2 x^2}\right )}{m+1}+\frac {x^m \, _2F_1\left (\frac {1}{2},-\frac {m}{2};1-\frac {m}{2};\frac {1}{a^2 x^2}\right )}{a m}-\frac {4 x^m \, _2F_1\left (\frac {3}{2},-\frac {m}{2};1-\frac {m}{2};\frac {1}{a^2 x^2}\right )}{a m} \]

[Out]

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

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Rubi [A]  time = 1.05, antiderivative size = 150, normalized size of antiderivative = 1.00, 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[x^m/E^(3*ArcCoth[a*x]),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 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 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]

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 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]]

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*}

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Mathematica [C]  time = 0.27, size = 192, normalized size = 1.28 \[ \frac {x^{m+1} \left (-3 (m+1) \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {\frac {a x-1}{a^2}} F_1\left (m;-\frac {1}{2},\frac {1}{2};m+1;a x,-a x\right )+2 (m+1) \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {\frac {a x-1}{a^2}} F_1\left (m;-\frac {1}{2},\frac {3}{2};m+1;a x,-a x\right )+m \sqrt {1-a x} \sqrt {x^2-\frac {1}{a^2}} \, _2F_1\left (-\frac {1}{2},-\frac {m}{2}-\frac {1}{2};\frac {1}{2}-\frac {m}{2};\frac {1}{a^2 x^2}\right )\right )}{m (m+1) \sqrt {1-a x} \sqrt {x^2-\frac {1}{a^2}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F]  time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a x - 1\right )} x^{m} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(a*x+1)]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^m\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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