3.1359 \(\int \frac{e^{n \tanh ^{-1}(a x)} x^m}{(c-a^2 c x^2)^2} \, dx\)

Optimal. Leaf size=42 \[ \frac{x^{m+1} F_1\left (m+1;\frac{n+4}{2},2-\frac{n}{2};m+2;a x,-a x\right )}{c^2 (m+1)} \]

[Out]

(x^(1 + m)*AppellF1[1 + m, (4 + n)/2, 2 - n/2, 2 + m, a*x, -(a*x)])/(c^2*(1 + m))

________________________________________________________________________________________

Rubi [A]  time = 0.098707, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6150, 133} \[ \frac{x^{m+1} F_1\left (m+1;\frac{n+4}{2},2-\frac{n}{2};m+2;a x,-a x\right )}{c^2 (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[(E^(n*ArcTanh[a*x])*x^m)/(c - a^2*c*x^2)^2,x]

[Out]

(x^(1 + m)*AppellF1[1 + m, (4 + n)/2, 2 - n/2, 2 + m, a*x, -(a*x)])/(c^2*(1 + m))

Rule 6150

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 -
a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p
] || GtQ[c, 0])

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])

Rubi steps

\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)} x^m}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac{\int x^m (1-a x)^{-2-\frac{n}{2}} (1+a x)^{-2+\frac{n}{2}} \, dx}{c^2}\\ &=\frac{x^{1+m} F_1\left (1+m;\frac{4+n}{2},2-\frac{n}{2};2+m;a x,-a x\right )}{c^2 (1+m)}\\ \end{align*}

Mathematica [F]  time = 0.44451, size = 0, normalized size = 0. \[ \int \frac{e^{n \tanh ^{-1}(a x)} x^m}{\left (c-a^2 c x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(E^(n*ArcTanh[a*x])*x^m)/(c - a^2*c*x^2)^2,x]

[Out]

Integrate[(E^(n*ArcTanh[a*x])*x^m)/(c - a^2*c*x^2)^2, x]

________________________________________________________________________________________

Maple [F]  time = 0.21, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}{x}^{m}}{ \left ( -{a}^{2}c{x}^{2}+c \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctanh(a*x))*x^m/(-a^2*c*x^2+c)^2,x)

[Out]

int(exp(n*arctanh(a*x))*x^m/(-a^2*c*x^2+c)^2,x)

________________________________________________________________________________________

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{1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*x^m/(-a^2*c*x^2+c)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{m} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*x^m/(-a^2*c*x^2+c)^2,x, algorithm="fricas")

[Out]

integral(x^m*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2), x)

________________________________________________________________________________________

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(exp(n*atanh(a*x))*x**m/(-a**2*c*x**2+c)**2,x)

[Out]

Timed out

________________________________________________________________________________________

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{1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*x^m/(-a^2*c*x^2+c)^2,x, algorithm="giac")

[Out]

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