3.137 \(\int e^{-2 \coth ^{-1}(a x)} x^m \, dx\)

Optimal. Leaf size=36 \[ \frac{x^{m+1}}{m+1}-\frac{2 x^{m+1} \text{Hypergeometric2F1}(1,m+1,m+2,-a x)}{m+1} \]

[Out]

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

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Rubi [A]  time = 0.0412358, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6167, 6126, 80, 64} \[ \frac{x^{m+1}}{m+1}-\frac{2 x^{m+1} \, _2F_1(1,m+1;m+2;-a x)}{m+1} \]

Antiderivative was successfully verified.

[In]

Int[x^m/E^(2*ArcCoth[a*x]),x]

[Out]

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

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rule 6126

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> Int[(x^m*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x] /; Fre
eQ[{a, m, n}, x] &&  !IntegerQ[(n - 1)/2]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 64

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m +
1, m + 2, -((d*x)/c)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-(d/(b*c)), 0])))

Rubi steps

\begin{align*} \int e^{-2 \coth ^{-1}(a x)} x^m \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} x^m \, dx\\ &=-\int \frac{x^m (1-a x)}{1+a x} \, dx\\ &=\frac{x^{1+m}}{1+m}-2 \int \frac{x^m}{1+a x} \, dx\\ &=\frac{x^{1+m}}{1+m}-\frac{2 x^{1+m} \, _2F_1(1,1+m;2+m;-a x)}{1+m}\\ \end{align*}

Mathematica [A]  time = 0.0103053, size = 27, normalized size = 0.75 \[ \frac{x^{m+1} (1-2 \text{Hypergeometric2F1}(1,m+1,m+2,-a x))}{m+1} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m/E^(2*ArcCoth[a*x]),x]

[Out]

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

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Maple [C]  time = 0.412, size = 93, normalized size = 2.6 \begin{align*}{a}^{-1-m} \left ({\frac{{x}^{m}{a}^{m} \left ( amx-m-1 \right ) }{ \left ( 1+m \right ) m}}+{x}^{m}{a}^{m}{\it LerchPhi} \left ( -ax,1,m \right ) \right ) -{a}^{-1-m} \left ({\frac{{x}^{m}{a}^{m}}{m}}+{\frac{{x}^{m}{a}^{m} \left ( -1-m \right ){\it LerchPhi} \left ( -ax,1,m \right ) }{1+m}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^(-1-m)*(x^m*a^m*(a*m*x-m-1)/(1+m)/m+x^m*a^m*LerchPhi(-a*x,1,m))-a^(-1-m)*(x^m*a^m/m+1/(1+m)*x^m*a^m*(-1-m)*L
erchPhi(-a*x,1,m))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*x - 1)*x^m/(a*x + 1), 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}}{a x + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a*x - 1)*x^m/(a*x + 1), x)

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Sympy [C]  time = 2.37547, size = 119, normalized size = 3.31 \begin{align*} \frac{a m x^{2} x^{m} \Phi \left (a x e^{i \pi }, 1, m + 2\right ) \Gamma \left (m + 2\right )}{\Gamma \left (m + 3\right )} + \frac{2 a x^{2} x^{m} \Phi \left (a x e^{i \pi }, 1, m + 2\right ) \Gamma \left (m + 2\right )}{\Gamma \left (m + 3\right )} - \frac{m x x^{m} \Phi \left (a x e^{i \pi }, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} - \frac{x x^{m} \Phi \left (a x e^{i \pi }, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*m*x**2*x**m*lerchphi(a*x*exp_polar(I*pi), 1, m + 2)*gamma(m + 2)/gamma(m + 3) + 2*a*x**2*x**m*lerchphi(a*x*e
xp_polar(I*pi), 1, m + 2)*gamma(m + 2)/gamma(m + 3) - m*x*x**m*lerchphi(a*x*exp_polar(I*pi), 1, m + 1)*gamma(m
+ 1)/gamma(m + 2) - x*x**m*lerchphi(a*x*exp_polar(I*pi), 1, m + 1)*gamma(m + 1)/gamma(m + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*x - 1)*x^m/(a*x + 1), x)