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

Optimal. Leaf size=34 \[ \frac {x^{m+1} F_1\left (-m-1;\frac {1}{6},-\frac {1}{6};-m;\frac {1}{x},-\frac {1}{x}\right )}{m+1} \]

[Out]

x^(1+m)*AppellF1(-1-m,1/6,-1/6,-m,1/x,-1/x)/(1+m)

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Rubi [A]  time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6173, 133} \[ \frac {x^{m+1} F_1\left (-m-1;\frac {1}{6},-\frac {1}{6};-m;\frac {1}{x},-\frac {1}{x}\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(1 + m)*AppellF1[-1 - m, 1/6, -1/6, -m, x^(-1), -x^(-1)])/(1 + 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])

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]

Rubi steps

\begin {align*} \int e^{\frac {1}{3} \coth ^{-1}(x)} x^m \, dx &=-\left (\left (\left (\frac {1}{x}\right )^m x^m\right ) \operatorname {Subst}\left (\int \frac {x^{-2-m} \sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {x^{1+m} F_1\left (-1-m;\frac {1}{6},-\frac {1}{6};-m;\frac {1}{x},-\frac {1}{x}\right )}{1+m}\\ \end {align*}

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Mathematica [F]  time = 0.83, size = 0, normalized size = 0.00 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^m \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[E^(ArcCoth[x]/3)*x^m, x]

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fricas [F]  time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (x + 1\right )} x^{m} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{x - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))^(1/6)*x^m,x, algorithm="fricas")

[Out]

integral((x + 1)*x^m*((x - 1)/(x + 1))^(5/6)/(x - 1), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))^(1/6)*x^m,x, algorithm="giac")

[Out]

integrate(x^m/((x - 1)/(x + 1))^(1/6), x)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (\frac {-1+x}{1+x}\right )^{\frac {1}{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((-1+x)/(1+x))^(1/6)*x^m,x)

[Out]

int(1/((-1+x)/(1+x))^(1/6)*x^m,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))^(1/6)*x^m,x, algorithm="maxima")

[Out]

integrate(x^m/((x - 1)/(x + 1))^(1/6), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^m}{{\left (\frac {x-1}{x+1}\right )}^{1/6}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/((x - 1)/(x + 1))^(1/6),x)

[Out]

int(x^m/((x - 1)/(x + 1))^(1/6), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\sqrt [6]{\frac {x - 1}{x + 1}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))**(1/6)*x**m,x)

[Out]

Integral(x**m/((x - 1)/(x + 1))**(1/6), x)

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