3.1370 \(\int \frac{e^{2 \tanh ^{-1}(a x)} x^2}{(c-a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=31 \[ -\frac{1-2 a x}{6 a^3 c^3 (1-a x)^3 (a x+1)} \]

[Out]

-(1 - 2*a*x)/(6*a^3*c^3*(1 - a*x)^3*(1 + a*x))

________________________________________________________________________________________

Rubi [A]  time = 0.0879717, antiderivative size = 31, 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, 81} \[ -\frac{1-2 a x}{6 a^3 c^3 (1-a x)^3 (a x+1)} \]

Antiderivative was successfully verified.

[In]

Int[(E^(2*ArcTanh[a*x])*x^2)/(c - a^2*c*x^2)^3,x]

[Out]

-(1 - 2*a*x)/(6*a^3*c^3*(1 - a*x)^3*(1 + a*x))

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 81

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*
x)^(n + 1)*(e + f*x)^(p + 1)*(2*a*d*f*(n + p + 3) - b*(d*e*(n + 2) + c*f*(p + 2)) + b*d*f*(n + p + 2)*x))/(d^2
*f^2*(n + p + 2)*(n + p + 3)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && NeQ[n + p + 3,
0] && EqQ[d*f*(n + p + 2)*(a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1)))) - b*(d*e*(n + 1)
+ c*f*(p + 1))*(a*d*f*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2))), 0]

Rubi steps

\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac{\int \frac{x^2}{(1-a x)^4 (1+a x)^2} \, dx}{c^3}\\ &=-\frac{1-2 a x}{6 a^3 c^3 (1-a x)^3 (1+a x)}\\ \end{align*}

Mathematica [A]  time = 0.0306313, size = 30, normalized size = 0.97 \[ \frac{1-2 a x}{6 a^3 c^3 (a x-1)^3 (a x+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*ArcTanh[a*x])*x^2)/(c - a^2*c*x^2)^3,x]

[Out]

(1 - 2*a*x)/(6*a^3*c^3*(-1 + a*x)^3*(1 + a*x))

________________________________________________________________________________________

Maple [A]  time = 0.034, size = 54, normalized size = 1.7 \begin{align*}{\frac{1}{{c}^{3}} \left ( -{\frac{1}{16\,{a}^{3} \left ( ax+1 \right ) }}-{\frac{1}{12\,{a}^{3} \left ( ax-1 \right ) ^{3}}}-{\frac{1}{8\,{a}^{3} \left ( ax-1 \right ) ^{2}}}+{\frac{1}{16\,{a}^{3} \left ( ax-1 \right ) }} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)*x^2/(-a^2*c*x^2+c)^3,x)

[Out]

1/c^3*(-1/16/a^3/(a*x+1)-1/12/a^3/(a*x-1)^3-1/8/a^3/(a*x-1)^2+1/16/a^3/(a*x-1))

________________________________________________________________________________________

Maxima [A]  time = 0.9849, size = 66, normalized size = 2.13 \begin{align*} -\frac{2 \, a x - 1}{6 \,{\left (a^{7} c^{3} x^{4} - 2 \, a^{6} c^{3} x^{3} + 2 \, a^{4} c^{3} x - a^{3} c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^2/(-a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

-1/6*(2*a*x - 1)/(a^7*c^3*x^4 - 2*a^6*c^3*x^3 + 2*a^4*c^3*x - a^3*c^3)

________________________________________________________________________________________

Fricas [A]  time = 1.90582, size = 97, normalized size = 3.13 \begin{align*} -\frac{2 \, a x - 1}{6 \,{\left (a^{7} c^{3} x^{4} - 2 \, a^{6} c^{3} x^{3} + 2 \, a^{4} c^{3} x - a^{3} c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^2/(-a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

-1/6*(2*a*x - 1)/(a^7*c^3*x^4 - 2*a^6*c^3*x^3 + 2*a^4*c^3*x - a^3*c^3)

________________________________________________________________________________________

Sympy [A]  time = 0.567505, size = 49, normalized size = 1.58 \begin{align*} - \frac{2 a x - 1}{6 a^{7} c^{3} x^{4} - 12 a^{6} c^{3} x^{3} + 12 a^{4} c^{3} x - 6 a^{3} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)*x**2/(-a**2*c*x**2+c)**3,x)

[Out]

-(2*a*x - 1)/(6*a**7*c**3*x**4 - 12*a**6*c**3*x**3 + 12*a**4*c**3*x - 6*a**3*c**3)

________________________________________________________________________________________

Giac [A]  time = 1.1264, size = 61, normalized size = 1.97 \begin{align*} -\frac{1}{16 \,{\left (a x + 1\right )} a^{3} c^{3}} + \frac{3 \, a^{2} x^{2} - 12 \, a x + 5}{48 \,{\left (a x - 1\right )}^{3} a^{3} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*x^2/(-a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

-1/16/((a*x + 1)*a^3*c^3) + 1/48*(3*a^2*x^2 - 12*a*x + 5)/((a*x - 1)^3*a^3*c^3)