3.1358 \(\int \frac{e^{n \tanh ^{-1}(a x)} x^m}{c-a^2 c x^2} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.102289, 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} \[ \frac{x^{m+1} F_1\left (m+1;\frac{n+2}{2},1-\frac{n}{2};m+2;a x,-a x\right )}{c (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(1 + m)*AppellF1[1 + m, (2 + n)/2, 1 - n/2, 2 + m, a*x, -(a*x)])/(c*(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}{c-a^2 c x^2} \, dx &=\frac{\int x^m (1-a x)^{-1-\frac{n}{2}} (1+a x)^{-1+\frac{n}{2}} \, dx}{c}\\ &=\frac{x^{1+m} F_1\left (1+m;\frac{2+n}{2},1-\frac{n}{2};2+m;a x,-a x\right )}{c (1+m)}\\ \end{align*}

Mathematica [B]  time = 0.195915, size = 106, normalized size = 2.52 \[ \frac{x^m \left (e^{-2 \tanh ^{-1}(a x)}-1\right )^m \left (e^{-2 \tanh ^{-1}(a x)}+1\right )^m \left (-e^{-4 \tanh ^{-1}(a x)} \left (e^{2 \tanh ^{-1}(a x)}-1\right )^2\right )^{-m} e^{n \tanh ^{-1}(a x)} F_1\left (-\frac{n}{2};m,-m;1-\frac{n}{2};-e^{-2 \tanh ^{-1}(a x)},e^{-2 \tanh ^{-1}(a x)}\right )}{a c n} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.203, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}{x}^{m}}{-{a}^{2}c{x}^{2}+c}}\, 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),x)

[Out]

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

[Out]

-integrate(x^m*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^2 - c), 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^{2} c x^{2} - c}, 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),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{x^{m} e^{n \operatorname{atanh}{\left (a x \right )}}}{a^{2} x^{2} - 1}\, dx}{c} \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),x)

[Out]

-Integral(x**m*exp(n*atanh(a*x))/(a**2*x**2 - 1), x)/c

________________________________________________________________________________________

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}}{a^{2} c x^{2} - c}\,{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),x, algorithm="giac")

[Out]

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