\(\int e^{-2 \coth ^{-1}(a x)} x^m \, dx\) [137]

Optimal result

Integrand size = 12, antiderivative size = 36 \[ \int e^{-2 \coth ^{-1}(a x)} x^m \, dx=\frac {x^{1+m}}{1+m}-\frac {2 x^{1+m} \operatorname {Hypergeometric2F1}(1,1+m,2+m,-a x)}{1+m} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6302, 6261, 81, 66} \[ \int e^{-2 \coth ^{-1}(a x)} x^m \, dx=\frac {x^{m+1}}{m+1}-\frac {2 x^{m+1} \operatorname {Hypergeometric2F1}(1,m+1,m+2,-a x)}{m+1} \]

[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 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x)^(m + 1)/(b*(m + 1)))*Hypergeometr
ic2F1[-n, m + 1, m + 2, (-d)*(x/c)], 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])))

Rule 81

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 6261

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 6302

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]

Rubi steps \begin{align*} \text {integral}& = -\int e^{-2 \text {arctanh}(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} \operatorname {Hypergeometric2F1}(1,1+m,2+m,-a x)}{1+m} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int e^{-2 \coth ^{-1}(a x)} x^m \, dx=\frac {x^{1+m} (1-2 \operatorname {Hypergeometric2F1}(1,1+m,2+m,-a x))}{1+m} \]

[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)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 5.

Time = 0.49 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.58

method	result	size
meijerg	\(a^{-m -1} \left (\frac {x^{m} a^{m} \left (a m x -m -1\right )}{\left (1+m \right ) m}+x^{m} a^{m} \operatorname {LerchPhi}\left (-a x , 1, m\right )\right )-a^{-m -1} \left (\frac {x^{m} a^{m}}{m}+\frac {x^{m} a^{m} \left (-m -1\right ) \operatorname {LerchPhi}\left (-a x , 1, m\right )}{1+m}\right )\)	\(93\)

[In]

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

[Out]

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

Fricas [F]

\[ \int e^{-2 \coth ^{-1}(a x)} x^m \, dx=\int { \frac {{\left (a x - 1\right )} x^{m}}{a x + 1} \,d x } \]

[In]

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

[Out]

integral((a*x - 1)*x^m/(a*x + 1), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.44 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.22 \[ \int e^{-2 \coth ^{-1}(a x)} x^m \, dx=\frac {a m x^{m + 2} \Phi \left (a x e^{i \pi }, 1, m + 2\right ) \Gamma \left (m + 2\right )}{\Gamma \left (m + 3\right )} + \frac {2 a x^{m + 2} \Phi \left (a x e^{i \pi }, 1, m + 2\right ) \Gamma \left (m + 2\right )}{\Gamma \left (m + 3\right )} - \frac {m x^{m + 1} \Phi \left (a x e^{i \pi }, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} - \frac {x^{m + 1} \Phi \left (a x e^{i \pi }, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int e^{-2 \coth ^{-1}(a x)} x^m \, dx=\int { \frac {{\left (a x - 1\right )} x^{m}}{a x + 1} \,d x } \]

[In]

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

[Out]

integrate((a*x - 1)*x^m/(a*x + 1), x)

Giac [F]

\[ \int e^{-2 \coth ^{-1}(a x)} x^m \, dx=\int { \frac {{\left (a x - 1\right )} x^{m}}{a x + 1} \,d x } \]

[In]

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

[Out]

integrate((a*x - 1)*x^m/(a*x + 1), x)

Mupad [F(-1)]

Timed out. \[ \int e^{-2 \coth ^{-1}(a x)} x^m \, dx=\int \frac {x^m\,\left (a\,x-1\right )}{a\,x+1} \,d x \]

[In]

int((x^m*(a*x - 1))/(a*x + 1),x)

[Out]

int((x^m*(a*x - 1))/(a*x + 1), x)